On some inequalities characterizing the exponential function. (English) Zbl 0735.39006

Alsina, Claudi; García Roig, Jaime L.

Arch. Math., Brno 26, No. 2-3, 67-71 (1990).

Some inequalities relating the slope of a function and mean values are solved and characterizations of the exponential functions are obtained. In particular, the following two theorems are proved.
1. A function \(f:\mathbb{R}\to\mathbb{R}\) satisfies \((f(y)-f(x))/(y- x)\leq(f(x)+f(y))/2\) for all \(x,y\in\mathbb{R}\) such that \(x<y\) iff \(f\) has the form \(f(x)=g(x)e^ x\), \(x\in\mathbb{R}\), where \(g:\mathbb{R}\to\mathbb{R}\) is a non- increasing function.
2. Let \(M:[0,+\infty)\times[0,+\infty)\to[0,+\infty)\) be a continuous function such that \(M(x,x)=x\) for all \(x\geq0\). A function \(f:\mathbb{R}\to[0,+\infty)\) satisfies \(M(f(x),f(y))\leq(f(y)-f(x))/(y- x)\leq(f(x)+f(y))/2\) for all \(x,y\in\mathbb{R}\) such that \(x<y\) iff \(f(x)=ce^ x\), \(x\in\mathbb{R}\), where \(c\in[0,+\infty)\).

Reviewer: K.Baron (Katowice)

MSC:

39B72	Systems of functional equations and inequalities
26D07	Inequalities involving other types of functions

Keywords:

functional inequalities in several variables; characterizations of the exponential functions