ON GENERALIZED K-UNIFORMLY CLOSE-TO-CONVEX FUNCTIONS OF JANOWSKI TYPE

This work is concerned with the class of analytic functions that maps open unit disk onto conic domains. Necessary condition, arc length, growth rate of coefficients, radius problems and property of some integral transformation under the class are examined.


Introduction
Let A be the class of functions f (z) whose series representation is given by f (z) = z + ∞ n=2 a n z n , (1.1) and regular in an open unit disk ∆ = {z : | z| < 1}. Let S denotes the class of univalent functions in ∆ and C(ρ), S * (ρ) and K(ρ), 0 ≤ ρ < 1 be the well known subclasses of S which consist of convex, starlike and close-to-convex functions of order ρ respectively. C(0) ≡ C, S * (0) ≡ S * and K(0) ≡ K are the classes of convex, starlike and close-to-convex functions in ∆ respectively. A function f ∈ A is subordinate to g ∈ A (written as f (z) ≺ g(z)) if there exists a function w(z) with |w(z)| < 1 and w(0) = 0 such that f (z) = g(w(z)). In addition, if g(z) is univalent in ∆, then f (0) = g(0) and f (∆) ⊂ g(∆) [4].
Let H be the class of functions p(z) = 1 + where h ∈ P[1, −1] = P, the class of functions with positive real part. This class of functions was first considered and study extensively by Janowski [6].
Kanas and Wisniowska [7,8] introduced the class of k-uniformly convex functions and k-uniformly starlike function denoted by k − U CV and k − U ST respectively, which were defined subject to the conic domain Ω k , k ≥ 0, given by This domain represents the right half plane for k = 0, the right branch of hyperbola for 0 < k < 1 and an ellipse when k > 1. The function p k (z) plays an extremal role for all functions that maps ∆ onto Ω k and it is given by tz , t ∈ (0, 1), z ∈ ∆ and t is chosen such that k = cosh( πR (t) 4R(t) ), R(t) is Legendre's complete elliptic integral of the first kind and R (t) is the complementary integral of R(t) [7].
We denote by P (p k ), the class of functions that are subordinate to p k (z). Ronning [20] proved that for p ∈ P (p k ), there exists a function h ∈ P such that p(z) = h γ (z) and γ is given as : It was also shown in [9] that for p k (z) = 1 + δ k z + · · · ∈ P (p k ), or alternatively, Geometrically, the effect of Ω k [A, B] on Ω k was described in [18].
where p k is given by (1.3). Also, it is worthy to note that p ∈ k − P [A, B] ⊂ P(β 1 ) which implies that [18]) where h 1 ∈ P and β 1 is given by We extend the class k − P [A, B] as follows : (1.11) For k = 0, A = 1, B = −1, α = 0, we have the class P µ introduced and studied in [22]. Also, when µ = 2, α = 0, we get the class k − P [A, B], which was first considered by K.I. Noor in [18]. The class is the same as the class k − P µ (γ * ) studied in [17].
We now define the following classes of functions.
It is obvious to note that is the class of functions of bounded boundary rotation(see [1], [22]) or equivalently as We note the following special cases.
, is the class of functions studied by Silvia [23].
is the class of close to convex functions examined by Kaplan [10].
is the class of uniformly close to convex of order ρ, −1 ≤ ρ < 1 that was taken into account in [24].
, is the class of functions that was studied by K.I.
where Re a > 0, Re(c − a) > 0, Γ denotes the Gamma function and G(a, b, c; z) is hypergeometric function.
Unless if otherwise stated, we lay down once and for all that

Some Preliminary Lemmas
We need the following lemmas.
[1] Every function f ∈ V µ is a close to convex function of order µ 2 − 1.
Then for µ > 3 there exists s 1 ∈ S * (0) and p ∈ P such that

Main Result
In this section, we present our main work.
Thus, there exist f 1 , f 2 ∈ C(β) such that Integrating and using the result due to Brannan [1], we have where γ is given by (1.4).
Proof. Let By logarithmic differentiation and using Lemma 2.1, it follows that Therefore, Re This means that θ2 θ1 Re Using Goodman result in [3], we have the following.  Re Then where γ is given by (1.4).
Proof. By Cauchy Theorem, we have for z = re iθ n|a n | ≤ 1 2πr n 2π 0 |zf (z)|dθ = 1 2πr n L r (f ). Now, by applying Theorem 3.6 and setting r = 1 − 1 n as n → ∞, we have the result. Proof. For f ∈ k − T µ [A, B, C, D, α], we write where s 1 ∈ S * , p 1 ∈ P and h 1 ∈ P [A, B]. Let z 1 be complex number with |z 1 | = r. Then by Cauchy Theorem, Since s 1 ∈ S * , then by a result due to Golusin [2], we have Now, using Distortion Theorem for starlike function s 1 (z), Holder's inequality, Corollary 3.8 and Lemma 2.3, we have where D 1 is a constant. Setting r = 1 − 1 n and z 1 = n n+1 , the proof is completed.  where and γ, β are respectively given by (1.4) and (3.1).
Proof. We set where φ ∈ V µ and h ∈ P [A, B]. By logarithmic differentiation, we have For h ∈ P [A, B] and φ ∈ V µ , it is known in [6] and [22] respectively that Using these results, we have that If we let Then Thus, r ∈ (0, 1). Hence, the theorem is proved.
onto a convex domain Using the well-known distortion theorems for φ ∈ V µ and h 1 ∈ P, we prove the following. and where To calculate I 1 , let ξ = r 1 u and r 1 = 1+r 1−r . Then Using (3.10), (3.11) and (3.12), the upper bound is obvious. Now, we proceed to calculate the lower bound. Let d r denotes the radius of the largest schlicht disk centered at the origin contained in the image of | z | < r under the function f (z). Then there exists a point The ray from 0 to f (z) lies entirely in the image of ∆ and the inverse image of this ray is a curve Γ in | z | < r. Thus, Let ν = 1−ρ 1+ρ . Then dρ = −2 (1+ν) 2 dν and 1 − ρ = 2ν 1+ν . Going through the same process as it has been done for the upper bound, we easily obtain the lower bound and this completes the proof.
Differentiating logarithmically and with some simple computations, we have .

This implies
Differentiating logarithmically once again, we have .