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
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
 (1) 
inside the circle. In other words,
 (2) 
for z≦ 1. For a fixed a on the unit circle, let C_{a} denote the class of functions f(z)
which are regular in z≦ 1, vanish at the point z = a, and for which
Any positive number A being given, it is clearly possible to construct a function f(z)
of the class C_{a} for which
i.e. ℳ_{f}(1) is not uniformly bounded for f ∈ C_{a}. If f(z) is restricted to a subclass of
C_{a} there may exist a uniform bound for ℳ_{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.
