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It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity…
Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequality for convex functions known as Sherman's inequality. We extend Sherman's result to the class of n-strongly convex functions using…
During the study of the topic of limit summability of functions (introduced by the author in 2001), we encountered some types of functions that are related to the mean value theorem. In this paper, we formally define mean value and…
The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent…
We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of…
We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric function with respect simultaneous shift of all its parameters. For a particular case of Heine's…
In this paper we extend some notions, previously defined for log-concave functions, to the larger domain of so-called {\alpha}-concave functions. We begin with a detailed discussion of support functions - first for log-concave functions,…
In this paper, we state some characterizations of $h$-convex function is defined on a convex set in a linear space. By doing so, we extend the Jensen-Mercer inequality for $h$-convex function. We will also define $h$-convex function for…
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and we point out the results for some special…
Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the…
We study a family of inequalities on pairs of measure spaces involving functions defined on product domains. Our main result establishes a Jensen-type inequality under a general product-measure framework, extending classical inequalities…
In this paper we prove some monotonicity, log--convexity and log--concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Tur\'an…
In this paper, two new classes of convex functions as a generalization of convexity which is called (h-s)_{1,2}-convex functions are given. We also prove some Hadamard-type inequalities and applications to the special means are given.
In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by…
We characterize the symmetric measures which satisfy the one dimensional convex Poincar\'e inequality. For these measures the tenesorization argument yields concentration inequalities for their products and convex sets in R^n.
This note aims to present some reverse inequalities about the power means and Karcher mean via the Kantorovich constant and some of these have been generalized to higher power. Also, we generalize the reverse weighted arithmetic-geometric…
Famous Redheffer's inequality is generalized to a class of anti-periodic functions. We apply the novel inequality to the generalized trigonometric functions and establish several Redheffer-type inequalities for these functions.
In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash…
We prove that the inequalities $\sin_{p,q}(\sqrt{rs})\geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ and $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ hold for all $p,q\in(1,\infty)$,…
The aim of this paper is to present some new Fejer-type results for convex functions. Improvements of Young's inequality (the arithmetic-geometric mean inequality) and other applications to special means are pointed as well.