INTEGRAL MEANS OF CERTAIN CLASS OF ANALYTIC FUNCTIONS

In this paper we discuss the following class of functions $(,3)={I(z): "f(z) 11<31l(z) 11 D}whereO_<_<l 0<3_<1 0_<<1 I-"’g(z) + ,z E and l(z)= z + _, n r’ is analytic in D= {z: zl < 1}, g(z) is a starlike function of order a. A subordination about this class is obtained, the integral means of functions in S,(a,3) and some extremal properties are studied.

We first gave a subordinate about this class, then we discuss the integral means of functions in $,X(a,3), from this we can get some extremal properties about S,X(a,3 ).We also discuss a subclass of 2. A SUBORDINATION ABOUT SA(a,3).
The definitions of u*(x) and the symmetric decreasing arrangement function can be found in [2].
PROOF.From the definition of SA(a, ), we know there exists a function #(z)e S*(a) such that the inequMity (1.1) holds.So we have, from Threm 2.1 A= PZ, A() Thus dO ( olpB,(reiO) dO, by mma a.a.
PROOF.We can easily know f(z) is univalent in D from the definition of $X(a,B), so Izl(-lzl) - The sharpness can bee seen from the function (1_ z)2(l_)(l_x) S,(a,).
From the univMence of I(z) we know ---lz # o, so we c define a single-vMued d Mytic brch of tog ---z.where the inequality holds only for f(z)= zp,,(rz), zl 1.

Finally
we obtain, by Lemma 3.1, s-_.. (,o, We can similarly prove the case of negative sign.The condition of the equality can easily be obtained.THEOREM 3.2.Let l(z) e SA(a,B), then for p > 0 we have and G. CHUNYI(O<r<l)    where the equality holds only for f(z) 1. ko(xz p,A(xz) ix1.

THEOREM 4 . 3 .
Let y(z)= z + anz" S(), then for reM number g we have the shp estimates: