As [R.sup.4]-valued functions, every such f is an

analytic function on [R.sup.2] everywhere.

Let H(U) be the class of

analytic functions in the open unit disk U:= {z [member of] C: |z| < 1} and H[a, n] be the subclass of H(U) consisting of the form

If f is an

analytic function in the interior of [[epsilon].sub.[rho]], then it can be expanded as

Let y be a nonzero

analytic function on [C.sup.N] and [PHI] be an analytic self-map of [C.sup.N].

A fixed point method use an iteration function (IF) which is an

analytic function mapping its domain of definition into itself.

For functions p and q, where p(0) = q(0) = 0, we write p [??] q (i.e., p is subordinate to q) if there exists an

analytic function [omega] with [omega] (0) = 0 and [absolute value of ([omega](z))] < 1 so that p(z) = q([omega](z)) in D.

Based on the

analytic function z([zeta]), the one-to-one corresponding relations between [theta] and [alpha] are determined as

In the field of Geometric Function Theory, various subclasses of the normalized

analytic function class A have been studied from different viewpoints.

Define an

analytic function p : D [right arrow] C by

By noticing that [p.sub.k] (x, y), [q.sub.k] (x, y) are both

analytic functions, we rewrite system (10) into the following form:

Since the hypergeometric series in (83) converges absolutely in D, it follows that F([beta], [gamma], [delta]; z) defines an

analytic function in D and plays an important role in the theory of univalent functions.

This class is called starlike class of

analytic function, let f [member of] [gamma].