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On Hardy-type integral inequalities with the gamma function.

Jianquan Liao1, Bicheng Yang1

  • 1Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 51003 P.R. China.

Journal of Inequalities and Applications
|July 7, 2017
PubMed
Summary

This study derives equivalent conditions for Hardy-type integral inequalities using real analysis and weight functions. The best possible constant factors involving the gamma function are established for non-homogeneous kernels.

Keywords:
Hardy-type integral inequalityequivalent formnormoperatorweight function

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Area of Science:

  • Mathematical Analysis
  • Inequalities Theory

Background:

  • Hardy-type integral inequalities are fundamental in analysis.
  • Investigating inequalities with non-homogeneous kernels presents unique challenges.

Purpose of the Study:

  • To establish equivalent conditions for two types of Hardy-type integral inequalities.
  • To determine the best possible constant factors for these inequalities.
  • To explore operator expressions and homogeneous kernel cases.

Main Methods:

  • Application of real analysis techniques.
  • Utilization of weight functions.
  • Analysis of non-homogeneous and homogeneous kernels.

Main Results:

  • Derivation of equivalent conditions for Hardy-type integral inequalities.
  • Proof that constant factors involving the gamma function are optimal.
  • Consideration of operator forms and specific kernel types.

Conclusions:

  • The established conditions provide a deeper understanding of Hardy-type inequalities.
  • The optimality of the gamma function-related constants is confirmed.
  • The study extends existing results to non-homogeneous kernels and operator contexts.