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

Kottakkaran Sooppy Nisar1, Feng Qi2,3,4, Gauhar Rahman5

  • 11Department of Mathematics, College of Arts and Science at Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Riyadh Region, Kingdom of Saudi Arabia.

Journal of Inequalities and Applications
|August 24, 2018
PubMed
Summary

This study introduces new inequalities for the extended gamma function and Kummer confluent hypergeometric k-function. It provides a novel proof for the log-convexity of the extended gamma function using Hölder inequality.

Keywords:
Confluent hypergeometric k-functionExtended gamma functionInequalityLogarithmic convexity

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Area of Science:

  • Mathematical Analysis
  • Special Functions

Background:

  • The extended gamma function and Kummer confluent hypergeometric k-function are crucial in various mathematical and scientific domains.
  • Classical inequalities provide foundational tools for establishing new mathematical properties.

Purpose of the Study:

  • To derive novel inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function.
  • To present a new proof for the log-convexity of the extended gamma function.
  • To introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function.

Main Methods:

  • Application of classical inequalities, including Chebychev's inequality for synchronous and asynchronous mappings.
  • Utilization of the Hölder inequality to establish log-convexity.
  • Development of a new Turán type mean inequality.

Main Results:

  • New inequalities involving the extended gamma function and Kummer confluent hypergeometric k-function have been established.
  • A new proof demonstrating the log-convexity of the extended gamma function is presented.
  • A Turán type mean inequality for the Kummer confluent k-hypergeometric function has been introduced.

Conclusions:

  • The findings extend the understanding of inequalities related to special functions.
  • The new proof of log-convexity offers an alternative approach using fundamental inequalities.
  • The introduced Turán type mean inequality contributes to the theory of hypergeometric functions.