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Entire Functions of Irregular Growth take every value

Entire Functions of Irregular Growth take every value ENTIRE FUNCTIONS OF IRREGULAR GROWTH TAKE EVERY VALUE J. M. WHITTAKER The following result will be established: THEOREM. Let p, X, p be the order, lower order and exponent of convergence of the zeros of an entire function. Then p = max (X, p ). (1) COROLLARY. NO value is exceptional B for an entire function of irregular growth. In this proposition some, or all, of p , X, p may be infinite. Denote the right-hand side of (1) by fx. Since X ^ p,p ^ p in all cases, \i ^ p and (1) is certainly true if X or p is infinite; so we may take them to be finite. Dividing /(z) , if necessary, by z we may suppose that/(0 ) # 0 and can then write/(z) in the form P(z) exp {Q{z)}, where P(z) is a canonical product of order p and Q(z) is entire. It is familiar [1; 273] that if we surround the zeros z of P(z) with circles of radii \z \~ , whereh > p then p u 1+ \P(z)\ > exp (-r" « ) (|z| = r > r (e)) in the region excluded from the circles. Enclose the circles http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Entire Functions of Irregular Growth take every value

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/4.2.130
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Abstract

ENTIRE FUNCTIONS OF IRREGULAR GROWTH TAKE EVERY VALUE J. M. WHITTAKER The following result will be established: THEOREM. Let p, X, p be the order, lower order and exponent of convergence of the zeros of an entire function. Then p = max (X, p ). (1) COROLLARY. NO value is exceptional B for an entire function of irregular growth. In this proposition some, or all, of p , X, p may be infinite. Denote the right-hand side of (1) by fx. Since X ^ p,p ^ p in all cases, \i ^ p and (1) is certainly true if X or p is infinite; so we may take them to be finite. Dividing /(z) , if necessary, by z we may suppose that/(0 ) # 0 and can then write/(z) in the form P(z) exp {Q{z)}, where P(z) is a canonical product of order p and Q(z) is entire. It is familiar [1; 273] that if we surround the zeros z of P(z) with circles of radii \z \~ , whereh > p then p u 1+ \P(z)\ > exp (-r" « ) (|z| = r > r (e)) in the region excluded from the circles. Enclose the circles

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1972

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