Fungrim entry: 4a2ac8

\left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma\!\left(x\right)\right|

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

References:

B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-3.

TeX:

\left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma\!\left(x\right)\right|

x \in \mathbb{R} \,\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
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("4a2ac8"),
    Formula(LessEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Abs(GammaFunction(x)))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR))),
    References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-3."))

Topics using this entry

Bounds and inequalities for the gamma function