Characterization of the Euler gamma function with the aid of an arbitrary mean

We prove that a continuous function $f:\left(0,\infty\right)\rightarrow\left(0,\infty\right)$ satisfying the functional equation $$ f\left(x+1\right)=xf\left(x\right),\qquad x>0,\quad f\left(1\right)=1, $$ is the Euler gamma function iff for some $a>0$ and a strict and continuous mean $M:\left(a,\infty\right)^{2}\rightarrow\left(a,\infty\right)$,the following inequality holds: $$ f\left(M\left(x,y\right)\right)f\left(\frac{xy}{M\left(x,y\right)}\right)\leq f\left(x\right)f\left(y\right),\qquad x,y\in\left(a,\infty\right). $$ Taking for $M$ the geometric mean $G\left(x,y\right)=\sqrt{xy}$, we obtain the result of [2] generalizing the classical Bohr—Mollerup theorem [1]. For $M=A$, where $A\left(x,y\right)=\frac{x+y}{2}$ is the arithmetic mean, the assumed inequality reduces to $f\left(A\left(x,y\right)\right)f\left(H\left(x,y\right)\right)\leq f\left(x\right)f\left(y\right)$ for all $x,y>a$, where $H$ is the harmonic mean, and the result gives a new characterization of the gamma function, involving the arithmetic and harmonic means.