Editing Lambert W function (section)

== Bounds and inequalities ==
A number of non-asymptotic bounds are known for the Lambert function.

Hoorfar and Hassani<ref>A. Hoorfar, M. Hassani, [https://www.emis.de/journals/JIPAM/article983.html?sid=983 Inequalities on the Lambert ''W'' Function and Hyperpower Function], JIPAM, Theorem 2.7, page 7, volume 9, issue 2, article 51. 2008.</ref> showed that the following bound holds for {{math|''x'' ≥ ''e''}}:
: <math>\ln x -\ln \ln x + \frac{\ln \ln x}{2\ln x} \le W_0(x) \le \ln x - \ln\ln x + \frac{e}{e - 1} \frac{\ln \ln x}{\ln x}.</math>

They also showed the general bound
: <math>W_0(x) \le \ln\left(\frac{x+y}{1+\ln(y)}\right),</math>
for every <math>y>1/e</math> and <math>x\ge-1/e</math>, with equality only for <math>x = y \ln(y)</math>.
The bound allows many other bounds to be made, such as taking <math>y=x+1</math> which gives the bound
: <math>W_0(x) \le \ln\left(\frac{2x+1}{1+\ln(x+1)}\right).</math>

In 2013 it was proven<ref name = "Chatzigeorgiou">
{{cite journal
 | last1 = Chatzigeorgiou| first1 = I.
 | title = Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation
 | journal = IEEE Communications Letters
 | volume = 17
 | issue = 8
 | pages = 1505–1508
 | year = 2013
 | arxiv = 1601.04895| doi = 10.1109/LCOMM.2013.070113.130972 | s2cid = 10062685
}}</ref> that the branch {{math|''W''<sub>−1</sub>}} can be bounded as follows:
: <math>-1 - \sqrt{2u} - u < W_{-1}\left(-e^{-u-1}\right) < -1 - \sqrt{2u} - \tfrac{2}{3}u \quad \text{for } u > 0.</math>
Roberto Iacono and John P. Boyd<ref name="doi.org">{{Cite journal |last1=Iacono |first1=Roberto |last2=Boyd |first2=John P. |date=2017-12-01 |title=New approximations to the principal real-valued branch of the Lambert W-function |url=https://doi.org/10.1007/s10444-017-9530-3 |journal=Advances in Computational Mathematics |language=en |volume=43 |issue=6 |pages=1403–1436 |doi=10.1007/s10444-017-9530-3 |s2cid=254184098 |issn=1572-9044}}</ref> enhanced the bounds as follows: 
: <math>\ln \left(\frac{x}{\ln x}\right) -\frac{\ln \left(\frac{x}{\ln x}\right)}{1+\ln \left(\frac{x}{\ln x}\right)} \ln \left(1-\frac{\ln \ln x}{\ln x}\right) \le W_0(x) \le \ln \left(\frac{x}{\ln x}\right) - \ln \left(\left(1-\frac{\ln \ln x}{\ln x}\right)\left(1-\frac{\ln\left(1-\frac{\ln \ln x}{\ln x}\right)}{1+\ln \left(\frac{x}{\ln x}\right)}\right)\right).</math>