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

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
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