By using the decision theorem and properties of the Schur-convex function, the Schur-geometric convex function and the Schur-harmonic function, the Schur- convexity, Schur-geometric convexity and ...Schur-harmonic convexity of a class of complete symmetric functions are studied. As applications, some symmetric function inequalities are established.
In the above article <xref ref-type="bibr" rid="ref1">1 , the funding mentioned was correct, but the article was missing an Acknowledgement section, which is required for funding. The correct ...Acknowledgement is listed below.
Some characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces LM(μ,X) endowed with the Orlicz norm being ...I‐convex as well as being Q‐convex.
Abstract
This article is about transient information regarding the functions that satisfy the conditions of monotonicity and convexity on a set, onset of matrices of particular patterns with finite ...or infinite order called operator monotone or operator convex functions. Data is gathered by investigating numerous existing articles, review papers, publications that show wide scattering in multiple dimensions according to the behavior and prolific results. This work emphasizes not only the origin, key results, and applications of convexity and monotonicity but also their extensions in different directions. It also takes us to the current research scenario in relating fields and may provide an opportunity to express some new ideas, methods, or concepts.
The present paper deals with h-convex functions introduced by S. Varošanec in 21. In our main result we give a characterization of h-convex functions satisfying the condition h(α)+h(1−α)=1, α∈0,1.
We derive C^sup 2^-characterizations for convex, strictly convex, as well as strongly convex functions on full dimensional convex sets. In the cases of convex and strongly convex functions this ...weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the case of strictly convex functions we weaken the well-known sufficient C^sup 2^-condition for strict convexity to a characterization. Several examples illustrate the results. PUBLICATION ABSTRACT
On w-Isbell-convexity Olela Otafudu, Olivier; Sebogodi, Katlego
Applied general topology,
04/2022, Volume:
23, Issue:
1
Journal Article
Peer reviewed
Open access
Chistyakov introduced and developed a concept of modular metric for an arbitrary set in order to generalise the classical notion of modular on a linear space. In this article, we introduce the theory ...of hyperconvexity in the setting of modular pseudometric that is herein called w-Isbell-convexity. We show that on a modular set, w-Isbell-convexity is equivalent to hyperconvexity whenever the modular pseudometric is continuous from the right on the set of positive numbers.
In the present paper we introduce a notion of (ω,t)-convexity as a natural generalization of the notion of usual t-convexity, t-strongly convexity, approximate t-convexity, delta t-convexity and many ...other. The main result of this paper establishes the necessary and sufficient conditions under which an (ω,t)-convex map can be supported at a given point by an (ω,t)-affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of (ω,t)-convexity are given.
In this paper, we prove the correct q-Hermite–Hadamard inequality, some new q-Hermite–Hadamard inequalities, and generalized q-Hermite–Hadamard inequality. By using the left hand part of the correct ...q-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some q-midpoint type integral inequalities through q-differentiable convex and q-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.
Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized ...subtractively by its cumulant or free energy function, or equivalently normalized divisively by its partition function. Both the cumulant and partition functions are strictly convex and smooth functions inducing corresponding pairs of Bregman and Jensen divergences. It is well known that skewed Bhattacharyya distances between the probability densities of an exponential family amount to skewed Jensen divergences induced by the cumulant function between their corresponding natural parameters, and that in limit cases the sided Kullback-Leibler divergences amount to reverse-sided Bregman divergences. In this work, we first show that the α-divergences between non-normalized densities of an exponential family amount to scaled α-skewed Jensen divergences induced by the partition function. We then show how comparative convexity with respect to a pair of quasi-arithmetical means allows both convex functions and their arguments to be deformed, thereby defining dually flat spaces with corresponding divergences when ordinary convexity is preserved.
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