Bound on Complex Values of Gamma Function

Theorem
Let $\map \Gamma z$ denote the Gamma function.

Then for any complex number $z = s + i t$, we have for $\size b \le \size t$:


 * $\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$

Proof
From the Euler Form of the Gamma Function:

Because $\size b \le \size t$, we have that:

Using this we obtain: