On univalence of certain infinite products

This paper is dedicated to Professor A. W. Goodman who has been, and continues to be, an inspiration to many mathematicians including both authors. At 80, Professor Goodman has certainly proved his message that was often delivered to many of us, "I may retire from teaching but never from mathematics." It is certainly a message for us all! We prove: Let f be an entire function of the form where g is an entire function with Reg′ (z) ⩾ α > 0 for all and p n .> 0 (no exponential factor appears in the product when p n = 0). If |arg f(z)|