A COVERING THEOREM FOR ODD TYPICALLY-REAL FUNCTIONS

An analytic function f ( z ) = z + a 2 z 2 + … in | z | 1 is typically-real if Im f ( z ) Im z ≥ 0 . The largest domain G in which each odd typically-real function is univalent (one-to-one) and the domain ⋂ f ( G ) for all odd typically real functions f are obtained.

An analytic function f(z) z + amZ2 +... in the unit disk E (Izl < i) is in the class T of typically-real functions if and only if there exists a nondecreas- ing function y on [0,7] such that y() i, y(0) 0, and f(z) I gay(t) 1 2z cos t + z 0 The function y when normalized on (0,7) by y(t) (y(t+) + y(t-))/2 is uniquely determined by f.

E.P. MARKES
The domain of unlvalence of the class T is known [23 to be (1.2) Brannon and Krwan [3] proved that the largest domain contained in f(G) for every function in T is [w < 1/4.
In this paper we obtain the corresponding results for the class T O of odd typlcally-real functions.Recently Goodman [4] determined the largest domain that is contained in f(E) for every f e T. The analog of this result for the class T O is an open problem.
2. The domain of unlvalence of T O THEOREM 2.1.The domain of unlvalence for T O is the domain G of (1.2).
PROOF.Since T O c T, each f e T O is univalent in G.The theorem is estab- lished, therefore, if we can show that there is a function f e T O that is not univalent in any domain D that properly contains G. Let f(z) : (1 :) + ;I(1 + ) /(I ) Since /(I-2) is not univalent in any domain that properly contains II < i, we conclude that f is not univalent in any domain that properly contains G.
3. A covering theorem for T O THEOREM 3.1.The largest domain U contained in f(G) for every f e T O is the 18 domain that includes the origin, is bounded in the right half-plane by w pe where 0 f (cs 0)/2 o < 1el < T/4, and is symmetric relative to the imaginary axis.
Since the convex hull H contains for each z e aG the values of 2f(z) for all f e T O and since every point of H is the value of 2f(z) for some f e T O we conclude that U is the exact domain covered by all f e T O when z G.

2 f
l z) + z/(l + z) This function is clearly in T O since T is a linear class.The function 2z(2.1) -2 l+z maps G onto II < i.By the change of variables (2.1), the function f has the form =I (z) z/(l 2z + z 2) + 1/2z/(l + 2z +