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.

Abstract.The authors exhibit two linearly independent real solutions, f\ and fi, to w" + Hw = 0 such that /1/2 is transcendental, f and fi have only real zeros and poles and H is a nonconstant rational function. This shows the sharpness of a recent result of Hellerstein and Rossi. Some related results are also proved.

Abstract.Let A be entire. Suppose that there exists an unbounded quasidisk D such that A is sufficiently small in D. We prove that then any nontrivial solution to y" + Ay = 0 has at most one zero in D. We show that if A = Q exp P where P and Q are polynomials, one can usually take D to be an angle of opening x/n where n is the degree of P .