Non-real Zeros of Derivatives of a Class of Real Entire Functions

In 1943, G. Polya conjectured that the number of non-real zeros the kth derivative of a real entire function of order greater than 2, with finitely many non-real zeros, tends to infinity as k goes to infinity. This was verified in 2005 by A. Eremenko and W. Bergweiler. A natural extension is whether the number of non-real zeros of the kth derivative increases monotonically as k goes to infinity. We show that the number of non-real zeros of the kth derivative of a function f increases monotonically with differentiation when f(z)=z^me^{K(z)} where m is a natural number and K is in one of several special classes of real polynomials