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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
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.
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.
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