= + z<j>'(z)]/cp(z) and D(z) = w(z) then B(z) 0 and con-

We can refine the conclusion of this last result by carefully sleeting the functions <j>, cp and w. If we let <j>(z) = g(z)/z and cp(z) = g'(z) in Theorem 4.3a we obtain the following result.

COROLLARY 4.3a.l. Let n be a positive integer and let y e C with

(4.3-4) Re [ z:~;;) + y + n -1 J > 0. For f, w e ..J./ [ 0, n ], define the function F by

(4.3-5) F(z) = y-~ I [f(t) + w(t)]g'(t)ty-l dt. z g(z)

(4.3-6) N = Sup { lzg' (z)I(M + lw(z)l) } lzl<l lzg' (z) + g(z)(y + n-1)1

4.3 INTEGRAL OPERA TORS WITH BOUNDED FUNCTIONS 209

From ( 4.3-6) we deduce

From ( 4.3-6) we deduce that

(4.3-8) {

210 SECOND-ORDER DIFFERENTIAL SUBORDINATIONS

1 "f 0 1 • 1 < y. 1 + y - rl/1,1

(4.3-9) 1/(z)l ~ lzl => 1-.A.z J f(t)t'Y-l dt ~ Nlzl. z1 (1-.A..t) 2

We next obtain another corollary of Theorem 4.3a by making the

COROLLARY 4.3a.2. Let n be a positive integer and let cj> E ;J)n satisfy

(4.3-10) [ zcj>'(z) J

4.3 INTEGRAL OPERATORS WITH BOUNDED FUNCTIONS 211

(4.3-13)

Theorem 4.3a and its two corollaries can still be used in this case. However,

in this special case we can replace conditions (4.3-1) and (4.3-3) of Theorem

4.3a by one simple condition as given in the next theorem. Note that this

new condition is also independent of n. The proof follows immediately from Corollary 4.1c.1 and will be omitted.