Stieltjes and other integral representations for functions of Lambert W

where lnk z = ln z+2πik, and ln z is the principal branch of natural logarithm [14]. This paper considers only the principal branch k = 0, which is the branch that maps the positive real axis onto itself, and therefore we shall usually abbreviate W0 as W herein. Many functions of W are members of a number of function classes, specifically, the classes of Stieltjes functions, Pick functions and Bernstein functions, including subclasses Thorin-Bernstein functions and complete Bernstein functions. This is mainly due to the fact that W is a real symmetric function, in the terminology of [5, p. 160] (see also [23, p. 155]), with positive values on the positive real line. The mentioned classes are of particular interest because they admit certain integral representations. A description of the classes can be found in a review paper [8] and a recently published book [19]. In this paper we show that many functions containing W are Stieltjes functions. Also, we extend the properties of the set of Stieltjes functions in Sections 1.2 and 4. In addition, we give one more proof of the fact [15] that W function is Bernstein. Moreover, we show that W is a complete Bernstein function.