A. M. Fink

The classical Hadamard-Hermite inequality requires that the measure be a symmetric and positive.We prove versions which require neither of these conditions.Furthermore, we prove that no such theorems exist with less … The classical Hadamard-Hermite inequality requires that the measure be a symmetric and positive.We prove versions which require neither of these conditions.Furthermore, we prove that no such theorems exist with less restrictions than ours, ie.they are best possible.

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Positive Solutions of Second Order Systems of Boundary Value Problems

1993-11-01

A. M. Fink , Juan A. Gatica

Approximation of Solutions of Singular Second Order Boundary Value Problems

1991-03-01

A. M. Fink , Juan A. Gatica , Gastón E. Hernández +1 more

The boundary value problems \[ ({\text{P}}_1 )\begin{array}{*{20}c} {y'' + \frac{{n - 1}}{x}y' + f(x,y) = 0,} \\ {y'(0) = y(1) = 0,} \\ \end{array} \] and \[ \begin{gathered} y'' + … The boundary value problems \[ ({\text{P}}_1 )\begin{array}{*{20}c} {y'' + \frac{{n - 1}}{x}y' + f(x,y) = 0,} \\ {y'(0) = y(1) = 0,} \\ \end{array} \] and \[ \begin{gathered} y'' + f(x,y) = 0, \hfill \\ ({\text{P}}_2 )\quad \alpha y(0) - \beta y'(0) = 0, \hfill \\ \gamma y(1) + \delta y'(1) = 0 \hfill \\ \end{gathered} \] with the function $f(x,y)$ satisfying conditions that allow for singularities to be present are studied with the view of obtaining general existence, uniqueness, and approximation of positive solutions. Furthermore, the behavior of solutions near $x = 1$ is described for the special case $f(x,y) = a(x)y^{ - p} $.

Liapunov functions and almost periodic solutions for almost periodic systems

1969-03-01

A. M. Fink , George Seifert

Positive almost periodic solutions of some delay integral equations

1990-01-01

A. M. Fink , Juan A. Gatica

Comparing two integral means for absolutely continuous mappings whose derivatives are in L∞[a,b] and applications

2002-07-01

Neil S Barnett , Pietro Cerone , Sever S Dragomir +1 more

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A Treatise on Grüss’ Inequality

1999-01-01

A. M. Fink

Some Inequalities for Monotone Functions

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Inequalities Involving Kernels

1991-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

On Chebyshev's other inequality

1984-01-01

A. M. Fink , Max Jodeit

We formulate the notion of a best possible inequality.This involves finding the largest class of functions and measures for which an inequality is true.We give two examples of Chebyshev's inequality, … We formulate the notion of a best possible inequality.This involves finding the largest class of functions and measures for which an inequality is true.We give two examples of Chebyshev's inequality, e.g.f£ dμ Jafgdμ 2= flϊfdμ f£ gdμ for all pairs (f,g) which are increasing if and only if /" du ^ 0, /* du 2= 0 for all x.Other examples include Jensen's inequality. Introduction.Let μ be a probability measure on the real line and/and g increasing functions.Then

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Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign

1995-01-01

Nguyên Phuong Các , A. M. Fink , Juan A. Gatica

We find a solution to the radial Laplacian equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y plus StartFraction upper N minus 1 Over x EndFraction y Superscript prime Baseline plus lamda a … We find a solution to the radial Laplacian equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y plus StartFraction upper N minus 1 Over x EndFraction y Superscript prime Baseline plus lamda a left-parenthesis x right-parenthesis f left-parenthesis y right-parenthesis equals 0 comma y prime left-parenthesis 0 right-parenthesis equals y left-parenthesis 1 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>x</mml:mi> </mml:mfrac> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">y + \frac {{N - 1}}{x}y’ + \lambda a(x)f(y) = 0,y’ (0) = y(1) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <italic>a</italic> may change sign and is "sufficiently positive". The function <italic>f</italic> is qualitatively like <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>y</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{e^y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we conclude solutions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to lamda less-than-or-equal-to lamda 0"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>≤</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \leq \lambda \leq {\lambda _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Integral Inequalities Involving Functions with Bounded Derivatives

1991-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Almost periodic functions

1974-01-01

A. M. Fink

Ultimate boundedness does not imply almost periodicity

1971-03-01

A. M. Fink , Paul O. Frederickson

Almost automorphic and almost periodic solutions which minimize functionals

1968-01-01

A. M. Fink

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On an inequality of Nehari

1969-03-01

A. M. Fink , Donald F. St. Mary

In a private communication with one of the authors, Professor Nehari has indicated that the inequality is undecided since the argument given in [I] that Theorem II implies Theorem I … In a private communication with one of the authors, Professor Nehari has indicated that the inequality is undecided since the argument given in [I] that Theorem II implies Theorem I is incorrect. It is the purpose of this note to show that (1) is correct for n =2. In fact, we prove a stronger result for (2) y+gy'+fy =O. THEOREM. Let a and b be successive zeros of a nontrivial solution to (2) where f and g are integrable. Then

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Discrete Inequalities of Generalized Wirtinger Type

1974-02-01

A. M. Fink

Compact Families of Almost Periodic Functions and an Application of the Schauder Fixed-Point Theorem

1969-11-01

A. M. Fink

Jensen inequalities for functions with higher monotonicities

1990-12-01

A. M. Fink , Max Jodeit

Nonnegative Solutions of Quasilinear Elliptic Problems with Nonnegative Coefficients

1997-02-01

Nguyên Phuong Các , A. M. Fink , Juan A. Gatica

Discrete inequalities of generalized wirtinger type

1973-06-01

A. M. Fink

The Centroid Method in Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Semi-separated conditions for almost periodic solutions

1972-03-01

A. M. Fink

Extensions of almost automorphic sequences

1969-09-01

A. M. Fink

Hadamard's inequality for log-concave functions

2000-09-01

A. M. Fink

Bernoulli’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Conjugate Inequalities for Functions and Their Derivatives

1974-05-01

A. M. Fink

We consider finding the best possible constants $C = C(n,\alpha ,\beta ,p,q)$ such that $\| f \|_p \leqq C(b - a)^{n + 1/p -1/q} \| {f^{(n)} } \|_q $ under … We consider finding the best possible constants $C = C(n,\alpha ,\beta ,p,q)$ such that $\| f \|_p \leqq C(b - a)^{n + 1/p -1/q} \| {f^{(n)} } \|_q $ under several differen sets of boundary conditions. Specifically in one case, f is required to have a zeros at $a $ and $\beta $ zeros at b where $n \leqq \alpha + \beta \leqq 2n$, $\alpha \leqq n$ and $\beta \leqq n$. In the other case, f has $\alpha $ zeros at $a $, $\beta $ zeros at b and $\int _a^b f(x)x^r (x - a)^\alpha (x - \beta )^\beta dx = 0$, $r = 0, \cdots ,n - 1 - \alpha - \beta $, where $0 \leqq \alpha $, $0 \leqq \beta $ and $\alpha + \beta < n$. The various problems are related and the numbers C are calculated exactly in some cases. Finally, we show how some of these can be applied to disconjugacy problems in differential equations.

Almost Periodic Functions Invented for Specific Purposes

1972-10-01

A. M. Fink

This is an expository survey of new definitions of almost periodic functions which were developed from researches on almost periodic differential equations. Some of these are not well known but … This is an expository survey of new definitions of almost periodic functions which were developed from researches on almost periodic differential equations. Some of these are not well known but are very useful for showing the almost periodicity of solutions of differential equations.

A generalized sturm comparison theorem and oscillation coefficients

1969-06-01

A. M. Fink , Donald F. St. Mary

Convergence and Almost Periodicity of Solutions of Forced Lienard Equations

1974-01-01

A. M. Fink

We consider the Lienard equation $x'' + f(x)x' + g(x) = p(t)$ and find conditions under which solutions are bounded or under which all solutions converge. Further, if p is … We consider the Lienard equation $x'' + f(x)x' + g(x) = p(t)$ and find conditions under which solutions are bounded or under which all solutions converge. Further, if p is almost periodic, we find sufficient conditions for the existence of an almost periodic solution. The most easily stated criterion for the existence of almost periodic solutions is $0 < 2\inf g'(x)\leqq 2\sup g'(x)\leqq \inf f^2 (x)\leqq \sup f^2 (x) < \infty $. The case when $f(x) \equiv c$ was treated previously by Loud [1]. Our method is based on using the $l_1 $-norm as a Lyapunov function.

Nonoscillation theorems for differential equations with general deviating arguments

1983-01-01

A. M. Fink , Takaŝi Kusano

Uniqueness theorems and almost periodic solutions to second order differential equations

1968-10-01

A. M. Fink

A generalization of the arithmetic-geometric means inequality

1976-12-01

A. M. Fink , Max Jodeit

It is shown that the arithmetic mean of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 w 1 comma ellipsis comma x Subscript n Baseline w Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow … It is shown that the arithmetic mean of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 w 1 comma ellipsis comma x Subscript n Baseline w Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_1}{w_1}, \ldots ,{x_n}{w_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exceeds the geometric mean of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 comma ellipsis comma x Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_1}, \ldots ,{x_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unless all the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s are equal, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w 1 comma ellipsis comma w Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{w_1}, \ldots ,{w_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depend on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 comma ellipsis comma x Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_1}, \ldots ,{x_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and satisfy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-slanted-equals w Subscript i Baseline greater-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \leqslant {w_i} > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unless <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript i Baseline equals min x Subscript k"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo movablelimits="true" form="prefix">min</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_i} = \min {x_k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This inequality is then applied to an integral inequality for functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\;\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^{(k)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> convex and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-slanted-equals k greater-than n"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \leqslant k > n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Steffensen type inequalities

1982-12-01

A. M. Fink

Dedicated to Dedicated to

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On Opial's Inequality for f (n)

1992-05-01

A. M. Fink

We prove inequalities of the type ∫ 0 h |f (i) (x)f (j) (x)|dx≤C(n,i,j,p)h 2n−i−j+1−2/p (∫ h |f (n) (x)| p dx) 2p when f(0)=f'(0)=...=f (n−1) (0)=0. We assume that … We prove inequalities of the type ∫ 0 h |f (i) (x)f (j) (x)|dx≤C(n,i,j,p)h 2n−i−j+1−2/p (∫ h |f (n) (x)| p dx) 2p when f(0)=f'(0)=...=f (n−1) (0)=0. We assume that f (n−1) is absolutely continuous and f (n) ∈L p (0,h), with p≥1, n≥2, and 0≤i≤j≤n−1

Some opial, Lyapunov, and De IaValée poussin inequalities with nonhomogeneous boundary conditions

2000-01-01

Richard C. Brown , A. M. Fink , Don Hinton

Young’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Wendroff's inequalities

1981-08-01

A. M. Fink

How to Polish off Median Polish

1988-09-01

A. M. Fink

Tukey’s median polish is an algorithm for smoothing data in two-way tables. Each iteration lowers the $L_1 $ norm of the residual. For commensurable data the algorithm converges in a … Tukey’s median polish is an algorithm for smoothing data in two-way tables. Each iteration lowers the $L_1 $ norm of the residual. For commensurable data the algorithm converges in a finite number of steps. It does not, in general, converge to the least $L_1 $ norm residual. We provide an algorithm that converges in a finite number of steps for any real data and gives the least $L_1 $ residual.

Bessel’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Nonresonance Conditions for the Existence of Almost Periodic Solutions of Almost Periodic Systems

1971-09-01

A. M. Fink , George Seifert

For complex systems of the form \[ x' = Ax + \varepsilon g( {t,x,\varepsilon } ) , \] where A is a square matrix, g is analytic in the components … For complex systems of the form \[ x' = Ax + \varepsilon g( {t,x,\varepsilon } ) , \] where A is a square matrix, g is analytic in the components of x and almost periodic in t, and $\varepsilon $ is a complex parameter, we give conditions on the Fourier exponents of g and the eigenvalues of A such that for $\| \varepsilon |$ sufficiently small, the system will have an almost periodic solution. Cases where the eigenvalues of A have zero real parts are dealt with in this result. This paper corrects and clarifies previously published work along this line; cf. [1], [2], [3].

Nonoscillation theorems for a class of perturbed disconjugate differential equations

1983-01-01

A. M. Fink , Takaŝi Kusano

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Shannon’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

98.22 Polynomials with all real roots

2014-11-01

A. M. Fink

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98.30 The isoperimetric inequality for quadrilaterals

2014-11-01

A. M. Fink

Andersson's inequality

2003-01-01

A. M. Fink

Andersson's Inequality gives a lower obund for the integral of a product of convex functions in terms of the averages of each factor.We show that this result holds for a … Andersson's Inequality gives a lower obund for the integral of a product of convex functions in terms of the averages of each factor.We show that this result holds for a wider class of functions and for some signed measures.

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Comparing two integral means for absolutely continuous mappings whose derivatives are in L∞[a,b] and applications

2002-07-01

Neil S Barnett , Pietro Cerone , Sever S Dragomir +1 more

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Low order quadrature for convex functions

2002-01-01

A. M. Fink

Hadamard's inequality for log-concave functions

2000-09-01

A. M. Fink

Inequalities for averages of divided differences

2000-01-01

A. M. Fink

We provide upper bounds for the averages of divided differences in terms of the norms of an appropriate derivative.These generalize a result of Ostrowski. We provide upper bounds for the averages of divided differences in terms of the norms of an appropriate derivative.These generalize a result of Ostrowski.

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Some opial, Lyapunov, and De IaValée poussin inequalities with nonhomogeneous boundary conditions

2000-01-01

Richard C. Brown , A. M. Fink , Don Hinton

A Treatise on Grüss’ Inequality

1999-01-01

A. M. Fink

On some inequalities for convex functions of higher order

1999-01-01

Josip ‎Pečarić , Vera Čuljak , A. M. Fink

A best possible Hadamard inequality

1998-01-01

A. M. Fink

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Shapiro’s Inequality

1998-01-01

A. M. Fink

Nonnegative Solutions of Quasilinear Elliptic Problems with Nonnegative Coefficients

1997-02-01

Nguyên Phuong Các , A. M. Fink , Juan A. Gatica

Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign

1995-01-01

Nguyên Phuong Các , A. M. Fink , Juan A. Gatica

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MAJORIZATION FOR FUNCTIONS WITH MONOTONE NTH DERIVATIVES

1994-07-01

A. M. Fink

Classical and new inequalities in analysis

1993-12-01

D. S. Mitrinović , J. E. Pečarić , A. M. Fink

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Positive Solutions of Second Order Systems of Boundary Value Problems

1993-11-01

A. M. Fink , Juan A. Gatica

Eigenvalues of generalized Gelfand models

1993-06-01

A. M. Fink , Juan A. Gatica , Gastón E. Hernández

Quasilinearization Methods for Proving Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Abel’s and Related Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

More on Norm Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Shannon’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Hölder’s and Minkowski’s Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Mathieu’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Turán’s Inequalities from the Power Sum Theory

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

The Centroid Method in Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Continued Fractions and Padé Approximation Method

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Convex Minimax Inequalities-Equalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Bernoulli’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Young’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Convex Functions and Jensen’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Some Determinantal and Matrix Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Some Inequalities for Monotone Functions

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Connections between General Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Cyclic Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Grüss’ Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Norm Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Triangle Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Classical and New Inequalities in Analysis

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Gram’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Generalized Hölder and Minkowski Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Dynamic Programming and Functional Equation Approaches to Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Bessel’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Steffensen’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Fejér-Jackson’s Inequalities and Related Results

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Some Recent Results Involving Means

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Čebyšev’s Inequality

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Cauchy’s and Related Inequalities

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

For Every Answer There are Two Questions

1992-06-01

A. M. Fink , Juan A. Gatica

Duality is a concept that dominates modem optimization theory. The most familiar one is that associated with linear programming. Here we look at a different one. Most students are introduced … Duality is a concept that dominates modem optimization theory. The most familiar one is that associated with linear programming. Here we look at a different one. Most students are introduced to optimization in a first course in calculus. We propose to show that the idea and use of duality can be introduced in such a course. In a general way, duality asserts the existence of two solutions of extrema problems that are intimately related. Ideally, the two problems should use the same data; one problem should have a solution if and only if the other one does; and the solution of one should give information about the solution of the other. Consider by way of example two problems that are in every calculus text. I. Find the rectangle of maximum area when the perimeter is fixed. II. Find the rectangle of minimum perimeter when the area is fixed. An alert calculus student will observe that the extremal rectangle in both problems is a square. Is this an accident? One way to begin is to observe that Every time you solve an optimization problem you have proved an This is hardly a deep statement, but it is a useful principle. For example, if one has solved Problem I and discovered that the answer is a square then one can formulate a useful inequality. For let A be the area of any rectangle of perimeter P. Then A <A1 where A1 is the area of the square of perimeter P. But A1 = (P/4)2 so that

On Opial's Inequality for f (n)

1992-05-01

A. M. Fink

Coauthor	Papers Together
D. S. Mitrinović	50
Josip ‎Pečarić	50
Juan A. Gatica	7
Max Jodeit	3
George Seifert	3
Donald F. St. Mary	2
Nguyên Phuong Các	2
Gastón E. Hernández	2
Takaŝi Kusano	2
W. R. Madych	2
Pietro Cerone	1
Paul O. Frederickson	1
Michael W. Smiley	1
Sever S Dragomir	1
Gary H. Meisters	1
Wolfgang Kliemann	1
Don Hinton	1
Vera Čuljak	1
Richard C. Brown	1
Richard K. Miller	1
Neil S Barnett	1
Paul Waltman	1
J. E. Pečarić	1

Jensen inequalities for functions with higher monotonicities

1990-12-01

A. M. Fink , Max Jodeit

Partial Differential Equations

1988-01-01

Shiing-Shen Chern

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Analytic Inequalities

1970-01-01

D. S. Mitrinović

Classical and new inequalities in analysis

1993-12-01

D. S. Mitrinović , J. E. Pečarić , A. M. Fink

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Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitati

1955-12-01

Luigi Amerio

Integral Inequalities Involving a Function and Its Derivative

1971-08-01

P. R. Beesack

Beitr�ge zur Theorie der fastperiodischen Funktionen

1927-12-01

S. Bochner

Inequalities: Theory of Majorization and its Applications.

1981-06-01

Barry C. Arnold , Albert W. Marshall , Ingram Olkin

Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, … Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.

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On certain discrete inequalities and their continuous analogs

1966-07-01

Allen M. Pfeffer

In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous … In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequalities. We generalize their techniques to mixed and higher derivatives and inequalities with weight functions in the integrals. We have also considered analogues of inequalities of Muller and Redheffer and have used these inequalities to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the k th derivative of some periodic function and the second number is the square norm of the m th derivative of the same periodic function.

Some further remarks on the Ostrowski generalization of Čebyšev's inequality

1987-04-01

Josip ‎Pečarić

Generalization of an inequality of Ky Fan

1964-02-01

Norman Levinson

Conjugate Inequalities for Functions and Their Derivatives

1974-05-01

A. M. Fink

Integral Inequalities Involving a Function and its Derivative

1971-08-01

Paul R. Beesack

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsP. R. BeesackPaul Beesack wrote his Ph.D. thesis at Washington University under Z. Nehari. He spent 5 years … Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsP. R. BeesackPaul Beesack wrote his Ph.D. thesis at Washington University under Z. Nehari. He spent 5 years at McMaster before transferring to Carleton University in 1960. He spent a sabbatical leave at the Defence Research Board of Canada, and he has been active in preparing secondary school texts. His main research interest is differential and integral equations and inequalities. Editor.

On Chebyshev's other inequality

1984-01-01

A. M. Fink , Max Jodeit

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Best constants in a class of integral inequalities

1969-08-01

David W. Boyd

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Almost periodic properties of ordinary differential equations

1967-12-01

W. A. Eoppel

Notes on some inequalities for convex sequences and for k-convex functions

1980-01-01

Petar Vasić , Josip ‎Pečarić

Inequalities involving integrals of functions and their derivatives

1967-02-01

Donald C Benson

Inequalities involvingf_pandf(n)qforfwithnzeros

1972-08-01

James Brink

Let || II,, denote the L^-norm.This paper determines the smallest possible constants C which satisfy ll/ILsί C(δ-α) ||/< > ||, for certain classes of w-times continuously differentiable functions having n … Let || II,, denote the L^-norm.This paper determines the smallest possible constants C which satisfy ll/ILsί C(δ-α) ||/< > ||, for certain classes of w-times continuously differentiable functions having n zeros on some interval [a, b].Particular interest is placed on functions having a zeros at a and n -a zeros at b.It is shown that smallest possible constants exist for all positive integers n, for all extended real numbers p and q not less than one, and for a = 0, ... ,n providing the exponent s is chosen properly.Moreover, these constants can be used to determine best possible constants when the n zeros are restricted only by the condition that a are at a and β < na are at b. (2.1) Il/H, <ς C (δ -α)'||/< > || ff , / e C*<[a, b], a,

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Differential and Integral Inequalities

1970-01-01

Wolfgang Walter

An Inequality for Convex Functions

1954-11-01

E. M. Wright

On the zeros of solutions of y(n) + py = 0

1969-01-01

W.J Kim

�ber das Maximum des absoluten Betrages von $$\frac{1}{{b - a}}\int\limits_a^b {f\left( x \right)} g\left( x \right)dx - \frac{1}{{\left( {b - a} \right)^2 }}\int\limits_a^b {f\left( x \right)dx} \int\limits_a^b g \left( x \right)dx$$

1935-12-01

Gerhard Gr�ss

Discrete analogs of inequalities of Wirtinger

1955-06-01

Ky Fan , Olga Taussky , John A. Todd

On Certain Discrete Inequalities involving Partial Sums

1969-01-01

Paul R. Beesack

Our aim in this paper is to prove inequalities of the form 1 or 2 for all real values of the parameters α, β and all non-negative (in some cases … Our aim in this paper is to prove inequalities of the form 1 or 2 for all real values of the parameters α, β and all non-negative (in some cases all positive) x i . Obviously, a n is finite in all cases, and we shall show that A n is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants a n , A n (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive x i , otherwise the x i may be non-negative. Although we always write x i ≧ 0 in the following, this should be read as x i > 0 in case α < 0; similar remarks apply to the parameter t introduced below.

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The Laplace Transform. By D. V. Widder. Pp. x, 406. 36s. 1941. Princeton Mathematical series, 6. (Princeton University Press; Humphrey Milford)

1943-02-01

T. A. A. B.

An integral inequality with an application to ordinary differential operators

1978-01-01

W. N. Everitt

Synopsis This paper is concerned with integral inequalities of the form where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [ a, b ] … Synopsis This paper is concerned with integral inequalities of the form where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [ a, b ] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

Functional equation approach to inequalities III

1981-03-01

Chung-Lie Wang

A Note on Periodic Functions and Their Derivatives

1943-07-01

Richard Bellman

Classical and New Inequalities in Analysis

1993-01-01

D. S. Mitrinović , Josip ‎Pečarić , A. M. Fink

Discrete analogues of certain integral inequalities

1957-10-01

H. D. Block

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Inequalities in Fourier Analysis

1975-07-01

William Beckner

Almost periodic functions

1955-12-01

J. D. Weston

Compact Families of Almost Periodic Functions and an Application of the Schauder Fixed-Point Theorem

1969-11-01

A. M. Fink

On integral inequalities and compact embeddings associated with ordinary differential expressions

1979-05-01

R. J. Amos , W. N. Everitt

Opial-Type Inequalities that Involve Higher Order Derivatives

1984-01-01

Carl H. Fitzgerald

Some oscillation and comparison theorems for (r(t) y′)′ + p(t) y = 0

1969-03-01

Donald F. St. Mary

Almost Automorphic Functions on Groups

1965-07-01

William A. Veech

Introduction. S. Bochner has observed in various coiitexts that a certain property enjoyed by the almost periodic functions on a group G canl be used with advantage in obtaininig simpler … Introduction. S. Bochner has observed in various coiitexts that a certain property enjoyed by the almost periodic functions on a group G canl be used with advantage in obtaininig simpler and conceptually more natural proofs of certain theorems concerninig these functions. ([2], [4], [5].) Bochner calls his property almiost automorphy because it first arose in work on differential geometry. Taking G for the present to be the group of integers (= Z) an almost function f has the property that from any sequence {s'} C Z may be extracted a subsequence {cf} such that both limf(t+ cn) =g(t) and limg(tn) ==f(t) hold for each tC Z and some n -oo t-oo function g, but not necessarily uniformly. Bochner has observed that almost periodic functions are almost automorphic, but the converse is not true. ([5], [18].) However we will show in the present paper that the almost functiolls on a group can be characterized in terms of the almost periodic functions. A function f on G is almost if and only if it is the pointwise limit of a jointly almost automorphic net of almost periodic functions. (A consequence of this result is that a group is maximally (minimally) almost if and only if it is maximally (minimally) almost periodic.) Conversely one can characterize almost periodicity in terms, of almost automorphy: A function f is almost periodic if and only if lim f (t + ,) = g (t) is almost whenever the limit exists. This lb

Integral inequalities of the Wirtinger type

1958-09-01

Paul R. Beesack

Inequalities with three functions

1966-11-01

R. M. Redheffer

Differential and Integral Inequalities

2019-01-01

Dorin Andrica , Themistocles M. Rassias

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Integral inequalities of Čebyšev type

1985-11-01

Paul R. Beesack , Josip ‎Pečarić

Kolmogorov-Landau inequalities for monotone functions

1982-11-01

A. M. Fink

Hardy’s inequality and its extensions

1961-03-01

Paul R. Beesack

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�ber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert

1937-12-01

Alexander Ostrowski

On some Inequalities

1934-01-01

Shin-ichi Izumi , K. Kobayashi , Tatsuo Takahashi

On the Ostrowski generalization of C̆ebyšev's inequality

1984-09-01

Josip ‎Pečarić

Canonical forms and principal systems for general disconjugate equations

1974-01-01

William F. Trench

It is shown that the disconjugate equation (1) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L x identical-to left-parenthesis 1 slash beta Subscript n Baseline right-parenthesis left-parenthesis d slash d t right-parenthesis … It is shown that the disconjugate equation (1) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L x identical-to left-parenthesis 1 slash beta Subscript n Baseline right-parenthesis left-parenthesis d slash d t right-parenthesis dot left-parenthesis 1 slash beta Subscript n minus 1 Baseline right-parenthesis midline-horizontal-ellipsis left-parenthesis d slash d t right-parenthesis left-parenthesis 1 slash beta 1 right-parenthesis left-parenthesis d slash d t right-parenthesis left-parenthesis x slash beta 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mo>≡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <italic>a</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than t greater-than b"> <mml:semantics> <mml:mrow> <mml:mo>></mml:mo> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">> t > b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript i Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\beta _i} > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and (2) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript i Baseline element-of upper C left-parenthesis a comma b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\beta _i} \in C(a,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, can be written in essentially unique canonical forms so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Overscript b Endscripts beta Subscript i Baseline d t equals normal infinity left-parenthesis integral Underscript a Endscripts beta Subscript i Baseline d t equals normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mo largeop="false">∫</mml:mo> <mml:mi>b</mml:mi> </mml:msup> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo largeop="false">∫</mml:mo> <mml:mi>a</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i less-than-or-equal-to n minus 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>i</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq i \leq n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. From this it follows easily that (1) has solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 comma ellipsis comma x Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_1}, \ldots ,{x_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are positive in (<italic>a, b</italic>) near <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b left-parenthesis a right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">b(a)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and satisfy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow b Endscripts minus x Subscript i Baseline left-parenthesis t right-parenthesis slash x Subscript j Baseline left-parenthesis t right-parenthesis equals 0 left-parenthesis limit Underscript t right-arrow a Endscripts plus x Subscript i Baseline left-parenthesis t right-parenthesis slash x Subscript j Baseline left-parenthesis t right-parenthesis equals normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i greater-than j less-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>j</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq i > j \leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Necessary and sufficient conditions are given for (1) to have solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y 1 comma ellipsis comma y Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{y_1}, \ldots ,{y_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow b Endscripts minus y Subscript i Baseline left-parenthesis t right-parenthesis slash y Subscript j Baseline left-parenthesis t right-parenthesis equals limit Underscript t right-arrow a Endscripts plus y Subscript j Baseline left-parenthesis t right-parenthesis slash y Subscript i Baseline left-parenthesis t right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i greater-than j less-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>j</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq i > j \leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 prime right-parenthesis beta Subscript i element-of upper C Superscript left-parenthesis n minus i right-parenthesis Baseline left-parenthesis a comma b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2’){\beta _i} \in {C^{(n - i)}}(a,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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