Generalizations of the Jensen functional involving diamond integrals via Abel–Gontscharoff interpolation

In this paper we obtain several refinements of the Jensen inequality on time scales by generalizing Jensen’s functional for n-convex functions. We also investigate the bounds for the identities related to the new improvements obtained.

Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

Global Bounds for the Generalized Jensen Functional with Applications

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

Stability of Bi-Additive Mappings and Bi-Jensen Mappings

Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation f(x+y,z+w)=f(x,z)+f(y,w) and the bi-Jensen functional equation 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w).

Non-Archimedean hyperstability of Cauchy–Jensen functional equations on a restricted domain

Let X be a normed space, {U\subset X\setminus\{0\}} a non-empty subset, and {(G,+)} a commutative group equipped with a complete ultrametric d that is invariant (i.e., {d(x+z,y+z)=d(x,y} ) for {x,y,z\in G} ). Under some weak natural assumptions on U and on the function {\gamma\colon U^{3}\to[0,\infty)} , we study the new generalized hyperstability results when {f\colon U\to G} satisfies the inequality d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)% \biggr{)}\leq\gamma(x,y,z) for all {x,y,z\in U} , where {\frac{x+y}{\alpha}+z\in U} and {\alpha\geq 2} is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.