Improvement and generalization of some results related to the class of harmonically convex functions and applications

Volume 22, Issue 3, pp 282--294 http://dx.doi.org/10.22436/jmcs.022.03.07

Publication Date: August 31, 2020 Submission Date: March 31, 2020 Revision Date: May 07, 2020 Accteptance Date: June 21, 2020

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Authors

Imran Abbas Baloch - Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan. - Higher Education Department, Govt. College Gulberg, Punjab, Pakistan. Aqeel Ahmad Mughal - Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan. Yu Ming Chu - Department of Mathematics, Huzhou University, Huzhou, China. Absar Ul Haq - Department of Basic Sciences and Humanities, University of Engineering and Technology (Narowal Campus), Lahore 54000, Pakistan. Manuel De La Sen - Institute of Research and Development of Processors, University of the Basque Country, campus of Leioa (Bizkaia), 48940 Leioa, Spain.

Abstract

Discrete Jensen-type inequality for a harmonically convex function was established by Dragomir in [S. S. Dragomir, RGMIA Monographs, Victoria University, (2015)]. In [I. A. Baloch, A. H. Mughal, Y. M. Chu, M. De la Sen, Accepted in Aims Mathematics], Baloch et al. presented a variant of discrete Jensen-type inequality for harmonically convex functions. Moreover, they established a Jensen-type inequality for harmonically \(h\)-convex functions, and then they proved the variant of Jensen-type inequality for harmonically \(h\)-convex functions. Our results generalize and improve some earlier results in the literature (for example see [S. S. Dragomir, RGMIA Monographs, Victoria University, (2015)] and [I. A. Baloch, A. H. Mughal, Y. M. Chu, M. De la Sen, Accepted in Aims Mathematics]) for the said class. Additionally, using them gives us more interesting results.

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Keywords

• Harmonically convex functions
• harmonically h-convex functions
• Jensen-type inequality
• a variant of Jensen-type inequality
• harmonic series
• HM-GM-AM inequality
• a variant of HM-GM-AM

MSC

•  26D15
•  26A51
•  26D10
•  26A15

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