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The Best Constant For Inequality Involving Sum Of The Reciprocals And Product Of Positive Numbers With Unit Sum

Yagub N. Aliyev

Abstract

In the paper we study a special parameter containing algebraic inequality involving sum of reciprocals and product of positive real numbers whose sum is 1. We determine the best values of the parameter using a new optimization argument. In the case of three numbers the algebraic inequality have some interesting geometric applications involving a generalization of Euler's inequality about the ratio of radii of circumscribed and inscribed circles of a triangle.

The Best Constant For Inequality Involving Sum Of The Reciprocals And Product Of Positive Numbers With Unit Sum

Abstract

In the paper we study a special parameter containing algebraic inequality involving sum of reciprocals and product of positive real numbers whose sum is 1. We determine the best values of the parameter using a new optimization argument. In the case of three numbers the algebraic inequality have some interesting geometric applications involving a generalization of Euler's inequality about the ratio of radii of circumscribed and inscribed circles of a triangle.
Paper Structure (3 sections, 70 equations, 2 figures)

This paper contains 3 sections, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Appendix

Figures (2)

  • Figure 1: Geometric application of case $n=3$.
  • Figure 2: Parametric space curve (blue and green) representing intersection of plane $x+y+z=\beta$ (not shown) and surface $xyz=\alpha$ (not shown).