Generalized k-Uniformly Close-to-Convex Functions Associated with Conic Regions

and Applied Analysis 3 Using 1.7 , the following family of linear operators, see 7–9 , is defined in terms of the Hadamard product as Jλ,μf z Hλ,μ z ∗ f z , 1.8


Introduction
Let A denote the class of analytic functions f defined in the unit disc E {z : |z| < 1} and satisfying the condition f 0 0, f 0 1.Let S, S * γ , C γ and K γ be the subclasses of A consisting of functions which are univalent, starlike of order γ, convex of order γ, and close-to-convex of order γ, respectively, 0 ≤ γ < 1.Let S * 0 S * , C 0 C and K 0 K.For analytic functions f z ∞ n 0 a n z n and g z ∞ n 0 b n z n , by f * g we denote the convolution Hadamard product of f and g, defined by 1.1 We say that a function f ∈ A is subordinate to a function F ∈ A and write f z ≺ F z if and only if there exists an analytic function w z , w 0 0, |w z | < 1 for z ∈ E such that f z F w z , z ∈ E.
If F is univalent in E, then For k ∈ 0, 1 , define the domain Ω k as follows, see 1 : For fixed k, Ω k represents the conic region bounded, successively, by the imaginary axis k 0 , the right branch of hyperbola 0 < k < 1 , a parabola k 1 .Related with Ω k , the domain Ω k,γ is defined in 2 as follows: The functions which play the role of extremal functions for the conic regions Ω k,γ are denoted by p k,γ z with p k,γ 0 1, and p k,γ 0 > 0 are univalent, map E onto Ω k,γ , and are given as 2   2 π arc cos k arc tanh √ z , 0 < k < 1 .

1.5
It has been shown 3, 4 that p k,γ z is continuous as regards to k and has real coefficients for all k ∈ 0, 1 .
Let P p k,γ be the class of functions p z which are analytic in E with p 0 1 such that p z ≺ p k,γ z for z ∈ E. It can easily be seen that P p k,γ ⊂ P , where P is the class of Caratheodory functions of positive real part.
The class P m p k,γ is defined in 5 as follows.
Let p z be analytic in For k 0, γ 0, the class P m p 0,0 coincides with the class P m introduced by Pinchuk in 6 .Also P 2 P .
The generalized Harwitz-Lerch Zeta function 7 φ z, λ, μ is given as Abstract and Applied Analysis 3 Using 1.7 , the following family of linear operators, see 7-9 , is defined in terms of the Hadamard product as and φ z, λ, μ is given by 1.7 .From 1.7 and 1.8 , we can write For the different permissible values of parameters λ and μ, the operator J λ,μ has been studied in 3, 4, 7, 10-12 .
We observe some special cases of the operator 1.10 as given below We remark that J 1,1 f z is the well-known Libera operator and J 1,μ f z is the generalized Bernardi operator, see 13, 14 .Also J λ,1 f z L λ f z represents the operator closely related to the multiplier transformation studied by Flett 3 .We define the operator I λ,μ : A → A as see 15 .This gives us From 1.12 , the following identity can easily be verified i For k ∈ 0, 1 , we note that the domain Ω k given by 1.3 represents the following hyperbolic region: 1.14 The extremal function p k,γ z , for 0 < k < 1, can be written as where p k z , in a simplified form, is given below and the branch of √ z is chosen such that Im √ z ≥ 0. It is easy to see that, for h ∈ P p k,γ , Re h z > k γ / k 1 , k ∈ 0, 1 .That is and the order k γ / 1 k is sharp with the extremal function p z 1 − γ p k z γ, where p k z is given by 1.16 .
It can easily be verified that Re p 1 z > 1/2 and, in this case, the order 1/2 is sharp.
We now define the following.
Definition 1.2.Let f ∈ A and let the operator I λ,μ f be defined by 1.12 .
We note the following.
ii 0 − ∪R 0 m 0, μ R m is the class of functions of bounded radius rotation, see 13, 14 .
iii We denote k − ∪R and we note that, for 0 ii For k γ λ 0, we obtain the class T m introduced and discussed in 17 .

Preliminary Results
We need the following results in our investigation.Lemma 2.1 see 18 .Let q z be convex in E and j : In the following, one gives an easy extension of a result proved in 1 .
Lemma 2.2 see 5 .Let k ≥ 0 and let β, δ be any complex numbers with β / 0 and Re βk/ k and q k,γ z is an analytic solution of and q k,γ z is the best dominant of 2.3 .
where Coh E denotes the convex hull of h E .
Lemma 2.4 see 18 .Let u u 1 i u 2 , v v 1 iv 2 and let ψ u, v be a complex-valued function satisfying the conditions: Lemma 2.7.Let p ∈ P m p k,γ and p z 1 where Now the proof follows immediately by using the well-known Rogosinski's result, see 21 .

Main Results
We shall assume throughout, unless stated otherwise, that We note H z is analytic in E with H 0 1.

8
Abstract and Applied Analysis From 3.1 , we have That is Logarithmic differentiation of 3.4 and simple computations give us where q k,γ z is the best dominant and is given as

3.10
Consequently it follows, from 3.2 , that H ∈ P m p k,γ and For k 0, γ 0, we have the following special case.

3.11
Proof.We write 3.12 and proceeding as in Theorem 3.1, we obtain

3.13
We construct the functional ψ u, v by taking u H i z ,v zH i z , as The first two conditions of Lemma 2.4 can easily be verified.For condition iii , we proceed as follows:

3.15
where The right-hand side of 3.15 is less than equal to zero when A ≤ 0 and B ≤ 0. From A ≤ 0, we obtain γ 1 as given by 3.11 , and B ≤ 0 ensures that γ 1 ∈ 0, 1 .
This shows that all the conditions of Lemma 2.4 are satisfied and therefore Re H i z > 0. This implies H ∈ P m and consequently By taking α 1, β 0, λ 0, and m 2, we obtain a well-known result that every convex function is starlike of order 1/2.Also, for β 1, λ 0, α 1, and m 2, we obtain from 3.1 the Libera operator and in this case we obtain a known result with γ 1 2/ 3 √ 17 for starlike functions, see 18 .
Assigning permissible values to different parameters, we obtain several new and known results from Theorem 3.1 and Corollary 3. Proof.We can write 3.1 as where

3.21
We use Lemma 2.3 with h i ≺ p k,γ , i 1, 2, to have As a special case we note that, for λ 0 k, the subclass T γ m ⊂ T m is invariant under the integral operator defined by 3.1 .

Theorem 3.4. One has
where H z is analytic in E and is defined by 3.2 .Then, from 1.13 , we have

3.25
Applying similar technique used before, we have from 3.2 and 3.7 for i 1, 2

3.26
Thus, using Lemma 2.2, it follows that H i ≺ p k,γ , i 1, 2 and z ∈ E, consequently H ∈ P m p k,γ in E and this completes the proof.
As special cases, we have the following.
i Let m 2, λ ≥ 0.Then, from Theorem 3.4, it easily follows that

3.31
Now, on using 1.13 , we have

3.32
Abstract and Applied Analysis 13 Differentiation of 3.30 gives us z z I λ,μ f z I λ,μ g z zh z h z h 0 z , 3.33 and using 3.33 in 3.32 , we obtain

Applications of Theorem 3.6
The classes k − ∪R γ 2 λ, μ and k − ∪T γ m λ, μ are preserved under the following integral operators: 1 The proof is immediate since φ i z is convex in E for i 1, 2, 3, 4.
With essentially the same method together with Lemma 2.7, we can easily prove the following sharp coefficient results.a n z n .

3.37
Then where ρ n is Pochhamer symbol defined, in terms of Gamma function Γ, by and δ k,γ is as given by 2.9 .
As special case, one notes that i λ 0, m 2, then one has see [2].

3.41
This coefficient bound is well known for m 2, see 2 .
Using Theorem 3.8 with m 2, the following result can easily be proved.
where δ k,γ is as given by 2.9 .
By assigning different permissible values to the parameters, we obtain several known results, see 2, 22 .
We now prove the following.

Theorem 3 . 8 .
Let f ∈ k − ∪R γ m λ, m and let it be given by

γm⊂z s 1 z m 2 / 4 s 2
R m k γ / 1 k , and it follows from a result proved in 5 that there exist s 1 , s 2 ∈ S * k γ / k 1 such that F
μ , we have with 1/H 0 z {h 0 z μ} ∈ P , Lemma 2.1, we have h i z ≺ p k,γ z in E. This shows h ∈ P m p k,γ in E and consequently f ∈ k − ∪T γ m λ, μ .As a special case, we note that f ∈ k − ∪T A n z n , A n n μ / 1 μ λ a n .Since f ∈ k −∪R