THE PERMUTATION METHOD AND ITS APPLICATION TO THE PROOF OF INEQUALITIES

Abstract

Inequality is a fundamental concept in mathematics, with many theoretical facts and established statements (well-known inequalities) having broad applications across various branches of the field. One such application is in solving problems related to finding the range of values of a function, where inequalities play a crucial role in determining the function's maximum or minimum values and its limitations. Geometric problems that require finding the maximum or minimum value of a certain characteristic (linear size, area, volume, etc.) of a figure with variable elements can also be effectively addressed using inequalities. The Cauchy inequality, which establishes the relation of arithmetic and geometric means, is one such example. This article focuses on a widely applicable inequality that can be referred to as a "permutation inequality". The article presents proofs for both special and general cases of this inequality and demonstrates its usefulness through solving numerous problems, including well-known inequalities such as the Cauchy and Chebyshev's inequality, as well as the inequality expressing the relation of arithmetic and quadratic means. The application of permutation inequalities in proving numerous inequalities highlights its importance as a method of proving inequalities. Overll, the use of permutation inequalities offers an effective approach to solving problems related to inequalities, making it a valuable tool for mathematicians.At the end of the article, some inequalities are suggested to be proved independently by the specified method.