Related papers: A convexity-type functional inequality with infini…

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

Error bounds are central objects in optimization theory and its applications. They were for a long time restricted only to the theory before becoming over the course of time a field of itself. This paper is devoted to the study of error…

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other…

In this work, several sharp bounds for the \v{C}eby\v{s}ev functional involving various type of functions are proved. In particular, for the \v{C}eby\v{s}ev functional of two absolutely continuous functions whose first derivatives are both…

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision…

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of…

The condition number of a differentiable convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex…

In this paper we improve results related to Normalized Jensen Functional for convex functions and Uniformly Convex Functions.

We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems,…

In this paper, strongly $(\alpha,T)$-convex functions, i.e., functions $f:D\to \R$ satisfying the functional inequality $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t\alpha\big((1-t)(x-y)\big)-(1-t)\alpha\big(t(y-x)\big)$$ for $x,y\in D$ and $t\in…

John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish…

We prove conditions for the existence of a continuous linear right inverse for a surjective convolution operator in spaces of germs of analytic functions on convex subsets of the complex plane. Considered convex sets have a countable…

Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:C\longrightarrow [m,M]$ is a bounded convex function. We then rewrite the…

By considering a fixed point in unit disk $\Delta$, a new class of univalent convex functions is defined. Coefficient inequalities, integral operator and extreme points of this class are obtained.

We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers…

In this paper, we prove some new inequalities of Simpson's type for functions whose derivatives of absolute values are h-convex and h-concave functions. Some new estimations are obtained. Also we give some sophisticated results for some…

This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex…

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body…

Let $D$ be a domain in a finite-dimensional Euclidean space, and $H$ be a convex subcone in the convex cone of all subharmonic functions on $D$. We obtain a criterion for the existence of a lower envelope from $H$ for an arbitrary function…