ON THE SHARP CONSTANT FOR STARLIKENESS

We obtain a sharp constant of the sufficient condition for p-valently starlikeness, which had been studied by Nunokawa (1991), Obradovíc and Owa (1989), and Li (1993). 2000 Mathematics Subject Classification. Primary 30C45.


Introduction. Let A(p) denote the class of functions of the form
which are analytic in U = {z : |z| < 1}.A function f (z) in A(p) is said to be p-valently starlike if and only if Let S(p) denote the subclass of A(p) consisting of all functions f (z) which are p-valently starlike in U (cf. [1]).For a function g(z) in A(p), the interesting problem is to find the best constant A such that g(z) is in S(p) whenever In 1989, Obradović and Owa [6] obtained that A = 5/4 for the case of p = 1.For the general case, Nunokawa [5] gained that A = log 4. Recently, Li [2] improved these results and obtained that A = 3/2.In this paper, we will solve this problem completely and give the sharp constant A = 1.80898 ..., where A is the unique solution of the equation (1.4) For proving our result, we should recall the concept of subordination between analytic functions.Given two analytic functions f (z) and F(z), the function f (z) is said to be subordinate to F(z) if F(z) is univalent in U, f (0) = F(0), and f (U) ⊂ F(U).We denote this subordination by f (z) ≺ F(z) (see [7]).
Suppose that h(z) is analytic in U, and that Φ(z) is analytic in an appropriate domain D, we consider the following first-order differential subordination where p(z) is analytic in U, β is a complex constant.Changing the "≺" of (1.5) to "=", we get the corresponding first-order differential equation 2. Main results.Our results rest on the following lemma, which is the special case of [3,Theorem 3].
Lemma 2.1.Suppose that h(z) is a starlike function in U, Φ(z) is analytic in the domain D and p(z), q(z) are two analytic functions in U.If p(z) satisfies the relation (1.5), q(z) is a univalent solution of the corresponding equation (1.6) and p(0) = q(0), then p(z) ≺ q(z).
Theorem 2.2.Let g(z) ∈ A(p), and suppose that where the constant A is given by (1.4).Then g(z) ∈ S(p) and the result is sharp.

This proves g(z) ∈ S(p).
For any A 1 > A = 1.80898 ..., we get a function q 1 (z) by replacing A in (2.6) with A 1 and choosing an appropriate branch of log(A 1 + z).We can easily observe that the real part of q 1 (z) is not always positive.Through the relations q 1 (z) = zf (z)/f (z) and f (z) = g (p−1) (z)/p!, we can construct an analytic function g(z) which belongs to A(p) and satisfies (2.1), but it is not in S(p).This completes the proof.
Taking p = 1 in Theorem 2.2, we easily have the following corollary.

Corollary 2.3. If f (z) ∈ A(1) and it satisfies the condition
where the constant A is given by (1.4), then f (z) is univalent and starlike in U.
The problem that Nunokawa proposed in [5] has been solved completely, but the converse proposition of Theorem 2.2 is not true.We find a simple example f (z) = z/(1 − z) which belongs to S(1), but it does not satisfy (2.12).The following theorem is better than (2.1) because it includes at least this example.

Proof.
Let