f(z)/z] > 0, and since + y > 0, if we define

Since the term on the right satisfies the subordination (5.5-19) we can apply

Theorem 3.1c with A = 1/(~ + y) to obtain

(5.5-22)

where

(5.5-23)

[Note that the application of Theorem 3.1c requires that the value of v in this equation be less than the root of equation (3.1-9). It is easy to show that the root of (3,1-9) is larger than one and that the root of (5.5-23) is less than one.]

The subordination in (5.5-22) is equivalent to

(5.5-24)

(5.5-25)

zF'(z) p(z) = F(z) '

then from (5.5-21) and (5.5-25) we obtain

From (5.5-21) and (5.5-25) we also obtain zG'(z) = j3G(z)·(p(z) - 1). Hence (5-26) can be rewritten as

where

when <1 ~- (1 + p 2 )/2, p E lR and z E U. Since Re 'lf(ip, <1; z)

we see that (5.5-28) will be satisfied if the discriminant of this last term

satisfies

5.5 FUNCTIONS WITH BOUNDED TURNING 305

By using (5.5-30) and (5.5-23) we obtain the equivalent condition

(5 5 31) e < -1 ~ -1 ( 1 ) -1 8 + ll v • - 0 _ tan u + tan A v = tan 1 _ 81l v.