The Extremal Values of the Ratio of Differences of Means

I am very pleased to share that my recent paper, “The Extremal Values of the Ratio of Differences of Means,” has just been published in Mediterranean Journal of Mathematics (Vol. 22, Article 163, 2025). In this work, I examine how the ratio

\frac{A_n - G_n}{P_\alpha - G_n}​​

behaves, where A_n​ is the arithmetic mean of n non-negative variables, G_n​ is their geometric mean, and P_\alpha​ is the power (or generalized) mean of order α. The paper determines the maximal and minimal possible values of this ratio for all admissible parameters and variables.

1. Background: Understanding the Family of Means

Means are among the most fundamental constructions in mathematics. They provide a concise way to represent or summarize a set of numbers, and the relationships between different means reveal deep structural information about inequalities, symmetry, and variation.

The arithmetic mean is familiar to everyone:

A_n = \frac{x_1 + x_2 + \cdots + x_n}{n}.

The geometric mean is multiplicative in nature:

G_n = (x_1 x_2 \cdots x_n)^{1/n}.

And the power mean of order α unifies these and many others:

P_\alpha = \Bigl(\frac{x_1^\alpha + x_2^\alpha + \cdots + x_n^\alpha}{n}\Bigr)^{1/\alpha}.

Depending on α, this mean reduces to well-known special cases:

• α=1: arithmetic mean

• α=0: geometric mean (in the limit)

• α=−1: harmonic mean

The classical inequality G_n≤A_n​ is known to every student, but the differences and ratios of means provide much finer information. Studying how far apart these means can be leads to sharper inequalities and to a better quantitative understanding of how the data’s spread or asymmetry affects averaging processes.

2. The Central Question

The main problem addressed in the paper is the determination of the extremal (largest and smallest) values of

\frac{A_n - G_n}{P_\alpha - G_n}.

This ratio compares two “gaps”:

• the gap between the arithmetic and geometric means (A_n - G_n), and

• the gap between the power mean of order α and the geometric mean (P_\alpha - G_n​).

By forming their ratio, we can ask a new kind of question: given the step from G_n​ to P_\alpha​, how large or small can the further step to A_n​ be? This measure provides a refined way of comparing the growth of different means relative to each other.

While inequalities between means have been widely studied, results involving the ratio of their differences are quite rare. Moreover, the dependence of this ratio on both n and α makes the problem substantially more delicate than the standard pairwise comparisons.

3. Methodology and Key Results

The main methodological contribution of the paper is a reduction of the full n-variable optimization problem to a single-variable one. Through a careful analysis of the symmetry and homogeneity of the expression, it is shown that the extremal configuration occurs when the variables take on only two distinct values—some equal to one number and the rest equal to another. This simplification allows the use of classical calculus and inequality tools to find the precise extremal points.

With this reduction in hand, the paper derives exact expressions for the upper and lower bounds of the ratio, determining when it lies within the interval

\left(\frac{n}{n-1}\right)^{\frac{1}{\alpha}-1} \quad \text{and} \quad n^{\frac{1}{\alpha}-1}.

It also identifies cases where the ratio exceeds or falls below this range, depending on the parameters n and α.

The results include:

• explicit formulas for the maximum and minimum values of the ratio for each pair (n,α);

• a complete description of the configurations of variables achieving these extremal values;

• and a study of how these constants behave as n increases, including their monotonicity and asymptotic limits.

In this way, the paper provides a unified and comprehensive picture of the behaviour of (A_n - G_n)/(P_\alpha - G_n) across the entire family of power means.

4. Why These Results Matter

Although the problem is purely theoretical, it connects naturally to many areas of mathematical analysis and applied science. Some of the key reasons this work may be of interest include:

1. Sharp inequalities.
In many mathematical and applied problems, having exact constants is crucial. For example, in probability and statistics, inequalities between different means describe how expectations behave under transformations; in numerical analysis, they can estimate errors in approximations; in information theory, they appear in the study of divergences and entropies. Knowing the sharp constants allows these applications to achieve optimal precision.

2. Unified framework.
The results hold simultaneously for all n and all real α, covering arithmetic, geometric, harmonic, quadratic, and many other intermediate means within a single treatment. This unification clarifies how these different means relate to one another quantitatively.

3. Methodological contribution.
The reduction from an n-variable extremal problem to a single-variable one may be useful in other inequality studies. Such symmetry-based reductions often reveal deeper structural principles that extend beyond the specific case studied here.

4. Asymptotic behaviour.
Understanding how the best constants depend on n helps connect finite inequalities to their limiting forms as n tends to infinity. This bridges discrete inequalities and continuous analogues, which often arise in functional and integral inequalities.

5. Conceptual significance.
The ratio studied can be viewed as a “sensitivity measure” describing how rapidly the sequence of power means moves from the geometric to the arithmetic mean. In this sense, it contributes to the broader theory of means as a continuous scale of averaging processes.

5. Broader Context

Research on means and inequalities has a long and rich history, from the classical results of Cauchy, Jensen, and Hölder to modern refinements involving power, Lehmer, or Gini means. The interplay between these families of means often uncovers elegant constants, extremal structures, and monotonicity properties. My paper adds one more piece to this landscape by characterizing precisely how the difference between arithmetic and geometric means compares with that between power and geometric means.

The techniques developed here may be adaptable to other related questions—such as ratios involving harmonic or logarithmic means, or to inequalities between weighted means, which play an important role in applied probability and optimization theory. In addition, the asymptotic analysis may inform the study of sequences of inequalities when the number of variables grows large, a setting that naturally appears in modern data analysis.

6. Conclusion and Outlook

To summarise, the paper investigates the extremal values of the ratio ((A_n - G_n)/(P_\alpha - G_n), finds the exact upper and lower bounds for every combination of n and α, identifies the configurations achieving these bounds, and analyses their monotonic and asymptotic behaviour. The work thus provides sharp constants and a unified framework for comparing differences of classical means.

I hope that these findings will be of interest to colleagues working on inequalities, analysis, and optimization. I also hope that the reduction technique employed here may inspire similar approaches in related extremal-value problems.

I would like to thank the ADA University Faculty Research and Development Fund for supporting this research. I look forward to continuing discussions with readers and researchers interested in the theory of means and its applications.

Published in the Mediterranean Journal of Mathematics (2025), Volume 22, Article 163.
🔗 Read the full paper here

Aliyev, Y.N. The Extremal Values of the Ratio of Differences of Means. Mediterr. J. Math. 22, 163 (2025). https://doi.org/10.1007/s00009-025-02937-9

I also gave a talk in AAM Seminars about the topic:

https://youtu.be/OHH2rkQEbuI?si=aG5IRpkgxJT0Q3iA