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Some Hermite-Hadamard type inequalities for harmonically s-convex functions.

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Parameterized hilbert-type integral inequalities in the whole plane.

Qiliang Huang1, Shanhe Wu2, Bicheng Yang1

  • 1Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China.

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Researchers derived new Hilbert-type integral inequalities using real analysis. These inequalities feature nonhomogeneous kernels and multiparameters, with best possible constant factors.

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Area of Science:

  • Real Analysis
  • Mathematical Inequalities

Background:

  • Hilbert-type integral inequalities are fundamental in analysis.
  • Existing inequalities often have limitations regarding kernel types and parameters.

Purpose of the Study:

  • To establish novel Hilbert-type integral inequalities in the whole plane.
  • To analyze inequalities with nonhomogeneous kernels and multiple parameters.
  • To determine the optimality of constant factors.

Main Methods:

  • Application of real analysis techniques.
  • Estimation of weight functions.
  • Derivation of integral inequalities.

Main Results:

  • New Hilbert-type integral inequalities in the whole plane were established.
  • Constant factors involving hypergeometric and beta functions were proven to be the best possible.
  • Equivalent forms, reverses, and special cases with homogeneous kernels were considered.

Conclusions:

  • The study provides significant advancements in the theory of integral inequalities.
  • The derived inequalities offer powerful tools for various mathematical applications.
  • The best possible nature of the constant factors enhances the utility of these inequalities.