Length problems for Bazilevič functions

LetH denote the class of functions f which are analytic in the unit disk D = {z ∈ C : |z| < 1}, and A be the subclass ofH consisting of functions normalized by f (0) = 0 = f ′(0) − 1. Let S ⊂ A be the class of functions univalent (i.e. one-to-one) inD. Denote by S* the subclass of S of starlike functions, i.e. the class of functions f ∈ A such that f (D) is starlike with respect to the origin. It is well-known, since the work of [1], that f ∈ S* if, and only if, f ∈ A and


Preliminaries
Let H denote the class of functions f which are analytic in the unit disk D = {z ∈ C : |z| < 1}, and A be the subclass of H consisting of functions normalized by f (0) = 0 = f ′ (0) − 1. Let S ⊂ A be the class of functions univalent (i.e. one-to-one) in D. Denote by S * the subclass of S of starlike functions, i.e. the class of functions f ∈ A such that f (D) is starlike with respect to the origin. It is well-known, since the work of [1], that f ∈ S * if, and only if, f ∈ A and Recall that a set E ⊂ C is said to be starlike with respect to to the origin if, and only if, the linear segment joining 0 to every other point w ∈ E lies entirely in E. By P we denote the class of Carathéodory functions p which are analytic in D, satisfying the condition Re {︀ p(z) }︀ > 0 for z ∈ D, with p(0) = 1. Suppose now that f ∈ A, then f is close-to-convex if, and only if, there exists α ∈ (−π/2, π/2), and a function g ∈ S * such that This class of close-to-convex functions was introduced in [2]. Functions defined by (1.1) with α = 0 were considered earlier by Ozaki [3], see also Umezawa [4,5]. Moreover, Lewandowski [6,7] defined the class of functions f ∈ A for which the complement of f (U) with respect to the complex plane is a linearly accessible domain in the large sense. The Lewandowski class is identical with the class of close-to-convex functions.
Here, we denote this class by K, and note that S * ⊂ K ⊂ S. The class of close-to-convex functions forms an important subclass of S. Length problems for close-to-convex functions were recently considered in [8]. A proper subset of K is the class of bounded boundary rotation of f such that f ′ (z) ≠ 0 in the unit disc and Another even larger subset of S is formed by the Bazilevič functions. Bazilevič [9] introduced a class of functions f ∈ A which are defined by the following where h ∈ P and g ∈ S * , α is any real number and β > 0. Bazilevič showed that all such functions are univalent in D. Putting α = 0 in (1) and differentiating it, we have and Thomas [10] called a function satisfying condition (1.2) a Bazilevič function of type β. For further works on Bazilevič functions we refer to [11]- [15]. It is easy to see that Bazilevič functions of type β = 1 are close-toconvex functions, univalent in D. Furthermore, the set of starlike functions is contained in the set of Bazilevič functions of type β.
Let C(r) denote the curve which is image of the circle |z| = r < 1 under the mapping f . Let L(r) be the length of C(r) and A(r) the area enclosed by the curve C(r). Furthermore M(r) = max |z|=r |f (z)|. In [16], Thomas has shown the following: Note that in [17], Thomas  Applying the result of [18], we give a negative partial result of the above open problem (1.3). Some related problems were considered in [19,20].

2
Remark 1. If f (z) = z + ∑︀ ∞ n=2 an z n is analytic and univalent in D, then it is trivial that S(r) = A(r) for 0 < r < 1 so in this case (2.1) becomes as r → 1.
Proof. From (2.2) and applying the same method as in the proof of Theorem 1 in [22], we have Then, from [18, p.338], and from Lemma 2.1, we have the following where δ is fixed 0 < δ < ρ ≤ r < 1. Applying the result of [22, p.277] and the same method as in the calculation (2.3), we have This completes the proof of Theorem 2.3.
From Theorem 2.3, we easily have the following corollary. Then there is no Bazilevič function of type β satisfying the condition (1.3).