Notes on certain subordinations
by
Shigeyoshi Owa
Kinki University

Let A be the class of functions f(z) of the form

 

f(z) = z +  \infty å n=2  anzn

  which are analytic in the open unit disk U={z in C:|z| < 1}. Let S^* be the subclass of A consisting of functions f(z) which are satisfy

 

Re{\fraczf'(z)f(z)}  >  0         (z in U).

  A function f(z) in S^* is said to be starlike in U. For functions f(z) and g(z) belonging to A, we say that f(z) is subordinate to g(z) if there exists the function w(z) which is analytic in U with w(0)=0 and |w(z)| < 1 (z in U) such that f(z)=g(w(z)). We denote this subordination by f(z)\prec g(z). By virtue of the definition for subordinations, we know that:
(i) The subordination f(z)\prec g(z) implies that f(0)=g(0) and f(U) subset g(U).
(ii) If g(z) is univalent in U, then the subordination f(z)\prec g(z) is equivalent to f(0)=g(0) and f(U) subset g(U).
For a function f(z) in A, we consider the integral operator I_\alpha.\beta(f(z)) given by

 

I\alpha.\beta(f(z)) =  ì í î \frac\alpha+\betaz\beta ó õ z 0  t\beta-1f(t)\alphadt ü ý þ 1/\alpha   ,

  where \alpha in C, \alpha =/= 0, and \beta in C.
In the present talk, we derive some subordination properties for the above integral operator I_\alpha.\beta(f(z)). Also an application of hypergeometric functions is considered.

Date received: June 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-90.