Bound on Complex Values of Gamma Function

Theorem
Let $\Gamma \left({z}\right)$ denote the Gamma function.

Then for any complex number $z = s + i t$, we have for $\left\lvert{b}\right\rvert \le \left\lvert{t}\right\rvert$:


 * $\left\lvert{\Gamma \left({s + i t}\right)}\right\rvert \le \dfrac {\left\lvert{s + i b}\right\rvert} {\left\lvert{s + i t}\right\rvert} \left\lvert{\Gamma \left({s + i b}\right)}\right\rvert$

Proof
From the Euler Form of the Gamma Function:

Because $\left\lvert{b}\right\rvert \le \left\lvert{t}\right\rvert$, we have that:

Using this we obtain: