Abstract
New results associated with Hermite-Hadamard inequalities for superquadratic functions are given. A set of Cauchy's type means is derived from these Hermite-Hadamard-type inequalities, and its log-convexity and monotonicity are proved.
Highlights
The following inequality: f ab 2 ≤ b1 −a b f a t dt ≤ f a f 2 b1.1 is holding for any convex function, that is, well known in the literature as the HermiteHadamard inequality see 1, page 137
In many areas of analysis applications of HermiteHadamard inequality appear for different classes of functions with and without weights; see for convex functions, for example, 2, 3
Some useful mappings are defined connected to this inequality see in 4–6
Summary
1.1 is holding for any convex function, that is, well known in the literature as the HermiteHadamard inequality see 1, page 137. In many areas of analysis applications of HermiteHadamard inequality appear for different classes of functions with and without weights; see for convex functions, for example, 2, 3. Some useful mappings are defined connected to this inequality see in 4–6. We focus on a class of functions which are superquadratic and analogs and refinements of 1.1 are applied to obtain results useful in analysis. Theorems, and results that we use in this paper. The following definition is given in 7
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