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On rational bounds for the gamma function.

Zhen-Hang Yang1,2, Wei-Mao Qian3, Yu-Ming Chu1

  • 1Department of Mathematics, Huzhou University, Huzhou, 313000 China.

Journal of Inequalities and Applications
|September 29, 2017
PubMed
Summary
This summary is machine-generated.

This study establishes new bounds for the gamma function, providing the best possible constants for inequalities involving this fundamental mathematical function and Euler-Mascheroni constant.

Keywords:
completely monotonic functiongamma functionpsi functionrational bound

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Area of Science:

  • Mathematical Analysis
  • Special Functions

Background:

  • The gamma function is a crucial tool in various scientific fields.
  • Establishing precise inequalities for the gamma function is essential for theoretical advancements.

Purpose of the Study:

  • To derive and verify a double inequality involving the gamma function.
  • To determine the sharpest constants for specific inequalities related to the gamma function.
  • To identify a critical value within a given interval that satisfies particular conditions for the gamma function.

Main Methods:

  • Utilizing analytical techniques to establish inequalities.
  • Employing calculus and properties of the gamma function.
  • Performing rigorous mathematical derivations to find optimal constants.

Main Results:

  • The double inequality [Formula: see text] is proven to hold for all [Formula: see text].
  • The best possible constants [Formula: see text] and [Formula: see text] are identified for the inequality [Formula: see text] for all [Formula: see text].
  • A specific value of [Formula: see text] is determined in the interval [Formula: see text] where the behavior of the inequality changes.

Conclusions:

  • The established inequalities provide tighter bounds for the gamma function.
  • The identified constants are optimal, enhancing the precision of mathematical analysis.
  • The findings contribute to a deeper understanding of the gamma function's properties and applications.