On the growth of entire functions of bounded index

W. J. Pugh and S. M. Shah

Vol. 33 (1970), No. 1, 191–201

DOI: 10.2140/pjm.1970.33.191

Abstract

A class E of entire functions of zero order and with widely spaced zeros has been defined and it is proved that if f ∈ E then f′,f′′,⋯ ∈ E. Furthermore f is of index one. This class includes many functions which are both of bounded index and arbitrarily slow growth. If f is any transcendental entire function then there is an entire function g of unbounded index with the same asymptotic behavior. When f is of infinite order then it is of unbounded index and we simply take g = f. When f is of finite order we give the construction for g.

Mathematical Subject Classification

Primary: 30.57

Milestones

Received: 16 October 1968

Revised: 11 November 1969

Published: 1 April 1970