Fungrim entry: e0b322

\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma\!\left(x\right)}{\sqrt{\cosh\!\left(\pi y\right)}}

Assumptions:

x \in \left[\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

References:

B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4.

TeX:

\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma\!\left(x\right)}{\sqrt{\cosh\!\left(\pi y\right)}}

x \in \left[\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("e0b322"),
    Formula(GreaterEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Div(GammaFunction(x), Sqrt(Cosh(Mul(ConstPi, y)))))),
    Variables(x, y),
    Assumptions(And(Element(x, ClosedOpenInterval(Div(1, 2), Infinity)), Element(y, RR))),
    References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4."))

Topics using this entry

Bounds and inequalities for the gamma function