Abstract
© 2020 by authors, all rights reserved. Hermite-Hadamard type inequalities related to convex functions are widely being studied in functional analysis. Researchers have refined the convex functions as quasi-convex, h-convex, log-convex, m-convex, (α,m)-convex and many more. Subsequently, the Hermite-Hadamard type inequalities have been obtained for these refined convex functions. In this paper, we firstly review the Hermite-Hadamard type inequality for both convex functions and log-convex functions. Then, the definition of composite convex function and the Hermite-Hadamard type inequalities for composite convex functions are also reviewed. Motivated by these works, we then make some refinement to obtain the definition of composite log-convex functions, namely composite-ϕ−1 log-convex function. Some examples related to this definition such as GG-convexity and HG-convexity are given. We also define k-composite log-convexity and k-composite-ϕ−1 log-convexity. We then prove a lemma and obtain some Hermite-Hadamard type inequalities for composite log-convex functions. Two corollaries are also proved using the theorem obtained; the first one by applying the exponential function and the second one by applying the properties of k-composite log-convexity. Also, an application for GG-convex functions is given. In this application, we compare the inequalities obtained from this paper with the inequalities obtained in the previous studies. The inequalities can be applied in calculating geometric means in statistics and other fields.
Highlights
IntroductionDragomir and Mond [1] refined the Hermite-Hadamard inequality in (1) for log-convex functions
Let be a convex function on J with c, d ∈ J such that c < d
The definition of composite convex function was given by Dragomir [2]
Summary
Dragomir and Mond [1] refined the Hermite-Hadamard inequality in (1) for log-convex functions. Is known as an inequality of Hermite-Hadamard type for log-convex functions. The definition of composite convex function was given by Dragomir [2]. Dragomir [2] introduced the concept of k-composite convexity by assuming that ψ : [c, d ] → I , where I is a real number interval and is continuous and strictly increasing on I. Dragomir [3] refined the concept of composite convex function and obtained the definition of composite h-convex function. The Hermite-Hadamard inequalities for composite h-convex function were obtained. Kashuri et al [4] defined composite preinvex functions and k-composite preinvex functions They obtained some Ostrowski inequalities for these functions
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