A Two-Function Minimax Theorem

Abstract

Let X, Y be two non-empty sets and f, g two real functions on X × Y. In this paper, we consider the problem of guaranteeing the validity of the inequality

$$ \mathop {\inf }\limits_{x \in X} \mathop {sup}\limits_{y \in Y} f(x,y) \leqslant \mathop {sup}\limits_{y \in Y} \mathop {\inf }\limits_{x \in X} g(x,y) $$

.