A Functional Inequality Arising in Combinatorics

Abstract

In this paper, we discuss the functional inequality $p(n + m){\kern 1pt} \le {\kern 1pt} (_{n}^{{n + m}}){\kern 1pt} p(n){\kern 1pt} p(m)$, which arises in tournament theory and other parts of combinatorics. A simple transformation removes the binomial coefficient, and then the solution set divides naturally into three classes of functions. One class consists of all the nonpositive functions since this inequality puts no restriction on such functions. The counting-function solutions, i.e., the nonnegative solutions, all lie in the other two classes and satisfy easily obtainable exponential growth bounds. This set of solutions also possesses a structure in the sense that various combinations of these solutions, e.g., sums and products, are again in the set. Various solution functions and properties of solutions are obtained by introducing a slack function to convert the functional inequality to a functional equation. The general solution to this functional equation is obtained by transforming it to another functional equation whose general solution is known. Solution functions found in this manner occur in pairs and are sometimes even from different solution classes. This slack-function concept has modifications, so it can be applied in other ways to the functional inequality and to other inequalities.