Fungrim entry: 214b1c

$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k \in \left\{1, -1\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) < 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) \ge 0\right)\right)$
TeX:
\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k \in \left\{1, -1\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) < 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) \ge 0\right)\right)
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
LambertW$W_{k}\!\left(z\right)$ Lambert W-function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Im$\operatorname{Im}(z)$ Imaginary part
Source code for this entry:
Entry(ID("214b1c"),
Assumptions(And(Element(z, CC), Or(And(Element(k, Set(1, -1)), GreaterEqual(Re(z), 0)), And(Equal(k, -1), Less(Im(z), 0)), And(Equal(k, 1), GreaterEqual(Im(z), 0))))))