Upper bound for the complex Beta function

Question

Is there any work or reference regarding upper bounds for the complex beta function defined by

\begin{equation} B(x,y)=\frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}, \end{equation} for $\Re{x} >0$ and $\Re{y}>0$.

Answer 1

The command of Mathematica

NMaximize[{ComplexExpand[Abs[Gamma[s + I*t]*Gamma[a + I*b]/Gamma[s + a + I*(t + b)]]],s > 0 &&a > 0},{s, t, a, b}]

outputs "NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded." and

{4.7857*10^8, {s -> 0.32798, t -> 0.720557, a -> 0., b -> -2.08956*10^-9}}

The command of Mathematica

s = 1; t = a; MaxLimit[ ComplexExpand[Abs[Gamma[s + I*t]*Gamma[a + I*b]/Gamma[s + a + I*(t + b)]]], {t,b} -> {0, 0},Direction -> "FromAbove"]

results in $\infty$, confiming it.