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Journal of the London Mathematical Society 1987 s2-36(1):95-101; doi:10.1112/jlms/s2-36.1.95
© 1987 by London Mathematical Society
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© Oxford University Press

The Bloch Constant of Bounded Analytic Functions

Flavia Colonna

The University of Bari 70125 Bari, Italy The University of Maryland, College Park Maryland 20742, USA

In this work we solve the extremal problem of characterizing all bounded analytic functions f: {Delta} -> C (where {Delta} is the open unit disk) for which the Bloch constant

ßf = sup{(1–|z|2)|f'(z)|: z {varepsilon} {Delta}

is a bound. Normalizing, we study f: {Delta} -> {Delta} with ßf = 1. We show that these are precisely the conformal automorphisms of {Delta} together with those functions whose zeros form an infinite sequence (zn)n{varepsilon}N such that


Formula
,

where g is the non-vanishing function such that f/g is a Blaschke product. In particular, non-vanishing inner functions, finite Blaschke products, and outer functions for the class H{infty}({Delta}) with image contained in {Delta} are not extremal functions.


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