Generalized Alpha-Close-to-Convex Functions

We define the classes as follows: if and only if, for , , ; ; , where is a function of bounded boundary rotation. Coefficient estimates, an inclusion result, arclength problem, and some other properties of these classes are studied.


Introduction
Let A be the class of functions of the form: A class T k of analytic functions related with the class V k was introduced and studied in 2 .A function f ∈ A is in T k , k ≥ 2, if and only if there exists a function g ∈ V k such that, for z ∈ E, Re{f z /g z } > 0. It is clear that T 2 ≡ K.

International Journal of Mathematics and Mathematical Sciences
Let P denote the class of analytic functions p defined by with Re p z > 0 for z ∈ E.
We denote K γ as the class of strongly close-to-convex functions of order γ in the sense of Pommerenke 3 .A function f ∈ A belongs to K γ if and only if there exists g ∈ S * such that |Arg zf z /g z | ≤ πγ/2, for z ∈ E and γ ≥ 0.
Clearly K 0 C, K 1 K, and when 0 ≤ γ < 1, K γ is a subset of K and hence contains only univalent functions.For γ > 1, f ∈ K γ can be of infinite valence; see 4 .
We now define the following.
In 1.4 , we choose this branch of argument which equals β, |β| < πγ/2, γ ∈ 0, 1 , when z 0. We note that the condition |α| ≤ 1 implies that G β α, k, γ is nonempty.From the normalization conditions f 0 φ 0 1, it follows from Definition 1.1 that Re e −iβ > 0 and therefore |β| < γπ/2.Also, it follows from 1.4 that if f ∈ G β α, k, γ , then f z / 0 for z ∈ E. Condition 1.4 is equivalent to the following f ∈ G β α, k, γ if and only if there exists p ∈ P such that We define G α, k, γ the class of generalized α-close-to-convex functions as If α 0 in 1.6 , then the class G 0, k, 1 is identical with the class T k and G α, 2, 1 is the class K of close-to-convex functions.Also G β α, 2, 1 in the class of close-to-convex function with argument β was defined by Goodman and Saff 5 .For details of special cases of G β α, 2, 1 with φ z z in 1.4 , we refer to 6 .The special case with γ 1 α, k 2, and φ z z in 1.4 leads to the class of functions convex in the direction of the imaginary axis having special normalization; see 7 .

Main Results
We now prove the main results as follows.
Proof.We will use an extended version of the method given in 8 to prove this result.For α 0, the result is obvious.Let f ∈ G α, k, γ .By 1.4 , 1.5 , and 1.6 , then there exists a function φ ∈ V k and a function p ∈ P , |β| < π/2 such that We choose in 2.3 this branch of argument which is equal −β when z 0. Since where γ 1 is given by 2.1 .The constant γ 1 γ, α cannot be smaller.Let α ∈ 0, 1 be fixed.Let us consider the point z 0 ∈ C with |z 0 | 1 and Arg 1 − α 2 z 2 − arcsin α 2 .Let φ 0 ∈ V k be such that φ 0 z 0 is finite.Then, let where

2.5
Now, for z ∈ E, and Arg e −iβ −β.Since p 0 maps the unit circle |z| 1 onto imaginary axis, we may choose Therefore γ 1 cannot be smaller.

2.13
Proof.To prove this result, we shall essentially use the similar method given by Kaplan 9 .

2.17
The functions ℘, τ, ψ, and V are continuous and periodic with period 2π.From 1.4 , we can choose the branches of argument of ℘ z and τ z as

2.19
From 2.16 , 2.17 , and 2.19 , we have where ψ r, θ and R are defined by 2.13 .This completes the proof.
We note that, for γ 1, k 2, α 0, we obtain the necessary condition for f to be close-to-convex in E, proved in 9 .
Remark 2.3.From Theorem 2.2, we can interpret some geometrical meaning for the functions in G α, k, γ .For simplicity, let us suppose that the image domain is bounded by an analytic curve Γ.At a point on Γ, the outward drawn normal turns back at most − γ k/2 − 1 − R π, where A is given by 2.13 .This is a necessary condition for a function The functions in K γ k/2 − 1 − R need not even be finitely valent in E for k 2 γ − R > 4.
i Denote by L r, f the length of the image of the circle |z| r under f and by A r, f the area of f |z| ≤ r .Then, for 0 The function F σ , defined by 2.25 , shows that this upper bound is sharp.
where c α, k, r is a constant depending upon α, k, and γ only.

2.27
For φ ∈ V k , it is known 10 that there exist s 1 , s

2.32
Using Theorem 2.8 and putting r 1 − 1/n, we prove this result.
, and C denote the subclasses of A which are univalent, close-to-convex, starlike, and convex in E, respectively.Let V k be the class of functions of bounded boundary rotation.Paate 1 showed that a function f, defined by 1.1 and f z / 0, is in V k if and only if, for z re iθ , It is geometrically obvious that k ≥ 2 and V 2 ≡ C.
International Journal of Mathematics and Mathematical Sciences see 13 .Now, from 2.27 , 2.28 , and 2.29 , we have 2 ∈ S * such that