Related papers: Estimates for approximately Jensen convex function…

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper…

In this paper, we are dealing with the solution of the functional equation $$ \varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown functions $\varphi,f$ and $F$ defined on a same open subinterval of the reals.…

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class $\Lambda_{f,g}(a, b)$ of mean values where $f, g$ are continuously differentiable convex functions satisfying the relation…

Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$…

In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In…

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$,…

Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost…

We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra. Reverses and refinements of Jensen's and Slater's type…

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.

New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex.…

In this paper, we are going to describe the solutions of the functional equation $$ \varphi\Big(\frac{x+y}{2}\Big)(f(x)+f(y))=\varphi(x)f(x)+\varphi(y)f(y) $$ concerning the unknown functions $\varphi$ and $f$ defined on an open interval.…

In this work, we introduce the class of $h$-${\rm{MN}}$-convex functions by generalizing the concept of ${\rm{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty\right)$ such…

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open…

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…

Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $\kappa\in \mathbb R$, in the sense of Alexandrov. Let $\mu$ be a Borel probability measure on $M$, such that $\mu\in\mathcal P_2(M)$, and that has at…

In this note we prove Jensen-type inequality for certain non-convex functions. We apply our idea to prove some inequalities which were suggested at some high-level math olympiades.

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…

Classically, Jensen's Inequality asserts that if $X$ is a compact convex set, and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure $\mu$ on $K$, that $f(\text{bar}(\mu))\le \int f\;d\mu$, where $\text{bar}(\mu)$ is…

We give a Jensen operator inequality for strongly convex functions. As a corollary, we improve operator Holder-McCarthy inequality under suitable conditions.

In this paper, we obtain some new inequalities for ({\alpha},m)-convex functions. The analysis used in the proofs is fairly elementary and based on the use of Power-mean inequality.