q -Hermite–Hadamard Inequalities for Generalized Exponentially s , m ; η -Preinvex Functions

In this article, we introduce a new extension of classical convexity which is called generalized exponentially s , m ; η -preinvex functions. Also, it is seen that the new definition of generalized exponentially s , m ; η -preinvex functions describes different new classes as special cases. To prove our main results, we derive a new q m κ 2 -integral identity for the twice q m κ 2 -differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the q m κ 2 -integral by utilizing differentiable exponentially s , m ; η -preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

Starshaped sets

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines.

Some Remarks on Convex Network Flows forK-Spiders

We consider convex functions modeling flows in an arborescent network. The notion of majorization on trees is used to study some classical convexity results in this framework.

GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.

A generalization of convex functions via support properties

With respect to a given family of functions F, a function is said to be F-convex, if it is supported, at each point, by some member of F. For particular choices of F one obtains the convex functions and the generalized convex functions in the sense of Beckenbach. F-convex functions are characterized and studied, retaining some essential results of classical convexity.