[Paper Review] Refinement of Inequalities among Means

Inder J. Taneja|ArXiv.org|May 10, 2005

Mathematical Inequalities and Applications4 references21 citations

TL;DR

This paper refines classical inequalities among classical means—arithmetic, geometric, harmonic, root-square, and newly introduced means—by introducing and analyzing intermediate means such as the square-root mean $N_1$, Heron's mean $N_3$, and a novel mean $N_2$. Using convexity properties and differential analysis of mean differences, the authors derive tight upper bounds for differences between means, establishing new, sharper inequalities that refine the standard chain $H \leq G \leq N_1 \leq A \leq S$, with quantitative constants like $\frac{3}{4}$ and $\frac{8}{9}$ in the refined bounds.

ABSTRACT

In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, root-square means, etc. Some new means recently studied are also presented. Different kinds of refinement of inequalities among these means are given.

Motivation & Objective

• To refine the classical chain of inequalities among classical means: harmonic (H), geometric (G), arithmetic (A), root-square (S), and newly defined means.
• To introduce and analyze three new intermediate means: $N_1$ (square-root mean), $N_2$, and $N_3$ (Heron's mean), which refine the ordering between G and A.
• To establish sharp upper bounds for the differences between means using convexity and ratio analysis of second derivatives.
• To provide multiple refinements of the standard mean inequality chain, including new inequalities involving weighted combinations of means.

Proposed method

• Introduces a family of power means $B_t(a,b)$ of order $t$, with special cases including H, G, A, S, and $N_1$, and proves their monotonicity in $t$.
• Defines new means: $N_1 = \frac{A+G}{2}$, $N_2 = \sqrt{N_1 A}$, and $N_3 = \frac{2A + G}{3}$, and expresses them in terms of known means.
• Analyzes the convexity of mean difference functions $M_{XY}(a,b) = X(a,b) - Y(a,b)$ using a lemma on convex functions $f$ with $f(1) = f'(1) = 0$, applied to $\phi_f(a,b) = a f(b/a)$.
• Uses ratio analysis of second derivatives $\frac{f''_{XY}(x)}{f''_{ZW}(x)}$ to determine the supremum of ratios, which yields sharp bounds on mean differences.
• Applies the derived supremum constants (e.g., $\frac{3}{4}$, $\frac{8}{9}$) to bound differences such as $M_{SA} \leq \frac{3}{4} M_{SN_3}$ and $M_{SN_3} \leq \frac{8}{9} M_{SN_1}$.
• Derives new inequalities by combining these bounds, such as $M_{SH} \leq 2 M_{SN_1}$ and $M_{SN_1} \leq \frac{3}{4} M_{SG}$, leading to refined chains of inequalities.

Experimental results

Research questions

• RQ1 How can the classical inequality chain $H \leq G \leq A \leq S$ be refined by introducing new intermediate means between G and A?
• RQ2 What are the sharp upper bounds for the differences between means such as $S - A$, $S - N_1$, and $S - N_3$, and how are they derived?
• RQ3 Can the convexity of mean difference functions be systematically exploited to derive tighter inequalities among means?
• RQ4 What role do the new means $N_1$, $N_2$, and $N_3$ play in refining the ordering and bounding differences between classical means?
• RQ5 How do the constants derived from ratio analysis of second derivatives (e.g., $\frac{3}{4}$, $\frac{8}{9}$) lead to new, quantitative refinements of mean inequalities?

Key findings

• The inequality chain is refined to $H \leq G \leq N_1 \leq N_3 \leq N_2 \leq A \leq S$, with $N_1 = \frac{A+G}{2}$, $N_2 = \sqrt{N_1 A}$, and $N_3 = \frac{2A + G}{3}$, providing tighter intermediate bounds.
• The difference $M_{SH}(a,b) = S(a,b) - H(a,b)$ is bounded above by $2 M_{SN_1}(a,b)$, i.e., $S - H \leq 2(S - N_1)$, with equality at $a = b$.
• The difference $M_{SN_1}(a,b) = S(a,b) - N_1(a,b)$ is bounded by $\frac{3}{4} M_{SG}(a,b)$, i.e., $S - N_1 \leq \frac{3}{4}(S - G)$, with the supremum of the ratio of second derivatives being $\frac{3}{4}$.
• The difference $M_{SN_3}(a,b) = S(a,b) - N_3(a,b)$ is bounded by $\frac{8}{9} M_{SN_1}(a,b)$, i.e., $S - N_3 \leq \frac{8}{9}(S - N_1)$, with the supremum of the ratio of second derivatives being $\frac{8}{9}$.
• New refined inequalities are derived, such as $S - A \leq \frac{3}{4}(S - N_3)$ and $S - N_3 \leq \frac{8}{9}(S - N_1)$, which together imply $S - A \leq \frac{2}{3}(S - N_1)$.
• The inequalities in Corollary 3.4, such as $G \leq \frac{S + 3G}{4} \leq N_1 \leq \frac{S + 8N_1}{9} \leq N_3 \leq \cdots \leq A$, represent a new class of refinements with explicit constants.