[Paper Review] Inequalities Among Logarithmic-Mean Measures

Inder J. Taneja|arXiv (Cornell University)|Mar 14, 2011

Mathematical Inequalities and Applications5 references3 citations

TL;DR

This paper establishes a comprehensive hierarchy of logarithmic-mean-related inequalities among classical means—harmonic, geometric, arithmetic, root-square, logarithmic, and newly defined means—using convexity analysis and functional inequalities. It proves that the logarithmic mean lies between the geometric and square-root means, and derives tight bounds for differences between these means, with key results showing that the difference between the root-square and logarithmic means is bounded by multiples of other mean differences, such as $ M_{SL}(a,b) \leq \frac{9}{10}M_{SL}(a,b) $ and $ M_{SL}(a,b) \leq \frac{5}{3}M_{AG}(a,b) $. The analysis relies on generating functions and second-order derivative conditions to verify convexity and inequality chains.

ABSTRACT

In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented

Motivation & Objective

• To establish a complete ordering of logarithmic-mean-related means, including harmonic, geometric, arithmetic, root-square, logarithmic, and newly defined means such as $ N_1, N_2, N_3 $.
• To analyze the convexity of mean difference functions such as $ M_{SL}(a,b) = S(a,b) - L(a,b) $, $ M_{AL}(a,b) = A(a,b) - L(a,b) $, and others, using generating functions and second-order derivatives.
• To derive quantitative bounds between differences of means, such as $ M_{SL}(a,b) \leq \frac{9}{10}M_{SN_1}(a,b) $, using functional inequalities and numerical verification.
• To clarify the relative ordering of means, particularly proving that $ L(a,b) \leq \frac{S(a,b)+9L(a,b)}{10} \leq \cdots \leq N_1(a,b) \leq \frac{5N_2(a,b)+L(a,b)}{6} $, and to show that certain intermediate expressions are not comparable.

Proposed method

• The paper uses a generating function approach via $ \phi_f(a,b) = a f(b/a) $, where $ f $ is a convex, differentiable function with $ f(1) = f'(1) = 0 $, to represent mean differences and analyze their convexity.
• It applies Lemma 1.1 to prove convexity of mean difference functions by verifying that their second-order derivatives are nonnegative.
• It employs Lemma 1.2, which compares two convex functions $ f_1 $ and $ f_2 $ via the ratio of their second derivatives, to derive inequalities between mean differences.
• The paper defines and analyzes new means: $ N_1(a,b) = \left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2 $, $ N_2(a,b) = \left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)\sqrt{\frac{a+b}{2}} $, and $ N_3(a,b) = \frac{a + \sqrt{ab} + b}{3} $, and integrates them into the mean ordering.
• It uses functional graphs and numerical evaluation of functions like $ f_{T_1}(x) $, $ f_{T_2}(x) $, and $ f_{T_3}(x) $ to verify non-negativity and validate inequality chains.
• It proves inequalities by constructing difference functions (e.g., $ T_1(a,b) = \frac{5}{3}M_{AG}(a,b) - M_{SL}(a,b) $) and analyzing their sign via $ a f(x) $-form representations.

Experimental results

Research questions

• RQ1 What is the complete ordering of logarithmic-mean-related means, including harmonic, geometric, arithmetic, root-square, logarithmic, and newly defined means $ N_1, N_2, N_3 $?
• RQ2 How can the differences between these means—such as $ S(a,b) - L(a,b) $, $ A(a,b) - L(a,b) $, and $ N_2(a,b) - L(a,b) $—be bounded using convexity and functional inequalities?
• RQ3 Are there tight multiplicative bounds between mean differences, such as $ M_{SL}(a,b) \leq \frac{9}{10}M_{SN_1}(a,b) $, and how are they proven?
• RQ4 Can the ordering between intermediate expressions like $ \frac{S(a,b)+5L(a,b)}{6} $ and $ \frac{2N_3(a,b)+3L(a,b)}{5} $ be determined, or are they incomparable?
• RQ5 What is the role of functional convexity and second-order derivative analysis in proving the convexity and non-negativity of mean difference functions?

Key findings

• The complete ordering $ H(a,b) \leq G(a,b) \leq L(a,b) \leq N_1(a,b) \leq N_3(a,b) \leq N_2(a,b) \leq A(a,b) \leq S(a,b) $ is established, with $ L(a,b) $ as the logarithmic mean.
• The difference $ M_{SL}(a,b) = S(a,b) - L(a,b) $ is convex in $ \mathbb{R}_+^2 $, and satisfies $ M_{SL}(a,b) \leq \frac{9}{10}M_{SN_1}(a,b) $, where $ M_{SN_1}(a,b) = S(a,b) - N_1(a,b) $.
• The inequality $ M_{SL}(a,b) \leq \frac{5}{3}M_{AG}(a,b) $ holds, where $ M_{AG}(a,b) = A(a,b) - G(a,b) $, proven via the non-negativity of $ T_1(a,b) = \frac{5}{3}M_{AG}(a,b) - M_{SL}(a,b) $.
• The inequality $ \frac{2N_3(a,b)+3L(a,b)}{5} \leq \frac{5A(a,b)+7L(a,b)}{12} $ is valid, as shown by the non-negative function $ f_{T_2}(x) $, confirming the chain $ \frac{S(a,b)+9L(a,b)}{10} \leq \frac{2N_3(a,b)+3L(a,b)}{5} \leq \cdots \leq N_1(a,b) $.
• The expressions $ \frac{S(a,b)+5L(a,b)}{6} $ and $ \frac{2N_3(a,b)+3L(a,b)}{5} $ are incomparable, as $ f_{T_3}(x) $ takes both positive and negative values, proving no universal ordering exists.
• The inequality $ N_2(a,b) \leq \frac{5N_2(a,b)+L(a,b)}{6} \leq N_3(a,b) $ holds, with the second inequality confirmed by the non-negative function $ f_{T_4}(x) $, validating the full chain of means.