Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$, $x\geq0$ 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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A subclass of log-concave functions satifying a sum inequalitySuppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$, $x\geq0$ and $\alpha,\beta>0$: $$ \sum\limits_{k+n=m}[f(x+\alpha+k)f(x+\beta+n)-f(x+k)f(x+\alpha+\beta+n)]\geq0. $$ 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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