On a Problem of Hellerstein, Shen and Williamson

Abstract

Abstract. Suppose that / is a nonentire transcendental meromorphic function, real on the real axis, such that / and /' have only real zeros and poles, and /' omits a nonzero value. Confirming a conjecture of Hellerstein, Shen and Williamson, it is shown that then/is essentially/(z) = tanz -Bz -C for suitable values of B and C.

Iteration and the Zeros of the Second Derivative of a Meromorphic Function

“…It has been suggested that all such functions might be quotients of entire functions in the Laguerre-P61ya class and hence of order not exceeding 2. The problem was solved in [10,11] under the additional assumption that/' omits a finite value. The only functions obtained in that case are of the form /(z) = Ag(az + b) where A, a, b are real constants and g{z) -tan z or g(z) = tan z -Cz -D for suitable real constants C and D.…”