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Several sharp inequalities about the first Seiffert mean.

Boyong Long1, Ling Xu1, Qihan Wang1

  • 1School of Mathematical Sciences, Anhui University, Hefei, China.

Journal of Inequalities and Applications
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PubMed
Summary
This summary is machine-generated.

This study establishes optimal bounds for the first Seiffert mean. It explores relationships with geometric combinations of logarithmic, Neuman-Sándor, and second Seiffert means.

Keywords:
Logarithmic meanNeuman–Sándor meanSeiffert mean

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Area of Science:

  • Mathematical Analysis
  • Inequalities
  • Mean Values

Background:

  • Seiffert means are important in mathematical inequalities.
  • Understanding bounds for these means is crucial for further research.
  • Logarithmic, Neuman-Sándor, and Seiffert means have interconnected properties.

Purpose of the Study:

  • To determine the sharpest possible bounds for the first Seiffert mean.
  • To investigate the relationship between the first Seiffert mean and specific geometric combinations of other means.
  • To contribute to the theory of inequalities involving generalized means.

Main Methods:

  • Analytical methods to derive inequalities.
  • Comparison of different mean values.
  • Establishing sharp bounds through theoretical proofs.

Main Results:

  • The paper provides precise inequalities bounding the first Seiffert mean.
  • New relationships are established between the first Seiffert mean and geometric combinations involving logarithmic, Neuman-Sándor, and second Seiffert means.
  • The derived bounds are demonstrated to be the best possible.

Conclusions:

  • The study successfully determined the optimal bounds for the first Seiffert mean.
  • The findings offer a deeper understanding of the interplay between various mathematical means.
  • This research advances the field of inequalities and mean value theory.