[Paper Review] Equivalent inequalities

P. S. Bullen|ArXiv.org|Sep 3, 2008

Mathematical Inequalities and Applications2 references4 citations

TL;DR

This paper establishes formal equivalences among fundamental classical inequalities—such as the AM-GM, Hölder, Minkowski, Bernoulli, and Liapunov inequalities—by demonstrating that they can be transformed into one another via variable substitutions, weight adjustments, and sign reversals. The key contribution is a systematic framework showing that these inequalities are not isolated results but different manifestations of the same underlying mathematical principle, with equality conditions preserved under transformation.

ABSTRACT

Equivalencies of many basic elementary inequalities are given

Motivation & Objective

• To unify seemingly disparate classical inequalities by proving their formal equivalence under specific transformations.
• To clarify the logical and structural relationships between well-known inequalities, especially in the discrete setting.
• To resolve ambiguities in inequality formulation by rigorously defining validity sets and equality conditions.
• To provide a foundational reference for researchers by compiling known and disguised equivalences of elementary inequalities.
• To emphasize that many inequalities are not independent but are equivalent forms of a single underlying principle, especially in weighted and generalized settings.

Proposed method

• Uses formal definitions of inequalities as triples (validity set, equality set, and inequality formula) to ensure logical consistency.
• Applies variable substitutions and weight transformations to show that inequalities like Hölder and Liapunov are equivalent under reparameterization.
• Demonstrates equivalence between reverse inequalities (e.g., ∼I) and original forms by sign reversal and function substitution.
• Relies on weighted power means and generalized means (M_n^{[r]}) to unify inequalities across different exponent values.
• Employs change-of-variables techniques to map one inequality (e.g., weighted Hölder) into another (e.g., Liapunov) via parameter redefinition.
• Uses the structure of weighted sums and products to show that inequalities like Radon’s and Hölder’s are equivalent under appropriate substitutions.

Experimental results

Research questions

• RQ1 Which classical inequalities are formally equivalent under transformation of variables and weights?
• RQ2 How can reverse inequalities (e.g., ∼I) be shown to be equivalent to their original forms?
• RQ3 To what extent can inequalities like Hölder, Minkowski, and Liapunov be derived from one another via substitution?
• RQ4 What role do equality conditions play in establishing formal equivalence between inequalities?
• RQ5 How do disguised forms of inequalities—such as those with complex variable substitutions—still preserve equivalence to simpler forms?

Key findings

• The weighted Hölder inequality is formally equivalent to the Liapunov inequality via the change of variables: $ p = \frac{r-t}{r-s} $, $ \mathbf{a} = \mathbf{x}^{t/p} $, $ \mathbf{b} = \mathbf{x}^{r/p'} $.
• The reverse inequality $ \sim \text{L}_{n;r,s,t} $ holds under the condition $ t < r < s $, $ r < s < t $, or $ s < t < r $, and is equivalent to the original via sign reversal.
• The equal-weighted form of the Hölder inequality is equivalent to its weighted version, with equality if and only if the sequence $ \mathbf{x} $ is constant.
• Radon’s inequality is shown to be a mere rephrasing of Theorem 3(a) in [3], highlighting how subtle notation changes obscure equivalence.
• The equivalence between inequalities is preserved under transformations that maintain the structure of weighted sums and products.
• The paper establishes that many inequalities are not independent but are equivalent forms of a single principle, especially when equality conditions are consistently maintained.