This note shows that level sets of a function characterize an M-convex function introduced by Murota.

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

Let + := {(x1, . . . , xN ) ; xi 0, i = 1, 2, . . . , N} and + := + . Assume that f : + → + is monotone which means that it is monotone with respect to each variable. We denote f ↓, when f is decreasing (= nonincreasing) and f ↑ when f is increasing (= nondecreasing). Throughout this paper ω, u, v are positive measurable functions defined on + , N 1. A function P on [0,∞) is called a modular fu...

A class of discrete convex functions that can efficiently be minimized has been considered by Murota. Among them are L\-convex functions, which are natural extensions of submodular set functions. We first consider the problem of minimizing an L\-convex function with a linear inequality constraint having a positive normal vector. We propose a polynomial algorithm to solve it based on a binary se...

In 1957, Chandler Davis proved that unitarily invariant convex functions on the space of hermitian matrices are precisely those which are convex and symmetrically invariant on the set of diagonal matrices. We give a simple perturbation theoretic proof of this result. (Davis’ argument was also very short, though based on completely different ideas). Consider an orthogonally invariant function f ...

In this paper, we introduce a definition of geometric-arithmetically \((s,m)\) convex function and give some new inequalities Hermite-Hadamard type for the function. Finally, discuss applications these to special means.

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope f of a given function f : Rn×m → R, i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington. Mathematics Subject Classification. 65K10, 74G15, 74G65, 74N99. Received: May 27...

let $omega_x$ be a bounded, circular and strictly convex domain of a banach space $x$ and $mathcal{h}(omega_x)$ denote the space of all holomorphic functions defined on $omega_x$. the growth space $mathcal{a}^omega(omega_x)$ is the space of all $finmathcal{h}(omega_x)$ for which $$|f(x)|leqslant c omega(r_{omega_x}(x)),quad xin omega_x,$$ for some constant $c>0$, whenever $r_{omega_x}$ is the m...

Abstract Under the new concept of s - $(\alpha,m)$ ( α , m ) -convex functions, we obtain some Hermite–Hadamard inequalities with an function. We use these to estimate Riemann–Liouville fractional integrals second-order differentiable convex ...