Inverse problems on the parity of meromorphic functions

Abstract

By observing the periodicity of transcendental entire solutions of the complex differential equation f(z)f″(z)=p(z)sin2⁡z, where p(z) is a non-zero polynomial with real coefficients and real zeros, Yang's Conjecture has been proposed and considered by many authors recently. In this paper, we consider the parity of transcendental entire solutions of the equation above. Moreover, given a positive integer n, an integer k and a non-zero constant q, we consider the parity of a transcendental meromorphic function f under the assumption that either its differential polynomial f(z)nf(k)(z), or one of the q-difference polynomials f(z)nf(qz) or f(z)n+f(qz) is an even or an odd function.