Let $x,y,a,b \geq 1$. I have the feeling that the following inequality is true: $$B(x+a,y+b)=\frac{\Gamma(x+a)\Gamma (y+b)}{\Gamma(x+y+a+b)}\leq \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=B(x,y),$$ where $\Gamma$ denotes the Gamma function and $B$ the Beta function. Nevertheless, I can not find the appropriate properties of these functions that help me conclude this.
Is there any smart idea to see this? Thanks a lot! :)
One may use a classic integral representation of the Euler beta function: $$ \begin{align} B(x+a,y+b)&=\int_0^1 t^{x+a-1}(1-t)^{y+b-1}dt \\\\&\le\int_0^1 t^{x-1}(1-t)^{y-1}dt, \qquad (a\ge0,b\ge0), \\\\&=B(x,y). \end{align} $$