Using

In a series of papers 12,3,4) T. Kobayashi has given some interesting charac-terisations of the exponential function by its property of having its a-points col-Iinear on a lirc)'(a) for several values of a. In [4] he proved Theorem A. Let G be a transcendental entire function of finite lower order Assume that the zero points of G lie on the line Re z:0 and thqt the one-points lie on Re z:1. Then G(z) : P(exP Cz) where P is a polynomial and C a non-zero real constant. Of course there are restrictions on the polynomials which can occur in Theorem A and the possible forms are given in [4]' Kobayashi asks whether Theorem A stilt holds if the assumption that G has finite order be omitted from the hypotheses. We prove the Theorem. Let f be a transcendental entire function such thqt all the zeros o/ f tie on Rez:0 and all the one-points on Rez:l. Then f has finite order, so by Theorem A there exist a polynomial P and a non-zero real constant C such that f (') : P(exP Cz)' The proof depends on a recent result of J. Miles (Lemma 6) which states that if an entire function has both infinite order and real zeros then the zeros are in a certain sense scarce. Applying this to f (-iz) and f (|-iz)- 1 and using Nevanlinna's second fundamental theorem gives the result after a number of sub-sidiary points have been checked. The proof follows in six lemmas and a concluding section' Lemma 1. Let g be analytic in H: Imz>0 and omit the ualues 0 and I in H. Then there exists a constant X:K(d such that (l) loglg(re'o)l<Kr/(sin0), r>1,0=0-n. Proof. The map 7:EQ):i(l:+t)l1-t) maps the disc D: ltl''l to the half-plane 1L Applying Schottky's theorem to g(E(t)) we obtain lgGr(r))l = exp {K/(1- lr l)}