Fungrim entry: 72712c

\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}

Assumptions:

\left(k = 0 \;\mathbin{\operatorname{and}}\; \left|z\right| \ge 1\right) \;\mathbin{\operatorname{or}}\; \left(k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| \ge 4 \left(\left|k\right| + 1\right)\right)

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}

\left(k = 0 \;\mathbin{\operatorname{and}}\; \left|z\right| \ge 1\right) \;\mathbin{\operatorname{or}}\; \left(k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| \ge 4 \left(\left|k\right| + 1\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("72712c"),
    Formula(LessEqual(Abs(LambertW(z, k, 1)), Div(1, Abs(z)))),
    Variables(k, z),
    Assumptions(Or(And(Equal(k, 0), GreaterEqual(Abs(z), 1)), And(Element(k, ZZ), Element(z, CC), GreaterEqual(Abs(z), Mul(4, Add(Abs(k), 1)))))))

Topics using this entry

Lambert W-function