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Inequalities of extended beta and extended hypergeometric functions
1Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Hasa, Hofuf, 31982 Saudi Arabia.
Journal of Inequalities and Applications
|January 24, 2017
Summary
This study explores the log-convexity of extended beta functions, leading to new Turán-type inequalities. These findings also reveal properties of extended hypergeometric functions, including their monotonicity and concavity.
Area of Science:
- Mathematical Analysis
- Special Functions
Background:
- Extended beta and hypergeometric functions are crucial in various mathematical and physical applications.
- Understanding their properties like convexity and monotonicity is essential for further research.
Purpose of the Study:
- To investigate the log-convexity of extended beta functions.
- To establish Turán-type inequalities for these functions.
- To deduce properties of extended hypergeometric functions using these inequalities.
Main Methods:
- Analysis of the log-convexity of extended beta functions.
- Derivation of Turán-type inequalities.
- Application of inequalities to establish monotonicity, log-convexity, and log-concavity of extended hypergeometric functions.
Main Results:
- Established the log-convexity of extended beta functions.
- Derived new Turán-type inequalities.
- Deduced monotonicity, log-convexity, and log-concavity for extended hypergeometric functions, including specific cases for confluent and Gaussian hypergeometric functions.
- Obtained reverses of Turán-type inequalities.
Conclusions:
- The log-convexity of extended beta functions provides a powerful tool for analyzing extended hypergeometric functions.
- The established inequalities offer new insights into the behavior of these special functions.
- The results extend existing knowledge and open avenues for further mathematical investigations.
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