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Maximum Modulus Theorems and Schwarz Lemmata for Sequence Spaces, II

Published online by Cambridge University Press:  20 November 2018

B. L. R. Shawyer
B. L. R. Shawyer*
Affiliation:
Department of Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B9
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In this note, we continue the investigations of [3], proving another analogue of the maximum modulus theorem, this time for the sequence space bv, and we investigate maximal functions for such theorems. As in [3], we use the notation f∈MM if f is analytic in the disk |z| <1, continuous for |z| ≤ 1 and satisfies |f(z)| ≤ 1 on |z| = 1. We also write f∈SL if f∈MM and f(0) = 0. Whenever x={xk} is a sequence of complex numbers, we write f(x) = {f(xk)}.

In [3], we proved analogues of the maximum modulus theorem for the sequence spaces 5, m and c, and analogues of the Schwarz Lemma for the sequence spaces c0, lp and bv0. We begin this note with the sequence space bv.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 79 - 84
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Dunford, N., and Schwartz, J. T., Linear Operators, Part I, Interscience, New York, Fourth Printing, 1967.Google Scholar
2. Conway, J. B., Functions of One Complex Variable, Springer-Verlag, New York, Second Printing, 1975.Google Scholar
3. Shawyer, B. L. R., Maximum Modulus Theorems and Schwarz Lemmata for Sequence Spaces, Canad. Math. Bull., 18 (1975), 593-596.Google Scholar