Related papers: Generalization of Subadditive, Monotone and Convex…

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…

Let $f\colon \mathbb{R}_+\to\mathbb{R}$ be a continuous and strictly monotone function. In the main result of this paper, we show that, for a fixed $n\geq 2$, the $n$-variable mean $\mathscr{A}_f \colon \mathbb{R}_+^n \to \mathbb{R}_+$…

Let $D$ be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function $f: D\to \mathbf{R}\cup \{+\infty\}$ is convex if and only if for all $x,y \in D$ there exists $\alpha=\alpha(x,y) \in (0,1)$ such…

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$ f(x)\leq f(y)+\Phi(y-x), $$ where $\Phi:[0,\ell(I)[\,\to\mathbb{R}_+$ is a given nonnegative…

Given an integer $a\ge 1$, a function $f: \mathbb{R}\to \mathbb{R}$ is said to be $a$-subadditive if $$ f(ax+y) \le af(x)+f(y) \,\,\,\text{ for all }x,y \in \mathbb{R}. $$ Of course, $1$-subadditive functions (which correspond to ordinary…

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-convex if, for all $x,y\in I$, $t\in[0,1]$ it satisfies $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+t\Phi\big((1-t)|x-y|\big)+(1-t)\Phi\big(t|x-y|\big), $$ where…

A sequence $\Big(u_n\Big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_n\leq u_{n-1}+u_{n+1}\qquad \mbox{for all}\qquad n\in\mathbb{N}. $$ We present several characterizations of convex sequences and…

In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone…

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, we then…

In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to $p$-subharmonicity, subsolutions…

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…

We introduce and study a new class of generalized convex functions termed star quasiconvex functions. This class includes convex, star-convex, quasiconvex, quasar-convex, and positively homogeneous functions of any degree $p>0$ as special…

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing while…

A subequation on an open subset $X\subset \mathbb R^n$ is a subset $F$ of the space of $2$-jets on $X$ with certain properties. A smooth function is said to be $F$-subharmonic if all of its $2$-jets lie in $F$, and using the viscosity…

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…

For a subshift $(X, \sigma_X)$ and a subadditive sequence $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on $X$, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim_{n\rightarrow\infty}(1/{n})\int \log f_n d \mu=\int…

In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators $A$ and $B$, we prove that…

A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in…

Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition…

Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…