In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient …
In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient conditions for Chebyshev type inequalities are derived.
We present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality …
We present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality is obtained. Sharpness of each introduced inequality is illustrated by suitable examples.
We show several inequalities for angles between vectors and subspaces in inner product spaces, where concave functions are involved.In specific situations, some of them can be interpreted as triangle inequalities …
We show several inequalities for angles between vectors and subspaces in inner product spaces, where concave functions are involved.In specific situations, some of them can be interpreted as triangle inequalities for natural metrics on complex projective spaces.In a consequence, we obtain a few operator generalizations of the famous Cauchy-Schwarz inequality, where powers grater than two occur.
The cases of equality for reverse Schwarz inequalities and Grüss type inequalities are detailed studied.Necessary and sufficient conditions for them are given.Moreover, we introduce an unification of two reverse Schwarz …
The cases of equality for reverse Schwarz inequalities and Grüss type inequalities are detailed studied.Necessary and sufficient conditions for them are given.Moreover, we introduce an unification of two reverse Schwarz inequalities obtained by S.
Abstract We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner …
Abstract We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.
Summary The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values …
Summary The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values of products of random variables. Special cases of this inequality include important classical inequalities such as the Cauchy–Schwarz and Chebyshev inequalities. The purpose of this article is to prove Fujiwara’s inequality in abstract probability spaces with more general assumptions for random variables than those associated with monotonicity as found in the literature. This newly introduced property of random variables will be called synchronicity. As a consequence, we obtain inequalities originated by Chebyshev, Hardy–Littlewood–Pólya and Jensen–Mercer under new conditions. The results are illustrated by some examples constructed for specific probability measures and specific random variables. In biometrics, the obtained inequalities can be used wherever there is a need to compare the characteristics of experimental data based on means or moments, and the data can be modeled using functions that are synchronous or convex.
In this paper, some vector inequalities for convex maps are proved. The obtained results refer to the famous Jensen inequality and generalize further classical inequalities of Jessen and McShane. In …
In this paper, some vector inequalities for convex maps are proved. The obtained results refer to the famous Jensen inequality and generalize further classical inequalities of Jessen and McShane. In addition, the Hahn–Banach theorems with sublinear and convex maps are considered and used to prove the theorem on the support of certain convex maps.
In this paper, some vector inequalities for convex maps are proved. The obtained results refer to the famous Jensen inequality and generalize further classical inequalities of Jessen and McShane. In …
In this paper, some vector inequalities for convex maps are proved. The obtained results refer to the famous Jensen inequality and generalize further classical inequalities of Jessen and McShane. In addition, the Hahn–Banach theorems with sublinear and convex maps are considered and used to prove the theorem on the support of certain convex maps.
Summary The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values …
Summary The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values of products of random variables. Special cases of this inequality include important classical inequalities such as the Cauchy–Schwarz and Chebyshev inequalities. The purpose of this article is to prove Fujiwara’s inequality in abstract probability spaces with more general assumptions for random variables than those associated with monotonicity as found in the literature. This newly introduced property of random variables will be called synchronicity. As a consequence, we obtain inequalities originated by Chebyshev, Hardy–Littlewood–Pólya and Jensen–Mercer under new conditions. The results are illustrated by some examples constructed for specific probability measures and specific random variables. In biometrics, the obtained inequalities can be used wherever there is a need to compare the characteristics of experimental data based on means or moments, and the data can be modeled using functions that are synchronous or convex.
Abstract We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner …
Abstract We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.
We show several inequalities for angles between vectors and subspaces in inner product spaces, where concave functions are involved.In specific situations, some of them can be interpreted as triangle inequalities …
We show several inequalities for angles between vectors and subspaces in inner product spaces, where concave functions are involved.In specific situations, some of them can be interpreted as triangle inequalities for natural metrics on complex projective spaces.In a consequence, we obtain a few operator generalizations of the famous Cauchy-Schwarz inequality, where powers grater than two occur.
The cases of equality for reverse Schwarz inequalities and Grüss type inequalities are detailed studied.Necessary and sufficient conditions for them are given.Moreover, we introduce an unification of two reverse Schwarz …
The cases of equality for reverse Schwarz inequalities and Grüss type inequalities are detailed studied.Necessary and sufficient conditions for them are given.Moreover, we introduce an unification of two reverse Schwarz inequalities obtained by S.
We present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality …
We present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality is obtained. Sharpness of each introduced inequality is illustrated by suitable examples.
In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient …
In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient conditions for Chebyshev type inequalities are derived.
The purpose of this survey is to give a comprehensive introduction to some classes of classical and recent analytic inequalities in Inner Product Spaces.
The purpose of this survey is to give a comprehensive introduction to some classes of classical and recent analytic inequalities in Inner Product Spaces.
The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with (CBS)− inequality and provide refinements and reverse results as well as to …
The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with (CBS)− inequality and provide refinements and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with
this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companion, reverse and related results both for real and complex numbers are also presented.
New reverses of the Schwarz inequality in complex inner products spaces with applications for bounded linear operators are given. Some Grüss' type inequalities and their applications for numerical radius and …
New reverses of the Schwarz inequality in complex inner products spaces with applications for bounded linear operators are given. Some Grüss' type inequalities and their applications for numerical radius and the operator norm are provided as well.
In the framework of a pre-inner product C*-module over a unital C*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. …
In the framework of a pre-inner product C*-module over a unital C*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. Elezović et al. [Math. Inequal. Appl., 8 (2005), no.2, 223-231]. We apply our results to give Klamkin-Mclenaghan, Shisha-Mond and Cassels type inequalities. We also present a Grüss type inequality.
We prove two new reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C * -algebra.We apply our …
We prove two new reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C * -algebra.We apply our results to get some norm and integral inequalities.As a consequence, we improve a celebrated reverse Cauchy-Schwarz inequality due to G. Pólya and G. Szegö.
Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in inner …
Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in inner product spaces. Some natural applications for integral inequalities are also pointed out.
In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient …
In [3] Chebyshev type inequalities are proved for separable finite sequences.In this paper, the applicability of that notion is extended and replaced by a relation of synchronicity.In consequence, new sufficient conditions for Chebyshev type inequalities are derived.
Let $G$ be a closed subgroup of the orthogonal group $O(n)$ acting on $R^n$. A real-valued function $f$ on $R^n$ is called $G$-monotone (decreasing) if $f(y) \geqq f(x)$ whenever $y …
Let $G$ be a closed subgroup of the orthogonal group $O(n)$ acting on $R^n$. A real-valued function $f$ on $R^n$ is called $G$-monotone (decreasing) if $f(y) \geqq f(x)$ whenever $y \precsim x$, i.e., whenever $y \in C(x)$, where $C(x)$ is the convex hull of the $G$-orbit of $x$. When $G$ is the permutation group $\mathscr{P}_n$ the ordering $\sim$ is the majorization ordering of Schur, and the $\mathscr{P}_n$-monotone functions are the Schur-concave functions. This paper contains a geometrical study of the convex polytopes $C(x)$ and the ordering $\precsim$ when $G$ is any closed subgroup of $O(n)$ that is generated by reflections, which includes $\mathscr{P}_n$ as a special case. The classical results of Schur (1923), Ostrowski (1952), Rado (1952), and Hardy, Littlewood and Polya (1952) concerning majorization and Schur functions are generalized to reflection groups. It is shown that a smooth $G$-invariant function $f$ is $G$-monotone iff $(r'x)(r'\nabla f(x))\leqq 0$ for all $x \in R^n$ and all $r \in R^n$ such that the reflection across the hyperplane $\{z\mid r'z = 0\}$ is in $G$. Furthermore, it is shown that the convolution (relative to Lebesgue measure) of two nonnegative $G$-monotone functions is again $G$-monotone. The latter extends a theorem of Marshall and Olkin (1974) concerning $\mathscr{P}_n$, and has applications to probability inequalities arising in multivariate statistical analysis.
The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities …
The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecaric inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski's inequalities for sequences or real numbers and the generalization of Wagner's inequality in inner product spaces are pointed out. Finally, some Gruss type inequalities for n-tuples of vectors in inner product spaces and their natural applications for the approximation of the discrete Fourier and Mellin transforms are given as well.
Harvey generalizes the Cauchy-Schwarz inequality to an inequality involving four vectors.Here we show a stronger result than the inequality.Moreover we generalize the result to an inequality involving any number of …
Harvey generalizes the Cauchy-Schwarz inequality to an inequality involving four vectors.Here we show a stronger result than the inequality.Moreover we generalize the result to an inequality involving any number of real or complex vectors.
t A set is convex if for every pair P, Q of points of the set the line segment PQ is contained in the set.* We call <f> convex on …
t A set is convex if for every pair P, Q of points of the set the line segment PQ is contained in the set.* We call <f> convex on K if 0(J(zi+z 2 ) ) Si(<f>(zi) +<Kz 2 ) ) for all z h z 2 in K.
We generalize the well-known Cauchy-Schwarz inequality involving any number of real or complex matrices, and also give a necessary and sufficient condition for the equality.This is an improvement of the …
We generalize the well-known Cauchy-Schwarz inequality involving any number of real or complex matrices, and also give a necessary and sufficient condition for the equality.This is an improvement of the two recent literatures due to N. Harvey and D. Choi.
In [6], by means of convex functions Φ : R → R , Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable …
In [6], by means of convex functions Φ : R → R , Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).
Reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in …
Reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in inner product spaces. Some natural applications for integral inequalities are also pointed out.
There is a lots of known complemented Cauchy-Bunyakowsky-Schwarz' inequalities in the literature.In the first part of this paper we shall deduce many of them using very simple technique of inner …
There is a lots of known complemented Cauchy-Bunyakowsky-Schwarz' inequalities in the literature.In the first part of this paper we shall deduce many of them using very simple technique of inner product space.The similar technique is applied in the second part to complemented Grüss' inequality.
We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the …
We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the Hadamard inequality for Gram determinants are also given.
Some new Gruss type inequalities in inner product spaces and applications for integrals are given.
Some new Gruss type inequalities in inner product spaces and applications for integrals are given.
I. Des fonctions convexes et concaves.Ddfinition.Exemples.Dans sa e~l~bre Analyse alg~brique (note IX, pp.457--59) CAucHY dgmontre que ,Aa moyenne gdom6trique entre plusieurs nombres est toujours infdrieure h leur moyenne alg~brique,.La mdfhode …
I. Des fonctions convexes et concaves.Ddfinition.Exemples.Dans sa e~l~bre Analyse alg~brique (note IX, pp.457--59) CAucHY dgmontre que ,Aa moyenne gdom6trique entre plusieurs nombres est toujours infdrieure h leur moyenne alg~brique,.La mdfhode employge par CAVCHY est extrgmement dldgante, et elle h passd sans changement dans tous los traitds d'analyse algdbrique.Elle eonsiste, comme on sait, en ceci, que, de l'in6galit6 --'(a+b), ~/~b < 2 oh a et b sont des nombres positifs, on est conduit ~t l'in6galitd analogue pour quatre hombres, savoir ~/.b~a < -' (a + b+ c + d), 4 et aux suivantes, pour 8, I6,..., 2 m nombres, apr~s quoi ee nombre, par un artifice, est r6duit ~ un hombre arbitraire inf6rieur, n.Cette m6thode simple a gt6 mon point de d6part dans les recherches suivantes, qui conduisent, par une voie en r6alit6 tr6s simple et 616mentaire, ~ des r6sultats 9 " imp generaux et non sans ort~nee. 1 Oonf6rence faite ~ la Soci6t6 math6matique danoise le x 7
Some majorisation type discrete inequalities for convex functions are established.Two applications are also provided.
Some majorisation type discrete inequalities for convex functions are established.Two applications are also provided.
G. E. Forsythe, who edited the translation of Kantorovich's paper, included the following remark about this footnote: It is not clear to me that Kantorovich's inequality really is a special …
G. E. Forsythe, who edited the translation of Kantorovich's paper, included the following remark about this footnote: It is not clear to me that Kantorovich's inequality really is a special case of that of Polya and Szego. Examining the relation between the two inequalities more closely we found that this remark is well justified and can be made even more specific in that the inequality of Polya and Szeg6 in the form (4) is a special case of the Kantorovich inequality