1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 587

Assuming that f is not a constant function, first we consider the case

f(O) = 0. Since A(r) is a strictly increasing function of r (Corollary 8.16),

we have A(r) > A(O) = 0 for r > 0. Then the function

f(z)

g(z) = 2A(R) - f(z) (8.13-10)

is analytic in izl ::::; Rand g(O) = 0. Letting f(z) = u +iv, we have

u2 + v2

lg(z)l2 = [2A(R) - u]2 + v2 ::::; 1

since 4A(R)[A(R) - u] ;::: 0. Thus lg(z)I ::::; 1, and the function g satisfies

all the conditions of Corollary 8.18. Hence it follows that

1

lg(z)I ::::; R lzl

for izl < R and, in particular,

r

lg(z)I::::; R

for lzl = r. Solving (8.13-10) for f(z), we get

f(z) = 2A(R)g(z)

1 + g(z)

and using (8.13-11), we find that

lf(z)I < 2A(r)lg(z)I < 2rA(R)

• 1 - lg(z)i - R-r

for lzl = r, from which (8.13-9) follows for the case f(O) = 0.

(8.13-11)

(8.13-12)

If f(O) f= 0, it suffices to apply (8.13-12) to the function F(z) = f(z) -

f (0). We obtain

2r

lf(z) - f(O)I ::::; R _ r i1;J!1[Re(f(z) - f(O))]

and it follows that

lf(z)I - if(O)i ::::; Jf(z) -f(O)J::::; R

2

~ r [A(R) + if(O)IJ

or

lf(z)I::::; R

2

~ r A(R) + ~ ~ ~ if(O)I

for lzl = r. Hence

2r R+r

M(r) = fzl~ if(z)I::::; R-r A(R) + R _ r if(O)i