Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function

Kottakkaran Sooppy Nisar; Feng Qi; Gauhar Rahman; Shahid Mubeen; Muhammad Arshad

doi:10.1186/s13660-018-1717-8

Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function

J Inequal Appl. 2018;2018(1):135. doi: 10.1186/s13660-018-1717-8. Epub 2018 Jun 19.

Authors

Kottakkaran Sooppy Nisar¹, Feng Qi^{2

3

4}, Gauhar Rahman⁵, Shahid Mubeen⁶, Muhammad Arshad⁵

Affiliations

¹ 1Department of Mathematics, College of Arts and Science at Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Riyadh Region, Kingdom of Saudi Arabia.
² 2Institute of Mathematics, Henan Polytechnic University, Jiaozuo, China.
³ 3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, China.
⁴ 4Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, China.
⁵ 5Department of Mathematics, International Islamic University, Islamabad, Pakistan.
⁶ 6Department of Mathematics, University of Sargodha, Sargodha, Pakistan.

Abstract

In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev's inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function.

Keywords: Confluent hypergeometric k-function; Extended gamma function; Inequality; Logarithmic convexity.