Vol. 23, No. 1, 1967

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Remarks on Schwarz’s lemma

Qazi Ibadur Rahman and Q. G. Mohammad

Vol. 23 (1967), No. 1, 139–142
Abstract

If f(z) is an analytic function, regular in |z|1,|f(z)|1 for |z| = 1 and f(O) = 0, then by Schwarz’s lemma

|f(rei𝜃)| ≦ r,(0 ≦ r ≦ 1).

More generally, if f(z) is regular inside and on the unit circle, |f(z)|1 on the circle and f(a) = 0, where |a| < 1, then

|f(z) ≦ |(z − a)∕(1 − az)|,
(1)

inside the circle. In other words,

                    --
|f(z)l(z − a)| < 1∕|1 − az|,
(2)

for |z|1. For a fixed a on the unit circle, let Ca denote the class of functions f(z) which are regular in |z|1, vanish at the point z = a, and for which

max |f(z)| = 1.
|z|=1

Any positive number A being given, it is clearly possible to construct a function f(z) of the class Ca for which

ℳf (1) = m|za|x=1|f(z)∕(z − a)| > A,

i.e. f(1) is not uniformly bounded for f Ca. If f(z) is restricted to a subclass of Ca there may exist a uniform bound for f(1) as f(z) varies within the subclass. It is clear that such is the case for the important subclass consisting of all polynomials of degree at most n vanishing at z = a. The problem is to find the uniform bound.

Mathematical Subject Classification
Primary: 30.10
Milestones
Received: 15 April 1966
Published: 1 October 1967
Authors
Qazi Ibadur Rahman
Q. G. Mohammad