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Integral Means and Zero Distributions of Blaschke Products

Published online by Cambridge University Press:  20 November 2018

C. N. Linden
C. N. Linden*
Affiliation:
University College of Swansea, Swansea, Wales
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A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only if

If 0 appears m times in {zn} then

is the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.

For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such that

as r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 755 - 760
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Flett, T. M., On the rate of growth of mean values of holomorphic and harmonic functions, Proc. London Math. Soc. 20 (1970), 749–768.Google Scholar
2. MacLane, G. R. and Rubel, L. A., On the growth of Blaschke products, Can. J. Math. 21 (1969), 595–601.Google Scholar
3. Linden, C. N., On Blaschke products of restricted growth, Pacific J. Math. 38 (1971), 501–513.Google Scholar