Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets

Abstract

Convex analysis is a discipline of mathematics dedicated to the explication of the properties of convex sets and convex functions. Convex functions are extremely useful in proving many famous inequalities in mathematics. It is widely known that inequalities defined by convex functions originated with the works done by Holder and Jensen among others and applied to modeling a variety of problems both in hard and soft sciences. It is said that the arsenal of an analyst is heavily stocked with inequalities. Studies related to convexity have kept occupying a central position in almost all areas of mathematics, especially in functional analysis and operations research. Following the emergence of fuzzy set theory, fuzzy convexity alongside a number of related concepts has been explicated. However, not much has been done regarding fuzzy inequalities defined by fuzzy convexity. In this paper, a novel attempt to study arithmetic mean-geometric mean is proposed. In this short note, we provide two proofs of the arithmetic mean-geometric mean inequality for convex fuzzy sets. Mathematics Subject Classification 2020: 03E72, and 94D05.