A Note on the Class of Functions with Bounded Turning

and Applied Analysis 3 In this note we pose the following subclass of bounded turning functions in the the unit disk: for given numbers > 0 and α > 0, let us consider the class B p : B ( p ) { f ∈ A : ∣f ′ z )α − 1∣ < f ′ z )α − , z ∈ U. 1.9 It is easy to see that f ∈ B p if and only if f ′ z ≺ p : ( 1 − k z )1/α , z ∈ U, k ≥ 2. 1.10 For the development of the paper we need to define a set Q by all points on the right halfplane as follows: Q {w ∈ C : |w − 1| < w − , α > 0}. 1.11 Thus it is easy to verify that its boundary satisfies the equation of a parabola v2 2u 1 − − 1 − , w u iv . 1.12 2. Main Results First, our result is in the following form. Theorem 2.1. A function f ∈ B p , > 0, if and only if there exists an analytic function p, p z ≺ p z : ( 1 − k z )1/α , k ≥ 2, 2.1


Introduction
Let U : {z : |z| < 1} be the unit disk in the complex plane C, and let H denote the space of all analytic functions on U. Here, we suppose that H is a topological vector space endowed with the topology of uniform convergence over compact subsets of U. Also, for a ∈ C and n ∈ N, let H a, n be the subspace of H consisting of functions of the form f z a a n z n a n 1 z n 1 · · · . 1.1

Further, let
and let S denote the class of univalent functions in A. A function f ∈ A is called starlike if f U is a starlike domain with respect to the origin, and the class of univalent starlike f z * g z z ∞ n 2 a n b n z n , 1.5 where a n and b n are the coefficients of f and g, respectively.
The pre-Schwarzian derivative T f of f is defined by with the norm It is known that T f < ∞ if and only if f is uniformly locally univalent. It is also known that T f ≤ 6 for f ∈ S and that T f ≤ 4 for f ∈ K. Moreover, it is showed that, when |T f | ≤ 3.05f is univalent in U and when |T f | ≤ 2.28329, f is starlike in U see 1 . Recently, the sharp norm estimates for well-known integral operators are determined see 2-4 . For 0 ≤ ν < 1, let B ν denote the class of functions f of the form 2.2 so that {f } > ν in U. The functions in B ν are called functions of bounded turning. By the Nashiro-Warschowski theorem, the functions in B ν are univalent and also close to convex in U. It is well known that B ν / ⊆ S * and S * / ⊆B ν . In 5 , Mocanu obtained a subclass of S * , which is contained in B ν . Recently, Tuneski generalized the class of convex functions with bounded turning see 6 : Different studies of the class of bounded turning functions can be found in 7-10 .

Abstract and Applied Analysis 3
In this note we pose the following subclass of bounded turning functions in the the unit disk: for given numbers > 0 and α > 0, let us consider the class B p : It is easy to see that f ∈ B p if and only if f z ≺ p : For the development of the paper we need to define a set Q by all points on the right halfplane as follows: Thus it is easy to verify that its boundary satisfies the equation of a parabola

Main Results
First, our result is in the following form. Moreover, if function f ∈ B p takes the form then the subordination relation holds.
Proof. Let f ∈ B p , and let
2 . If f is given in 2.2 with an analytic function p z ≺ p z , then by a differentiation of 2.2 we obtain that f z p z ; therefore, and consequently f ∈ B p . Now we proceed to prove that f ∈ B p . For this purpose we will show that the set Let w ∈ Q . Then Multiplying these inequalities, we obtain Therefore, w ∈ Q. Define a function q z , z ∈ U, as follows:

2.10
We suppose that and thus Hence, q z ≺ p z . Putting q z in 2.2 implies 2.3 . To prove the subordination relation 2.4 , first we show that f z /z is a convex function. We observe that λ α, z n ∈ A.

2.13
Abstract and Applied Analysis 5 Let us consider the function Also, a calculation implies that

2.17
The aim is to show that 1 zF z /F z has positive real part in the unit disk. For z > 0 and a suitable choice of 0 < ≤ 1 such that 0 < 1 − z/2 < 1 and by using 2.17 , we have Consequently, we obtain that F ∈ K, and therefore f z /z is a convex function. Now by using the fact that if F, G ∈ K, satisfy f ≺ F and g ≺ G, then f * g ≺ F * G and k z z/ 1 − z is a convex function, immediately we establish 2.4 . This completes the proof.
Next we consider another class of functions of bounded turning. We will estimate the upper bound of these functions by using the pre-Schwarzian norm. Proof. Let f ∈ B p , and let P f : f z . Then, there exists an analytic function w : U → U with w 0 0 and

2.21
Define a function F ∈ A such that P F −p , that is, We proceed to determine the quantities T F |z| and T f z . Thus, by triangle inequality and the Schwarz-Pick lemma, we obtain

2.26
Consequently, we have