On univalence of certain infinite products

Abstract

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)|<π in D and

then f is univalent in D. On the basis of this simple criteria, and substantially adjusted well known results, in conjunction with computer experiments we obtain the exact radius of univalence of the Airy's function Ai(z), the Gamma function Г(z+ 1), and certain other entire functions. Some additional geometric properties of each of these functions are determined.

AMS No.:

• Radius of univalence Airy's functions
• Gamma function infinite products
• 30C99
• 30D99
• 33A15
• 33A70