On integral inequalities with applications to the logarithmic mean

Type: Article
Publication Date: 2025-04-30
Citations: 0
DOI: https://doi.org/10.4171/em/557

Abstract

This short note aims to introduce and derive a sequence of integral inequalities based on the well-established Radon inequality. In particular, it includes a generalization of the Chebyshev and Dunkel integral inequalities. As a special case, these inequalities readily yield the well-known Arithmetic-Logarithmic-Geometric Mean Inequality.

Locations

  • Elemente der Mathematik
This article deals with two fundamental topics in mathematical analysis: the formulation of integral expressions and the derivation of integral inequalities. In particular, it introduces new one-parameter integral formulas and 
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An upper bound of the logarithmic mean is given by a convex conbination of the arithmetic mean and the geometric mean. In addition, a lower bound of the logarithmic mean 
 An upper bound of the logarithmic mean is given by a convex conbination of the arithmetic mean and the geometric mean. In addition, a lower bound of the logarithmic mean is given by a geometric bridge of the arithmetic mean and the geometric mean. In this paper, we study the bounds of the logarithmic mean. We give operator inequalities and norm inequalities for the fundamental inequalities on the logarithmic mean. We give monotonicity of the parameter for the unitarily invariant norm of the Heron mean, and give its optimality as the upper bound of the unitarily invariant norm of the logarithmic mean. We study the ordering of the unitarily invariant norms for the Heron mean, the Heinz mean, the binomial mean and the Lehmer mean. Finally, we give a new mean inequality chain as an application of the point-wise inequality.
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An upper bound of the logarithmic mean is given by a convex combination of the arithmetic mean and the geometric mean.In addition, a lower bound of the logarithmic mean is 
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"The Geometric, Logarithmic, and Arithmetic Mean Inequality." The American Mathematical Monthly, 94(6), pp. 527–528 "The Geometric, Logarithmic, and Arithmetic Mean Inequality." The American Mathematical Monthly, 94(6), pp. 527–528