Beta function inequality

Question

I need to find values of the positive real parameters $a,b$ when $$B(a,b)<1,$$ where $B$ is beta function.

Is $a,b>1$ solution of the above inequality?

Answer 1

$B(a,b)$ is a convex function with respect to both its parameters (see page 47 and beyond here).
Both $\frac{\partial }{\partial a}B(a,b)$ and $\frac{\partial }{\partial b}B(a,b)$ at $(a,b)=(1,1)$ are negative (equal to $\psi(1)-\psi(2)=-1$) and $B(1,1)=1$. Similarly, these partial derivatives are negative at any $(a,b)\in\{(x,y):x,y>1\}=E$, hence $B(a,b)<1$ holds over $E$ for sure.

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$$\frac{d}{da}B(a,b) = B(a,b)\left[\psi(a)-\psi(a+b)\right] = B(a,b)\sum_{n\geq 0}\left(\frac{1}{n+a+b}-\frac{1}{n+a}\right)<0. $$