SOME CLASSES OF ALPHA-QUASI-CONVEX FUNCTIONS

Let C[C,D], -I<D<C<I denote the class of functions g,g(O) 0 g’ (O)=I, analytic in the unit disk E such that -(zg’ (z)) is subordinate I+CZ g (z) to I+DZ’ zgE. We investigate some classes of Alpha-Quasi-Convex Functions f, with f(O)=f (O)-I=0 for which there exists a gEC[C,D] such that f’ (z) (zf’ (z)) I+AZ _I<B<A<I Integral rep(l-a)g+ g(z is subordinate to I+BZ’ resentation, coefficient bounds are obtained. It is shown that some of these classes are preserved under certain integral operators.


INTRODUCTION
Let S,K,S and C denote the classes of analytic functions f: n f(z)=z a z which are respectively univalent, close-to-convex, starlike n 2 (wit=respect to the origin) and convex in the unit disc E. In [i], a new subclass C of univalent functions was introduced and studied.A function f belongs to C if there exists a convex function g such that, for zgE, (zf' (z)) Re >0. g'(z) The functions in C are called quasi-convex functions and C_C ._KCS.
It is also sknown that fgC  if, and only if, zf'EK.
For complete study of C see Noor [2].
A new class Q of a-quasi-convex functions has been defined and dis- cussed in some details in [3].A function f belongs to the class Qe,a real, if and only if there exists a convex function g such that, for zeE

C
We note that Q0 K and Q1 In [4] Janowski introduced the calss P[A,B].For A and B, I<E<A<I, a function p, analytic in E with p(0)=l belongs to the class P[A,B],if p(z) I+AZ is subordinate to I+BZ Also, given C and D, -I<D<C<I,_ C[C,D] and S [C,D]  denote the classes of functions f analytic in E with f(z)=z + a z n n n=2 such that (zf' (z)) ePIC,D] and zf' (z) P[C,D respectively.For C=I f'(z) f(z) and D=-I we note that C[I,-I]= C and S [i,-i] S Silvia [5]  We now define the following: n Definition 1.2.
Let >0 be real and f: Then fgQ[A,B; C,D], -I<B<A<I; -I<D<C<I if and only if there exists a function gC[C,D] such that, for zeE,

MAIN RESULTS
We shall now study some of the basic properties of the class Qe[A,B;C,D].From the definition 1.2, we immediately have: ,where 0<e<l is real and zeE.
Then fEQ[A,B;C,D], -I<B<A<I; -I<D<C<I if and only if FEE [A,B;C,D].
We now give the integral representation for the functions in the class ,B;C,D].
A function fEQ[A,B;C,D],_ for >0, -I<B<A<I; -I<D<C<I, if and only if there exists a function FgK[A,B;C,D] such that, for zEE, From (2.1), it follows that and the result follows immediately from theorem 2.1.
fK[A,B;C,D] and hence is univalent.
Then PROOF.
Silvia [5] has proved that if fleK[A,B;C,D], then so is z i_i l+l t fl (t) dt Re l >0 (2 2) Fl(Z) l z 1 bsing this result and the integral representation (2 2) with i a for f. [A,B;C,D, we obtain the required result.
For our next theorem, we need the following result due to Silvia [5].In definition 1.2, if we put A=I-28 B= -i; C=l-2y, D -i, then we have the following: Definition 3.1.
A function f, analytic in E, is said to be alpha-quasi- convex of order 6 type y, if, and only if, there exists a function geC[l-2y,-l] such that REMARK 3. i.
We now have the following: n THEOREM 3.1.
Let feQ [1-28, -l;l-2y,-l] and be given by f(z)=z + Y. a z n n=2 Then we have, for n>2 lanl<2__(3-2y This result is sharp and the equality holds for the function f0 defined as Let 0<I<i and 0<8<i.