REMARKS ON A PAPER BY SILVERMAN

We improve a result in Silverman’s paper (1999) and answer a question he posed. We also consider a similar problem and obtain sufficient conditions for starlikeness. 2000 Mathematics Subject Classification. 30C45.


Introduction.
Let A be the class of analytic functions in the unit disc U = {z : |z| < 1} having expansion of the form and let S ⊂ A be the set of univalent functions in U. A function f ∈ S is said to be starlike of order α, 0 < α < 1, and is denoted by S * α if Re z(f (z)/f (z)) > α, z ∈ U and is said to be convex and is denoted by Silverman [2] investigated properties of the functions f ∈ A and the class Some of the results established by him and relevant to us are given in the following theorem.
His method did not extend to b > 1 and he expected the order of starlikeness of G b to decrease from 1/2 to 0 as b increases from 1 to some value b 0 after which functions in G b need not be starlike.
In this paper we establish the following theorems.
) and this order of starlikeness is sharp.Furthermore, for b > 1 the elements of G b need not be regular in U.
We notice that if we put p 3. An analytic function f (z) is said to be subordinate to another analytic function g(z), denoted symbolically as f (z) ≺ g(z), if f (0) = g(0) and there exists an analytic function ω(z) ∈ A, ω(0) = 0 and |ω(z (1.4) (1.5) . (1.6) The special case of (1.4 In the notation of subordination the class G b defined by (1.3) can equivalently be written as We need the following result from [3].
where q is a convex function.

Proof of Theorem
By integration from 0 to z and using p(0) = 1, we get From (2.2) using Schwartz lemma for ω(z), we get or equivalently, |z| = r and This is equivalent to ) and this is sharp because (2.6).The function p(z) given by (2.9) satisfies (1.7) even for b > 1.However, (2.9) shows that for b > 1 both p(z) and f (z) have a pole at z = −1/b and Re p(z) can be negative.Thus, the functions f ∈ G b for b > 1 need not even be regular.