On a conjecture of W.E. Clark and M.E.H. Ismail

Abstract

W.E. Clark and M.E.H. Ismail conjectured the inequality

$$\displaylines{\left[\frac{x^{n}}{1-e^{-x}}\right] ^{(n)}>0,\qquad 0<x<\infty, n=1,\ldots} $$

(1)

We prove the inequality for values of x in the interval (2ln 2,∞). The proof involves writing the derivatives as an infinite series of Laguerre polynomials.