# A subclass of log-concave functions satifying a sum inequality

Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$ and real $x\geq0$, $\alpha,\beta>0$: $$\sum\limits_{k+n=m}[f(x+\alpha+k)f(x+\beta+n)-f(x+k)f(x+\alpha+\beta+n)]\geq0.$$ Is anything known about this class of functions? If $m=0$ we have $$f(x+\alpha)f(x+\beta)-f(x)f(x+\alpha+\beta)\geq0$$ which is equivalent to log-concavity of $f$. Similar question about functions satisfying $$\sum\limits_{k+n=m}\binom{m}{k}[f(x+\alpha+k)f(x+\beta+n)-f(x+k)f(x+\alpha+\beta+n)]\geq0.$$ Is there a reasonably simple sufficient condition on $f$ so that it satisfiesthe above inequalities?

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