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In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h, p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for the class of twice differentiable functions, where the convexity property of the target function is not assumed in advance. They represent a refinement of this inequality in the case of convex/concave functions with numerous applications.

In this article, we give some improvements and generalizations of the famous Jensen's and Jensen-Mercer inequalities for twice differentiable functions, where convexity property of the target function is not assumed in advance. They represent a refinement of these inequalities in the case of convex/concave functions with numerous applications in Theory of Means and Probability and Statistics.

For maps of analytic functions f : D → D; f(0) = a ∈ D, we prove their Lipschitz continuity regarding the distance ratio metric with a Lipschitz constant C depending only on a. Question of the best possible C = C(a) remains open.

In this article we proved an interesting property of the class of continuous convex functions. This leads to the form of pre-Hermite-Hadamard inequality which in turn admits a generalization of the famous Hermite-Hadamard inequality. Some further discussion is also given.

We presented here a refinement of Hermite-Hadamard inequality as a linear combination of its end-points. The problem of best possible constants is closely connected with well known Simpson's rule in numerical integration. It is solved here for a wide class of convex functions, but not in general. Some supplementary results are also given.

We determine the form of the best possible global bounds for the Jensen functional on the real line. Thereby, previous results on this topic are essentially improved. Some applications in Analysis and Information Theory are also given.

For a discrete law F and large n , we investigate the asymptotic relation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$EX_n^\mu \ell (X_n ) \sim C_\mu (EX_n )^\mu \in (a,b),C_\mu > 0(EX_n \to \infty ),$$ \end{document}, where ℓ(·) is an arbitrary slowly varying function. By a result from [6], it follows that regularly varying moments for Power Series Distributions, generated by an entire function of finite order, satisfy the above relation with Cµ = 1 for each µ ∈ (0, ∞).

We give a positive solution to the conjectured moment inequality for probability distributions with support on R . A considerable improvement of the classical Jensen’s inequality follows.

In this article we found the form of best possible global upper bound for Jensen's inequality. Thereby, previous results on this topic are essentially improved. We also give some applications in Analysis and Information Theory.

We prove a global assertion on logarithmic convexity of Csiszar’s ƒ-divergence. It follows that the relative s-information measure is log-convex for s ∈ ℝ, wherefrom some new inequalities connecting Kullback-Leibler divergence and χ2 and Hellinger distances arise.

We establish new lower and upper bounds for Jensen’s discrete inequality. Applying those results in information theory, we obtain new and more precise bounds for Shannon’s entropy.

In this article we shall give some necessary and sufficient conditions for a discrete law F such that the asymptotic equivalence EX n μ � (EXn) � (EXn � �) takes place for all positive � . An extension to the class of regularly varying moments is also given.

For the Power Series Distributions generated by an arbitrary entire function of finite order, applying methods of Karamata's Theory of Regular Variation, we obtain asymptotic behavior of its moments. As an illustration, we calculate the moments of distributions generated by the class of Mittag-Leffler functions of which the well-known Poisson Law is just a special case.

We give some new inequalities concerning the differences of power means. .

We give another global upper bound for Jensen's discrete inequality which is better than already existing ones. For instance, we determine a new converses for generalized A-G and G-H inequalities.

In this paper we shall prove a very precise moments inequality which is a mixture of Jensen’s and Lyapunov’s inequalities. Some applications in Probability and Information Theory are also given.

For a discrete law F and large n, we investigate the asymptotic relation where l(·) is an arbitrary slowly varying function. Then we prove that regularly varying moments for Power Series Distributions, generated by an entire function of finite order, satisfy the above relation with C μ = 1 for each μ ∈ (0, ∞).

In this paper we shall give a global upper bound for Jensen's inequality without restrictions on the target convex function f. We also introduce a characteristic c(f) i.e. an absolute constant depending only on f, by which the global bound is improved.

We give a generalization of well-known Valiron-Titchmarsh theorem on entire functions with negative zeros. Namely, we prove that (-1)plogP(r)ЄER[p,p+1] where P(r) denotes the canonical product of an entire function with genus p and almost negative zeros and ER is the class of extended regular variation.

ON COMPLEX-VALUED MATRIX TRANSFORMS WITH SLOWLY VARYING SEQUENCES

We construct a non-standard class B of entire functions which are not written in Hadamard's canonical form. By the properties of this class we found extremals of an operator of second order.

For an arbitrary entire function g of finite order and with non-negative Taylor coefficients, we shall prove that its restriction to the positive part of real axis belongs to de Haan's class Γ . It follows that the laws F 1 ( x ) = 1 - 1 / g ( x ) and F 2 ( x ) = ∫ - ∞ x d t / g ( | t | ) / C are in the domain of attraction of the double exponential law with explicitly given norming constants.

We give some necessary and sufficient conditions for the class of Appell polynomials to satisfy well-known Turan's inequality. Among the other corollaries, we apply our results to some classes of orthogonal polynomials.

For the power series distribution, generated by an entire function of finite order, we obtain the asymptotic behavior of its regularly varying moments. Namely, we prove that EwX??(X)\sim(EwX)??(EwX), ? > 0 (w??), where ?(?) is an arbitrary slowly varying function.

In this article we construct a class of entire functions of arbitrary finite order and with non-zero Taylor coefficients, which is not written in the form of Hadamard's canonical product. Some basic properties of this class are also given.

We prove a theorem concerning asymptotic behavior of general complex-valued kernel convolutions with slowly varying functions in the sense of Karamata. In applications, we show that the content of some classical theorems can be naturally extended on some parts of complex z-plane.

We prove a theorem concerning asymptotic behavior of general complex-valued kernel convolutions with slowly varying functions in the sense of Karamata. In applications we showed that the content of some classical theorems can be naturally extended on some parts of complex z-plane.

The classes KR∞, MR∞, R∞ of rapidly varying functions are natural extensions of Karamata's concept of regular variation. In ref. 2 we introduced a new class K of perfect Karamata's kernels and its subclasses Θ and Σ. In this paper we study inclusion properties of these classes and, among other results, we prove KR∞ C MR∞ C Σ C Θ C K. .

We give a necessary and sufficient condition for a positive measurable kernel C(-) to satisfy r f(t)c(t)dt -/(x) r c(t)dt (x -oo) Ji Jj whenever Ji is from the class of Karamata's regularly varying functions.

We give a unified solution of problems posed by Karamata--Blanusa--Mitrinovic more than 50 years ago on the determination of algebraic functions with prescribed asymptotic and monotonicity properties. .

For a class of partially ordered means, we introduce a notion of the (nontrivial) cancelling mean. A simple method is given, which helps to determine cancelling means for the well-known classes of the Holder and Stolarsky means. MSC: Primary 39B22; 26D20

We give study the Lipschitz continuity of M\"obius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric., Comment: 10 pages

We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of M\"obius transformations of the unit ball in ${\mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained., Comment: 14 pages

For zero-balanced Gaussian hypergeometric functions $ F(a,b;a+b;x),$ $a,b>0,$ we determine maximal regions of $ab$ plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each $x\in (0,1)$. Thereby an exhausting answer is given to an open problem., 9 pages

Bernoulli type inequalities for functions of logarithmic type are given. These functions include, in particular, Gaussian hypergeometric functions in the zero-balanced case $F(a,b;a+b;x)\,.$, Comment: 15 pages

Since (x, y, z) is integral, the claim follows. (b) The formulas given for x, y, and z in terms of a, ,8, s, and t give valid points in N(a, b, c) also when s + t is even or gcd(s, t) > 1. (c) Let g(x) = (a/x)2 + (b/(c x))2. Since g is convex on (0, c), the equation g(x) = 1 has zero, one, or two solutions, depending on the minimum of g(x) over (0, c). Each solution uniquely determines a point (x, y, z) in S(a, b, c). Setting g'(x) = 0, the minimum occurs when x = ca2/3/(a2/3 + b2/3), where the value of g is ((a2/3 + b2/3)/C2/3)3. Thus IS(a, b, c)I e {2, 1, 0}, according to whether c2/3 is greater than, equal to, or less than a2/3 + b2/3. We conclude that N(a, b, c) has at most two points. Characterizing triples where N (a, b, c) has at most one point is therefore equivalent to characterizing triples where it has two. If x and k are the first coordinates of two such points, then there are integers s, t, s, and i such that