INCLUSION RELATIONS BETWEEN CLASSES OF HYPERGEOMETRIC FUNCTIONS

received special attention after the surprising application of such functions by de Branges in the proof of the 70-year old Bieberbach Conjecture. In this paper we consider certain classes of analytic functions and examine the distortion and containment properties of generalized hypergeometric hnctions under some operators in these classes.


INTRODUCTION
The expansions and generating functions involving associated Lagurre, Jacobi Polynomials, Bessel functions and their generalizations such as hypergeometric functions of one and several variables occur frequently in the seemingly diverse fields of Physics, Engineering, Statistics, Probability, Operations Research and other branches of applied mathematics (see e.g.Exton [7] and Schiff [19]).The use of hypergeometric functions in univalent function theory received special attention after the surprising application of such functions by de Branges [6] in the proof of the Bieberbach Conjecture [2]; also see [1].i'Received" September, 1993.Revised: January, 1994.2This work was completed while the second author was a Visiting Scholar at the University of California, Davis.
O.P. AHUJA and M. JAI-IANGII Merkes-Scott [11], Carlson-$haffer  [4] and Ruscheweyh-Singh [181 studied the starlikeness of certain hypergeometric functions.Miller-Mocanu [12] found a univalence criterion for such functions.The second author and Silvia [8] and more recently Noor [14] studied the behavior of certain hypergeometric functions under various operators.In the present paper we consider certain classes of analytic functions and examine the distortion and containment properties of generalized hypergeometric functions under some operators in these classes.
Let S be the family of functions of the form f(z) = z -t-.E anZ (z e u) (.a) that are analytic and univalent in U.
We recall that a function is convex in U if it is univalent conformal mapping of U onto a convex domain.It is well-known that I) is convex in U if and only if + > o (z e u) and ' O. Also, a function f is said to be close-to-convex in U if there exists convex function (I) in U such that Re(f'(z)/ff2'(z)) > 0 (z E U).For 0 a < 1, we define the following subclasses of A: A(p, q; a): { ,Fq We use the following lemma due to Chen [5] to prove our first theorem.
To prove our next theorem we need the following lemma due to MacGregor [10].f,(z) < + z and z + 2oa(Z + z I) < If(z)! < z 2tog(X z I).