The Order of Strongly Starlikeness of the Generalized α-Convex Functions

We consider the order of the strongly-starlikeness of the generalized α-convex functions. Some sufficient conditions for functions to be p-valently strongly-starlike are given.


Introduction
Let N, R and C denote the sets of positive integers, real numbers and complex numbers, respectively.Definition 1.A function f is called p-valent in a domain D ⊂ C if the equation f (z) = w has at most p roots in D for every complex number w, and there is a complex number w 0 such that f (z) = w 0 has exactly p roots in D.
Let A(p) denote the class of analytic functions in U = {z : z ∈ C and |z| < 1} of the form: For p = 1, we denote A := A(1).
Definition 2. A function f ∈ A(p) is said to be p-valently starlike in U if it satisfies: We denote by S * p the subclass of A(p) consisting of all p-valently starlike functions in U.
Definition 3. If f ∈ A(p) satisfies: for some β ∈ (0, 1], then the function f is called p-valently strongly-starlike of order β in U. We denote this class by SS * p (β).
We say that for functions f and g analytic in U, g is subordinate to f , written g ≺ f , if there exists a Schwarz function w such that g(z) = f (w(z)) for z ∈ U.In particular, if f is univalent in U, then: In [17], Mocanu first introduced the class: of α-convex functions, which give a continuous passage from convex to starlike functions.He proved that every α-convex function is starlike.Recently, Nunokawa, Sokól and Trabka-Wieclaw [8] considered the generalized α-convex function class: In this paper, we shall further study the properties of the generalized α-convex functions.Several sufficient conditions for functions to be p-valently strongly starlike are obtained.
The following lemmas will be required in our investigation.
Lemma 1 (See [18]).Let g be analytic and univalent in U. Furthermore, let θ and ϕ be analytic in a domain D ⊇ g(U) with ϕ(w) = 0 for w ∈ g(U).Put: and suppose that (i) Q is univalent starlike in U and (ii) Re If q is analytic in U with q(0) = g(0), q(U) ⊂ D and: then q(z) ≺ g(z) (z ∈ U).The function g is the best dominant of (5).

Main Results
Theorem 1.
If q is analytic in U with q(0) = 1 and satisfies: where: 7) The bound β in ( 8) is sharp for the function q defined by: Proof.We choose: in Lemma 1.Then, the function g is analytic and convex univalent in U and: It is clear that ϕ and θ are analytic in a domain D, which contains g(U) and q(U) with ϕ(w) = 0 for w ∈ g(U).The function Q given by: is univalent starlike.Further, we have: and so: Furthermore, for |a| ≤ 1 β , we find that: Therefore, it follows from ( 10)-( 12) that: The other conditions of Lemma 1 are also satisfied.Hence, we conclude that: and the function g is the best dominant of (6).Furthermore, for the function q defined by ( 9), we have: and it follows that the bound β in ( 8) is sharp.The proof of Theorem 1 is completed.
For the function f defined by: we find after some computations that f satisfies ( 14) and: which shows that the bound βπ 2 in ( 13) is the largest number such that (14) holds true.The proof of Theorem 2 is completed.
for z ∈ U, then: In particular, if δ ≥ 1, then f is p-valently strongly starlike of order 1 δ .
Proof.Define the function p(z) by (15).Then, the condition (20) becomes: We want to prove that: If there exists a point z 0 (|z 0 | < 1) such that: then from Lemma 2, we have: where p(z 0 ) = ±ai, a > 0 and: and: For the case arg{p(z 0 )} = − π 2 , we have l < 0 and: where: The function: Therefore, (23) becomes: For the case arg{p(z 0 )} = π 2 , applying the same method as the above, we have l > 0 and: This contradicts (21).Now, the proof of Theorem 3 is completed.
Applying the same method as the above, we can prove the following theorem.
for z ∈ U, then: In particular, if δ ≥ 1, then f is p-valently strongly starlike of order 1 δ .
The sharpness part of the proof is similar to that in the proof of Theorem 2, and so, we omit it.The proof of Theorem 5 is completed.
It is obvious that the condition (25) is a generalization of the condition (4).Defining the function p(z) by(15), the condition (25) becomes: