Bound on Complex Values of Gamma Function

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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:

\(\ds \size {\map \Gamma {s + i t} }\) \(=\) \(\ds \lim_{M \mathop \to \infty} \size {\dfrac 1 {s + i t} \prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }\)
\(\ds \) \(=\) \(\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \size {\dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }\)
\(\ds \) \(=\) \(\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^{s + i t} } } {\size {1 + \frac {s + i t} n} }\)
\(\ds \) \(=\) \(\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i t} n} }\) Modulus of Exponential of Imaginary Number is One


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

\(\ds b^2\) \(\le\) \(\ds t^2\)
\(\ds \leadsto \ \ \) \(\ds \paren {\dfrac b n}^2\) \(\le\) \(\ds \paren {\dfrac t n}^2\)
\(\ds \leadsto \ \ \) \(\ds \paren {1 + \dfrac s n}^2 + \paren {\dfrac b n}^2\) \(\le\) \(\ds \paren {1 + \dfrac s n}^2 + \paren {\dfrac t n}^2\)
\(\ds \leadsto \ \ \) \(\ds \size {1 + \frac {s + i b} n}\) \(\le\) \(\ds \size {1 + \frac {s + i t} n}\) Definition of Complex Modulus
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 {\size {1 + \frac {s + i b} n} }\) \(\ge\) \(\ds \dfrac 1 {\size {1 + \frac {s + i t} n} }\)


Using this we obtain:

\(\ds \size {\map \Gamma {s + i t} }\) \(=\) \(\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i t} n} }\)
\(\ds \) \(\le\) \(\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i b} n} }\)
\(\ds \) \(=\) \(\ds \dfrac {\size {s + i b} } {\size {s + i t} } \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i b} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac{s + i b} n} }\)
\(\ds \) \(=\) \(\ds \dfrac {\size {s + i b} } {\size {s + i t} } \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i b} } \size {\prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i b} } {1 + \frac{s + i b} n} }\)
\(\ds \) \(=\) \(\ds \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }\)

$\blacksquare$

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