Mathematics and Statistics Vol. 8(3), pp. 334 - 338
DOI: 10.13189/ms.2020.080312
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Hermite-Hadamard Type Inequalities for Composite Log-Convex Functions

Nik Muhammad Farhan Hakim Nik Badrul Alam ^1,*, Ajab Bai Akbarally ², Silvestru Sever Dragomir ^3
1 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Pahang, Malaysia
² Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Shah Alam, Malaysia
³ Mathematics, College of Engineering and Science, Victoria University, Australia

ABSTRACT

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, (a,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-^-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-^-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.

KEYWORDS
Convex Functions, Hermite-Hadamard Inequalities, Composite Log-Convex Functions, GG-Convex

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Nik Muhammad Farhan Hakim Nik Badrul Alam , Ajab Bai Akbarally , Silvestru Sever Dragomir , "Hermite-Hadamard Type Inequalities for Composite Log-Convex Functions," Mathematics and Statistics, Vol. 8, No. 3, pp. 334 - 338, 2020. DOI: 10.13189/ms.2020.080312.

(b). APA Format:
Nik Muhammad Farhan Hakim Nik Badrul Alam , Ajab Bai Akbarally , Silvestru Sever Dragomir (2020). Hermite-Hadamard Type Inequalities for Composite Log-Convex Functions. Mathematics and Statistics, 8(3), 334 - 338. DOI: 10.13189/ms.2020.080312.