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Abstract In this paper, we approach the monotonicity of three functions involving the confluent hypergeometric function of the second kind from the perspective of using the monotonicity rules. By employing … Abstract In this paper, we approach the monotonicity of three functions involving the confluent hypergeometric function of the second kind from the perspective of using the monotonicity rules. By employing the monotonicity of these functions, we establish bounds for $U^\prime(a,b,x)/U(a,b,x)$ and $U(a,b,y)/U(a,b,x)$, which are shown to be extremely tight for large values of $x$ and $y$. Furthermore, by using the relationships between the confluent hypergeometric function of the second kind and other special functions, we derive a series of results concerning the incomplete gamma function and the modified Bessel function of the second kind, including the bounds for $\Gamma(\gamma,x)$, $K_{\nu+1}(x)/K_{\nu}(x)$, and $K_{\nu}^\prime(x)/K_{\nu}(x)$.
In this paper, we introduce the concept of strongly completely monotonic functions on time scales and explore several properties of these functions. We then present key results that are applied … In this paper, we introduce the concept of strongly completely monotonic functions on time scales and explore several properties of these functions. We then present key results that are applied to analyze the cases for continuous, discrete, and quantum time scales. As applications, we prove that the Gauss hypergeometric functions $F(a,b;c;z)$ and the confluent hypergeometric functions of the first kind $M(a,c,z)$ are absolutely monotonic, while the confluent hypergeometric functions of the second kind $U(a,b;z)$ are both strongly completely monotonic and completely monotonic.
In this paper, we present some monotonicity rules for the ratio of two power series $x\mapsto \sum_{k=0}^\infty a_k x^k / \sum_{k=0}^\infty b_k x^k$ under the assumption that the monotonicity of … In this paper, we present some monotonicity rules for the ratio of two power series $x\mapsto \sum_{k=0}^\infty a_k x^k / \sum_{k=0}^\infty b_k x^k$ under the assumption that the monotonicity of the sequence ${a_k/b_k}$ changes twice. Additionally, we introduce a local monotonicity rule in this paper.
In this paper, we investigate the monotonicity of the functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow from bar StartFraction sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts a Subscript … In this paper, we investigate the monotonicity of the functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow from bar StartFraction sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts a Subscript k Baseline w Subscript k Baseline left-parenthesis t right-parenthesis Over sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts b Subscript k Baseline w Subscript k Baseline left-parenthesis t right-parenthesis EndFraction"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mfrac> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:munderover> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">t \mapsto \frac {\sum _{k=0}^\infty a_k w_k(t)}{\sum _{k=0}^\infty b_k w_k(t)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar StartFraction integral Subscript alpha Superscript beta Baseline f left-parenthesis t right-parenthesis w left-parenthesis t comma x right-parenthesis normal d t Over integral Subscript alpha Superscript beta Baseline g left-parenthesis t right-parenthesis w left-parenthesis t comma x right-parenthesis normal d t EndFraction"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mfrac> <mml:mrow> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>β<!-- β --></mml:mi> </mml:msubsup> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>β<!-- β --></mml:mi> </mml:msubsup> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">x \mapsto \frac {\int _\alpha ^\beta f(t) w(t,x) \mathrm {d} t}{\int _\alpha ^\beta g(t) w(t,x) \mathrm {d} t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, focusing on case where the monotonicity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript k Baseline slash b Subscript k"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">a_k/b_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis t right-parenthesis slash g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <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:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(t)/g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders.
The main objective of this paper is to establish the $Y$-function and L'Hospital-type monotonicity rules with nabla and diamond-alpha derivatives on time scales. The main objective of this paper is to establish the $Y$-function and L'Hospital-type monotonicity rules with nabla and diamond-alpha derivatives on time scales.
Using monotonicity rules, we establish the monotonicity of three kinds of functions involving the Gaussian hypergeometric function. This refines some existing results. We also present some bounds for the zero-balance … Using monotonicity rules, we establish the monotonicity of three kinds of functions involving the Gaussian hypergeometric function. This refines some existing results. We also present some bounds for the zero-balance Gaussian hypergeometric function.
In this paper, we establish a sufficient and necessary condition for a function involving the inverse hyperbolic tangent function as a best possible upper bound for the complete elliptic integral … In this paper, we establish a sufficient and necessary condition for a function involving the inverse hyperbolic tangent function as a best possible upper bound for the complete elliptic integral of the first kind. Equivalently, we obtain a lower bound involving the arithmetic mean and logarithmic mean for the Gauss arithmetic-geometric mean. This provides a positive answer to a conjecture proposed by Yang, Song and Chu in 2014.
In this paper, we aim to construct $n$ dimensional Jensen, Hardy and Hermite-Hadamard type inequalities for multiple diamond-alpha integral on time scales. The cases of Hardy type inequality with a … In this paper, we aim to construct $n$ dimensional Jensen, Hardy and Hermite-Hadamard type inequalities for multiple diamond-alpha integral on time scales. The cases of Hardy type inequality with a weighted function and Hermite-Hadamard type inequality with three variables are also considered minutely.
In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic … In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic in ν if and only if β≥1, where T ν,α,β (s)=K ν 2 (s)-βK ν-α (s)K ν+α (s) defined on s>0 and K ν (s) is the modified Bessel function of the second kind of order ν. Finally, we determine the necessary and sufficient conditions for the functions s↦T μ,α,1 (s)/T ν,α,1 (s), s↦(T μ,α,1 (s)+T ν,α,1 (s))/(2T (μ+ν)/2,α,1 (s)), and s↦d n 1 dν n 1 T ν,α,1 (s)/d n 2 dν n 2 T ν,α,1 (s) to be monotonic in s∈(0,∞) by employing the monotonicity rules.
In this paper, we investigate the monotonicity of the functions $t \mapsto \frac{\sum_{k=0}^\infty a_k w_k(t)}{\sum_{k=0}^\infty b_k w_k(t)}$ and $x \mapsto \frac{\int_\alpha^\beta f(t) w(t,x) \textrm{d} t}{\int_\alpha^\beta g(t) w(t,x) \textrm{d} t}$, focusing … In this paper, we investigate the monotonicity of the functions $t \mapsto \frac{\sum_{k=0}^\infty a_k w_k(t)}{\sum_{k=0}^\infty b_k w_k(t)}$ and $x \mapsto \frac{\int_\alpha^\beta f(t) w(t,x) \textrm{d} t}{\int_\alpha^\beta g(t) w(t,x) \textrm{d} t}$, focusing on case where the monotonicity of $a_k/b_k$ and $f(t)/g(t)$ change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two $\mathcal{Z}$-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders.
As an efficient mathematical tool, monotonicity rules play an extremely crucial role in the real analysis field. In this paper, we explore some monotonicity rules for quotient of Delta, Nabla … As an efficient mathematical tool, monotonicity rules play an extremely crucial role in the real analysis field. In this paper, we explore some monotonicity rules for quotient of Delta, Nabla and Diamond-Alpha integrals with variable upper limits and parameters on time scales, respectively. Moreover, we consider the monotonicity rules for quotient of the product of multiple Delta integrals with parameters on time scales. Power series is also concerned for being a special case of integral with parameters on time scales.
In this paper, we generalize psi and polygamma functions based on the Laplace transform in the field of time scales, and explore some properties of them. Next, we present the … In this paper, we generalize psi and polygamma functions based on the Laplace transform in the field of time scales, and explore some properties of them. Next, we present the concepts of $q$-complete monotonicity, $q$-logarithmically complete monotonicity and $q$-absolute monotonicity with delta derivative on time scales. At last, we prove that the function \begin{equation*} s\mapsto \alpha \psi_{\mathbb{R}_0,\mathbb{T}}(s)-\ln s+\frac{1}{2s}+\frac{1}{12s^2} \end{equation*} is $1$-complete monotonicity on $(0,\infty)$ if $\mathbb{T}=\mathbb{N}$ and $\alpha \in [\frac{3-2\sqrt{3}}{6},\frac{3+2\sqrt{3}}{6}]$, and it is decreasing on $(0,\infty)$ if $\mathbb{T}=h\mathbb{N}\cup\{1\} (h\geq1)$ and $\alpha=1$, where $\mathbb{R}_0=[0,\infty)$ and $\psi_{\mathbb{R}_0,\mathbb{T}}$ is a psi function on time scales.
In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary … In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillatory solution by employing fixed point theorem.
In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria. In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.
In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of … In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.
In this paper, we establish sufficient conditions for Qi type diamond-alpha integral inequalities and its generalized form on time scales. In this paper, we establish sufficient conditions for Qi type diamond-alpha integral inequalities and its generalized form on time scales.
In this paper, we generalize Opial inequality to higher dimensions on time scales.We prove the Opial Delta-nabla inequality of n variables, and then give two diamond-alpha dynamic inequalities of Opial … In this paper, we generalize Opial inequality to higher dimensions on time scales.We prove the Opial Delta-nabla inequality of n variables, and then give two diamond-alpha dynamic inequalities of Opial type of n variables.As well, we introduce some special cases.

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On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with … On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}\left … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the modified Bessel functions of the second kind of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar upper K Subscript u Baseline left-parenthesis x right-parenthesis upper K Subscript v Baseline left-parenthesis x right-parenthesis slash upper K Subscript left-parenthesis u plus v right-parenthesis slash 2 Baseline left-parenthesis x right-parenthesis squared"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>u</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">x\mapsto K_{u}\left ( x\right ) K_{v}\left ( x\right ) /K_{\left ( u+v\right ) /2}\left ( x\right ) ^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly decreasing on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( 0,\infty \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our study not only involves the Turán type inequalities, log-convexity or log-concavity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the conjecture posed by Baricz, but also yields various new results concerning the monotonicity and convexity of the ratios of the modified Bessel functions of the second kind. As applications of our main theorems, some new sharp inequalities involving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are presented, which contain sharp estimates for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and sharp bounds for the ratios <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K prime Subscript v Baseline left-parenthesis x right-parenthesis slash upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v}^{\prime }\left ( x\right ) /K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript v plus 1 Baseline left-parenthesis x right-parenthesis slash upper K Subscript v Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K_{v+1}\left ( x\right ) /K_{v}\left ( x\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
The development of time scales is still in its infancy, yet as inroads are made, interest is gathering steam. Of a great deal of interest are methods being intro duced … The development of time scales is still in its infancy, yet as inroads are made, interest is gathering steam. Of a great deal of interest are methods being intro duced for dynamic equations on time sc
Abstract In this paper, we present some new Cauchy–Schwarz inequalities for N -tuple diamond-alpha integral on time scales. The obtained results improve and generalize some Cauchy–Schwarz type inequalities given by … Abstract In this paper, we present some new Cauchy–Schwarz inequalities for N -tuple diamond-alpha integral on time scales. The obtained results improve and generalize some Cauchy–Schwarz type inequalities given by many authors.
In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic … In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function -T ν,α,β (s) is completely monotonic in s and absolutely monotonic in ν if and only if β≥1, where T ν,α,β (s)=K ν 2 (s)-βK ν-α (s)K ν+α (s) defined on s>0 and K ν (s) is the modified Bessel function of the second kind of order ν. Finally, we determine the necessary and sufficient conditions for the functions s↦T μ,α,1 (s)/T ν,α,1 (s), s↦(T μ,α,1 (s)+T ν,α,1 (s))/(2T (μ+ν)/2,α,1 (s)), and s↦d n 1 dν n 1 T ν,α,1 (s)/d n 2 dν n 2 T ν,α,1 (s) to be monotonic in s∈(0,∞) by employing the monotonicity rules.
In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha … In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.
In this paper, we generalize Opial inequality to higher dimensions on time scales.We prove the Opial Delta-nabla inequality of n variables, and then give two diamond-alpha dynamic inequalities of Opial … In this paper, we generalize Opial inequality to higher dimensions on time scales.We prove the Opial Delta-nabla inequality of n variables, and then give two diamond-alpha dynamic inequalities of Opial type of n variables.As well, we introduce some special cases.
Introduction.Les fonctions absolument monotones jouent le m~me rble fondamental duns la th6orie des fonctions analytiques d'une variable r6elle que les fonctions (simplement) monotones pour la classe g6n6rale des fonctions £ … Introduction.Les fonctions absolument monotones jouent le m~me rble fondamental duns la th6orie des fonctions analytiques d'une variable r6elle que les fonctions (simplement) monotones pour la classe g6n6rale des fonctions £ variation born6e.~ I1 semble doric qu'une dtude syst6matique des propri6t6s des fonctions absolument monotones est indispensable pour bien p6n6trer t~ nature des fonctions ~nalytiques r6eiles.Le pr6sent M6moire a pour but de contribuer £ cette 6rude.En ~tablissant des in~gMit~s g6n6rMes auxquelles satisfont les fonctions absolument monotones sur un segment limit6 ou infini, i'M 6tudi6 surtout, quel est le segment maximum, off une fonction, prenant avec un nombre fini ou infini de ses d6riv6es successives des valeurs donn6es en un point, peut rester absolument monotone, et d'autre part, dans quels cas ces donn6es suffisent pour la d6terminer compl~tement.Cette 6rude est naturellement li6e £ la th6orie des s6ries divergentes, et, comme consequence partieuli~re, nous en d6duisons une nouvelle mdthode, ind6pendante des fractions continues, pour r6soudre le probl~me g6ndral des moments.~ 1 S. BERNSTEIN, ,Le~,ons su~" les propridtgs extrdmales etc., (Collection de Monographies publide sons la direction de M. E. Borel), Premiere Note (pp.I93--I97).Je rappel!erai que la thdorie classiquc des moments de Stieltjes a dt5 compl~tde recemnlent par M. H. HAMBURGER, ,Stieltjessches Momentenproblem,,Math.Ann.Bd. 8~ (235--315) , Bd. 82 (I 2o--I64) et M. T. CARLEM)~, "Sur les dquations int~qrales singuli~res it noyaux rdel et symdtrique; je signalerai aussi lcs M~moires de M. HAUSDORFF ,Summalionsmethoden und Momentfolgen,; Math.Zeitschrift Bd. 9 et ,Momentproblem fi~r endlicl~es Intervall, (dont j'ai pris connaissance pendant la r6daction de ce travail) qui aborde le probleme des monlents par des m6thodes prg, sentant certaines analogies avec les miennes.1-~822.Acta math*matica.52. Imprlm6 le 27 f6vrier 1928.Sur les fonctions absolument monotones.PREMIER CHAPITRE.Ba--(x--I)! --Cx-I(X'-I) n-1 + Cu--I(X--2) n-i .... ],(5) eomme on s'en assure, en remarquant que d n X p d log x '~ --pn ~V.Ceci pos6, nous uvons la proposition suivante: P Thdor~mo I. Si f(o),f (o),...,f(~)(o) sont les d&iv&s d'une fonctlon absolument monotone sur le segment (--c, o), on a f(o), dlog c~, ..: [.i-io~c2-h] >--o, dlogg c '''" el log 6 2h+l -quel que soit h.Sur les fonctions absolument monotones.En effet, par hypo~h~se, on a un ddveloppement g coefficients non ndgatifs f(x)=a o+aa(x+c)+.+a~(x+c) ~+... (7) !
In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, … In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions.
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript nu"><mml:semantics><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">K_{\nu }</mml:annotation></mml:semantics></mml:math></inline-formula>be the modified Bessel functions of the second kind of order<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"><mml:semantics><mml:mi>ν</mml:mi><mml:annotation encoding="application/x-tex">\nu</mml:annotation></mml:semantics></mml:math></inline-formula>. The ratio<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q … Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript nu"><mml:semantics><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">K_{\nu }</mml:annotation></mml:semantics></mml:math></inline-formula>be the modified Bessel functions of the second kind of order<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"><mml:semantics><mml:mi>ν</mml:mi><mml:annotation encoding="application/x-tex">\nu</mml:annotation></mml:semantics></mml:math></inline-formula>. The ratio<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript nu Baseline left-parenthesis x right-parenthesis equals x upper K Subscript nu minus 1 Baseline left-parenthesis x right-parenthesis slash upper K Subscript nu Baseline left-parenthesis x right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">Q_{\nu }\left ( x\right ) =xK_{\nu -1}\left ( x\right ) /K_{\nu }\left ( x\right )</mml:annotation></mml:semantics></mml:math></inline-formula>appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 1 right-parenthesis Superscript n Baseline upper Q Subscript nu Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis x right-parenthesis greater-than left-parenthesis greater-than right-parenthesis 0"><mml:semantics><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mi>Q</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ν</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>&gt;</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\left ( -1\right ) ^{n}Q_{\nu }^{\left ( n\right ) }\left ( x\right ) &gt;\left ( &gt;\right ) 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="x greater-than 0"><mml:semantics><mml:mrow><mml:mi>x</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">x&gt;0</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2 comma 3"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n=2,3</mml:annotation></mml:semantics></mml:math></inline-formula>if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue nu EndAbsoluteValue greater-than left-parenthesis greater-than right-parenthesis 1 slash 2"><mml:semantics><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>ν</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>&gt;</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\left \vert \nu \right \vert &gt;\left ( &gt;\right ) 1/2</mml:annotation></mml:semantics></mml:math></inline-formula>holds for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n=2</mml:annotation></mml:semantics></mml:math></inline-formula>. This yields several new consequences and improves some known results. Finally, two conjectures are proposed.
In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln … In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln x+x- \ln \sqrt{2\pi } ) +1}-\frac{120}{7}x^{2} $$ is strictly increasing from $( 0,\infty ) $ onto $( 1,1860/343 ) $ . This not only yields some known and new inequalities for the gamma function, but also gives some completely monotonic functions related to the gamma function.
This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic … This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequali
Using the representation theorem and inversion formula for Stieltjes transforms, we give a simple proof of the infinite divisibility of the student $t$-distribution for all degrees of freedom by showing … Using the representation theorem and inversion formula for Stieltjes transforms, we give a simple proof of the infinite divisibility of the student $t$-distribution for all degrees of freedom by showing that $x^{-\frac{1}{2}}K_\nu(x^{\frac{1}{2}})/K_{\nu+1}(x^{\frac{1}{2}})$ is completely monotonic for $\nu \geqq -1$. Our approach proves the stronger and new result, that $x^{-\frac{1}{2}}K_\nu (x^{\frac{1}{2}}) /K_{\nu+1}(x^{\frac{1}{2}})$ is a completely monotonic function of $x$ for all real $\nu$. We also derive a new integral representation.
In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of … In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.
We show that certain functions involving quotients of gamma functions are completely monotonic. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions … We show that certain functions involving quotients of gamma functions are completely monotonic. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions whose Laplace transforms involve quotients of gamma functions.
We prove that the function ^~v Kli(bxl'*)Ku(axll'1)is the Laplace transform of an infinitely divisible probability distribution when v > ß > 0 and b > a > 0. This implies … We prove that the function ^~v Kli(bxl'*)Ku(axll'1)is the Laplace transform of an infinitely divisible probability distribution when v > ß > 0 and b > a > 0. This implies the complete monotonie ity of the function.We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when x lies in the right half-plane.We conjecture that rby-/Ili(ax1l3)Ii,(bx1l2) ©'/"(fceVa^iax1/3) also is the Laplace transform of an infinitely divisible probability distribution.It is known that in the limit as v -► oo, the infinite divisibility property holds for both functions. Introduction.A probability distribution on [0, oo) is said to be infinitely divisible if and only if for every natural number n, the nth root of its Laplace transform is a Laplace transform of a probability distribution.A function / denned on (0, oo) and of class C°° is said to be completely monotonie if (-l)n/(n)(x) > 0 on (0, oo) for all n.The connection between infinitely divisible distributions and completely monotonie functions is expressed in the following theorem (see [5, p. 450]):THEOREM 1.The function w is the Laplace transform of an infinitely divisible probability distribution on [0, oo) if and only ifw = e~h where h(0-\-) = 0 and hi is completely monotonie.In §2 we shall prove the following result: THEOREM 2. Let K\ be the modified Bessel function of the second kind.Then
The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention.Although the basic … The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention.Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to cases "in between".In this paper we present time scales versions of the inequalities: Hölder, Cauchy-Schwarz, Minkowski, Jensen,
In this paper, we establish sufficient conditions for Qi type diamond-alpha integral inequalities and its generalized form on time scales. In this paper, we establish sufficient conditions for Qi type diamond-alpha integral inequalities and its generalized form on time scales.
In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria. In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.
In this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition … In this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition for a function constructed by ratios of gamma functions to be completely mixed monotonic. From this, some new inequalities for gamma, psi, and polygamma functions are derived.
The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall … The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor's formula, l'Hôspital's rule, and Kneser's theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor's formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.
Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be … Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be expressed in terms of standard higher transcendental functions. Inverting a Laplace transform involves integrating a complex-valued function over a vertical line, and establishing the positivity of the resulting integral can be tricky. Sometimes asymptotic methods are helpful, see for example Fields and Ismail [ 6 ].
We define a new difference equation analogue of the Bessel differential equation and investigate the properties of its solution, which we express using a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 2 … We define a new difference equation analogue of the Bessel differential equation and investigate the properties of its solution, which we express using a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 2 Baseline upper F 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_2F_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hypergeometric function. We find analogous formulas for Bessel function recurrence relations, a summation transformation which is identical to the Laplace transform of classical Bessel functions, and oscillation.
In this paper, we provide some new generalizations of Feng Qi type integral inequalities on time scales by using elementary analytic methods. In this paper, we provide some new generalizations of Feng Qi type integral inequalities on time scales by using elementary analytic methods.
By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients.We give examples of dynamic … By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients.We give examples of dynamic equations to which previously known oscillation criteria are not applicable.
In the article, we prove that the double inequalities $$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$ hold for all $x>0$ if and only if $a\geq1/4$ and … In the article, we prove that the double inequalities $$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$ hold for all $x>0$ if and only if $a\geq1/4$ and $b=0$ if $a, b\in[0, \infty)$ , where $K_{\nu}(x)$ is the modified Bessel function of the second kind. As applications, we provide bounds for $K_{n+1}(x)/K_{n}(x)$ with $n\in\mathbb{N}$ and present the necessary and sufficient condition such that the function $x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$ is strictly increasing (decreasing) on $(0, \infty)$ .