[Paper Review] Bivariate log-convexity of the more extended means and its applications

Zhen-Hang Yang|arXiv (Cornell University)|Aug 10, 2014

Mathematical Inequalities and Applications23 references3 citations

TL;DR

This paper investigates the bivariate log-convexity of two-parameter homogeneous functions and applies it to derive log-convexity and log-concavity properties of extended means, including Stolarsky, Gini, two-parameter identric, and Heronian means. The key contribution is proving that these means are bivariate log-concave on $[0, \∞)^2$ and log-convex on $(-∞, 0]^2$ with respect to parameters, leading to new and classical inequalities for means.

ABSTRACT

In this paper, the bivariate log-convexity of the two-parameter homogeneous function in parameter pair is vestigated. From this the bivariate log-convexity of the more extended means with respect to a parameter pair is solved. It follows that Stolarsky means, Gini means, two-parameter identric (exponential) means and two-parameter Heronian means are all bivariate log-concave on $\mathbb{[}0\mathbb{,\infty )}^{2}$ and log-convex on $% \mathbb{(-\infty ,}0\mathbb{]}^{2}$ with respect to parameters. Lastly, some classical and new inequalities for means are given.

Motivation & Objective

• To investigate the bivariate log-convexity of two-parameter homogeneous functions in the parameter pair.
• To establish log-convexity and log-concavity properties of more extended means (e.g., Stolarsky, Gini, identric, Heronian) with respect to their parameters.
• To derive new and classical inequalities for means using the log-convexity framework.
• To unify and generalize known mean inequalities through the four-parameter homogeneous mean structure.
• To provide a systematic method for estimating ratios of means via convexity-based bounds.

Proposed method

• Defines the four-parameter homogeneous mean $\boldsymbol{F}(p,q;r,s;a,b)$ as a generalization of Stolarsky and Gini means using logarithmic and identric means.
• Establishes bivariate log-convexity of the two-parameter homogeneous function $\mathcal{H}_f(p,q;a,b)$ via analysis of the function $L(p,q)$, the logarithmic mean of $p$ and $q$.
• Applies the log-convexity result to derive inequalities involving weighted geometric means of means.
• Uses the identity $\mathcal{H}_D(p,q;a,b) = e^{1/L(p,q)} S_{p,q}(a,b)$ to relate Stolarsky means to the homogeneous function $\mathcal{H}_D$.
• Derives double inequalities for ratios of means by combining log-convexity with exponential and logarithmic transformations.
• Applies the results to generate new inequalities by substituting specific parameter values into the general bounds.

Experimental results

Research questions

• RQ1 Under what conditions is the two-parameter homogeneous function $\mathcal{H}_f(p,q;a,b)$ bivariate log-convex in $(p,q)$?
• RQ2 How do the log-convexity properties of $\mathcal{H}_f$ extend to specific families of means such as Stolarsky, Gini, identric, and Heronian means?
• RQ3 What inequalities can be derived from the bivariate log-convexity of these means with respect to their parameters?
• RQ4 How do the derived inequalities compare to known classical inequalities like Stolarsky-Yang and Sándor-Yang?
• RQ5 Can new inequalities be systematically generated from the general framework of parameter-wise convexity?

Key findings

• Stolarsky, Gini, two-parameter identric, and two-parameter Heronian means are bivariate log-concave on $[0, \infty)^2$ and log-convex on $(-\infty, 0]^2$ with respect to their parameters.
• The inequality $1 \leq \frac{S_{\alpha p_1 + \beta p_2, \alpha q_1 + \beta q_2}(a,b)}{S_{p_1,q_1}^\alpha(a,b) S_{p_2,q_2}^\beta(a,b)} \leq \exp\left(\frac{\alpha}{L(p_1,q_1)} + \frac{\beta}{L(p_2,q_2)} - \frac{1}{L(\alpha p_1 + \beta p_2, \alpha q_1 + \beta q_2)}\right)$ holds for all $a,b > 0$ and $\alpha, \beta > 0$ with $\alpha + \beta = 1$.
• A similar double inequality holds for Gini means: $1 \leq \frac{G_{\alpha p_1 + \beta p_2, \alpha q_1 + \beta q_2}(a,b)}{G_{p_1,q_1}^\alpha(a,b) G_{p_2,q_2}^\beta(a,b)} \leq \exp\left(\frac{\alpha}{L(p_1,q_1)} + \frac{\beta}{L(p_2,q_2)} - \frac{1}{L(\alpha p_1 + \beta p_2, \alpha q_1 + \beta q_2)}\right)$.
• Application 4.3 yields the Stolarsky-Yang inequality: $1 \leq I/A_{2/3} \leq \sqrt{8}e^{-1} \approx 1.0405$.
• Application 4.4 produces new inequalities, such as $16\sqrt{2}e^{-1}/9 \approx 0.9249 \leq I/A_{2/3}^3 He^{-2} \leq 1$.
• Another new result is $1 \leq I/\sqrt{I_{6/5} I_{4/5}} \leq e^{1/24} \approx 1.0425$, and similarly for the power-exponential mean $Z$.