On the order of strong starlikeness and the radii of starlikeness for of some close-to-convex functions

In this paper we show several sufficient conditions for close-to-convex functions to be strongly starlike of some order. The results continue the line of study from the first author’s paper on the order of strong starlikeness of strongly convex functions, (Nunokawa in Proc Japan Acad Ser A 69(7):234–237, 1993). Also it appears an small improvement of a certain classical results of Ch. Pommerenke. As an application, we also derive estimates for the radii of star-likeness for close-to-convex functions.

two points of E lies entirely in E. An univalent function f maps D onto a convex domain E if and only if [11] Re 1 + z f (z) f (z) > 0 for all z ∈ D.
Such a function f is said to be convex in D (or briefly convex). In [8] Sakaguchi proved that if f ∈ A and g ∈ S * , then This result was also generalized, see [2] and [7]. In [6] Pommerenke established a formula for β = β(α) such that where f (z) ∈ K α . This is a generalization of the relation of type (1.1) because is in the class K α α, 0 < α ≤ 1 whenever f (z) ∈ A and there exist a function g(z) ∈ K such that Here we understand that Argw is a number in (−π, π]. It is known that if f (z) ∈ K α , then f (z) is close-to-convex and so f (z) is univalent in D. The class f (z) ∈ K α is called the class of strongly close-to-convex functions of order α. This result has found many applications.
and it says that f (z) is a strongly starlike function of order α, 0 < α ≤ 1. The class of strongly starlike functions was introduced in [1,10], we denote this class here by S * (α). One can consider functions satisfying condition (1.3) with 0 < α < 2 and in this case we will named such functions also strongly starlike of order α, 0 < α < 2. It is known that if f (z) is strongly starlike of order α > 1, then f (z) need not to be univalent in D.
exists for all z ∈ D. It is known the following Pommerenke's result [6, Lemma 1, p. 180]: If f (z) is analytic and g(z) is convex in D, then (1.5) Applying this Lemma with z 1 = 0 gives We note that a result of the form related to (1.5) was proved in [5, Theorem 2.1].

7)
and Proof From the hypothesis h(z) = 0, so the function is in the class H. Furthermore, we have and h 1/α (z) is contained in the circle with the radius and center respectively. From this, we obtain (1.7) and (1.8).
If we take g(z) = z then Pommerenke's result (1.5) becomes the following corollary.
In the next Corollary we extend α to 1 < α < 2 : for some α, 1 < α ≤ 2, then Proof For arbitrary z ∈ D and from the hypothesis 1 < α ≤ 2, can connect the point f (z) and f (0) = 1 by a line segment as f (z) f (0) = f (z)1. Applying the same method as in the proof of Pommerenke [6, p. 180], we have and so, applying the property of integral mean, we have To prove the main results, we also need the following generalization of the Nunokawa's Lemma, [3,4].
Then p ∈ H and it is the form p(z) = 1 + ∞ n=2 a n z n−1 .
Theorem 1.8 Let f (z) = z + ∞ n=2 a n z n be in K α for some α, then f (z) is starlike in |z| < |z 0 |, where |z 0 | is the smallest positive root of the equation which has the form Therefore, we have Putting we have for |z| as in (1.24).

Remark 3
The radius of starlikeness in the class K α may be equal or larger than the above value. This is an open question.