Vol. 33, No. 1, 1970

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On the growth of entire functions of bounded index

W. J. Pugh and S. M. Shah

Vol. 33 (1970), No. 1, 191–201

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
Received: 16 October 1968
Revised: 11 November 1969
Published: 1 April 1970
W. J. Pugh
S. M. Shah