Abstract
The intention of this note is to investigate some new important estimates for the Jensen gap while utilizing a 4-convex function. We use the Jensen inequality and definition of convex function in order to achieve the required estimates for the Jensen gap. We acquire new improvements of the Hölder and Hermite–Hadamard inequalities with the help of the main results. We discuss some interesting relations for quasi-arithmetic and power means as consequences of main results. At last, we give the applications of our main inequalities in the information theory. The approach and techniques used in the present note may simulate more research in this field.
Highlights
The theory of convex functions performs an extremely significant and consequential role in several areas of pure and applied sciences
Let us begin this section with the following theorem, in which we acquire an upper bound for the Jensen gap
We proposed a novel technique of obtaining of some significant estimates for Jensen’s gap while utilizing a 4-convex function
Summary
The theory of convex functions performs an extremely significant and consequential role in several areas of pure and applied sciences. There are several well-known inequalities which are the direct consequences and applications of convexity [4, 5]. In this respect, some of the noted inequalities associated with the class of convex functions are majorization, Hermite–Hadamard and Jensen–Mercer inequalities [6]. Some of the noted inequalities associated with the class of convex functions are majorization, Hermite–Hadamard and Jensen–Mercer inequalities [6] Among these inequalities, one of the considerable and vital inequalities which are studied very widely in the literature is the Jensen inequality. For some more extensive literature concerning to the Jensen inequality, see [15, 16]
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