On Certain Classes of Meromorphic Functions Associated with Conic Domains

and Applied Analysis 3 For k ∈ 0, 1 , define the domain Ωk as follows, see 3 : Ωk { u iv : u > k √ u − 1 2 v2 } . 1.10 For fixed k, Ωk represents the conic region bounded successively, by the imaginary axis k 0 , the right branch of hyperbola 0 < k < 1 , and a parabola k 1 . Related with Ωk, the domain Ωk,γ can be defined as below, see 4 : Ωk,γ ( 1 − γΩk γ, ( 0 ≤ γ < 1. 1.11 The functions pk,γ z with pk,γ 0 1, p′ k,γ 0 > 0, univalent in E {z : |z| < 1}, map E onto Ωk,γ and are given as in the following: pk,γ z ⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪ ⎪ ⎪⎩ 1 ( 1 − 2γz 1 − z , k 0 , 1 2 ( 1 − γ π2 ( log 1 √ z 1 − √z )2 , k 1 , 1 2 ( 1 − γ 1 − k2 sinh 2 [( 2 π arc cos k ) arc tanh √ z ] , 0 < k < 1 . 1.12 The functions pk,γ z are continuous as regard to k, have real coefficients for k ∈ 0, 1 , and play the part of extremal ones for many problems related to Ωk,γ . Let P be the class of analytic functions with positive real part, and let P pk,γ ⊂ P be the class of functions p z which are analytic in E, p 0 1 such that p z ≺ pk,γ z for z ∈ E, where “≺” denotes subordination, and pk,γ z are given by 1.12 . We define the following. Definition 1.1. Let p z be analytic in E with p 0 1. Then p z is said to belong to the class Pm pk,γ , form ≥ 2, 0 ≤ γ < 1, k ≥ 0, if and only if there exist p1, p2 ∈ P pk,γ such that p z ( m 4 1 2 ) p1 z − ( m 4 − 1 2 ) p2 z , z ∈ E. 1.13 We note that i k 0, Pm p0,γ Pm γ and with m 2, P2 γ coincides with P γ and p ∈ P γ implies Re p z > γ in E; ii when k 0, γ 0, we have the class Pm introduced in 5 . Definition 1.2. Let f ∈ . Then f z is said to belong to the class k −MRm γ if and only if −zf ′/f ∈ Pm pk,γ in E. For k 0, m 2, we obtain the class ∑∗ γ of meromorphic starlike functions of order γ . We can define the class k −MVm γ by the following relation: f ∈ k −MVm ( γ ) if -zf ′ ∈ k −MRm ( γ ) . 1.14 When k 0,m 2, γ 0, we obtain ∑ c of meromorphic convex functions. 4 Abstract and Applied Analysis Definition 1.3. Let f ∈ . Then f ∈ k −MRm γ if and only if Df ∈ k −MRm γ for z ∈ E∗. Similarly f ∈ k −MVα m γ if and only if Df ∈ k −MVm γ . We note that the classes k −MRm γ and k −MVα m γ are related by relation 1.14 . For k 0, γ 0, we have 0 −MVm 0 MVm, the class of meromorphic functions of bounded boundary rotation which was studied in 6 . The functions f ∈ MVm have integral representation of the form f ′ z − 1 z2 exp ∫2π 0 log ( 1 − ze−it ) dμ t } , 1.15 where μ t is a real-valued function of bounded variation on 0, 2π satisfying the conditions


Introduction
Let denote the class of functions f of the form a n z n , 1.1 which are analytic in punctured unit disc E * {z : 0 < |z| < 1} E \{0}.At z 0, the function f z has a simple pole.Let, for 0 ≤ β < 1, * β and c β be well-known subclasses of consisting of functions meromorphic starlike and meromorphic convex of order β, respectively, see 1 .
Let f, g ∈ , g z 1/z ∞ n 0 b n z n and f be given by 1.1 .Then convolution Hadamard product f * g of f and g is defined by Robertson 2 showed that f * g also belongs to .

Abstract and Applied Analysis
Let f ∈ and define, for α > −1, z ∈ E * , D α : → as It can easily be seen that, for α n ∈ N 0 {0, 1, 2, . ..}, We note that and so Equation 1.6 can be verified as follows.Since 1.8 From 1.3 , we can readily obtain the following identity for f ∈ and α > −1: D α , α > 1 is known as generalized Ruscheweyh derivative for meromorphic functions.

Abstract and Applied Analysis 3
For k ∈ 0, 1 , define the domain Ω k as follows, see 3 : 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 , and a parabola k 1 .Related with Ω k , the domain Ω k,γ can be defined as below, see 4 : The functions p k,γ z with p k,γ 0 1, p k,γ 0 > 0, univalent in E {z : |z| < 1}, map E onto Ω k,γ and are given as in the following:

1.12
The functions p k,γ z are continuous as regard to k, have real coefficients for k ∈ 0, 1 , and play the part of extremal ones for many problems related to Ω k,γ .
Let P be the class of analytic functions with positive real part, and let P p k,γ ⊂ P be the class of functions p z which are analytic in E, p 0 1 such that p z ≺ p k,γ z for z ∈ E, where "≺" denotes subordination, and p k,γ z are given by 1.12 .
We define the following.
Definition 1.1.Let p z be analytic in E with p 0 1.Then p z is said to belong to the class P m p k,γ , for m ≥ 2, 0 ≤ γ < 1, k ≥ 0, if and only if there exist p 1 , p 2 ∈ P p k,γ such that We note that i k 0, P m p 0,γ P m γ and with m 2, P 2 γ coincides with P γ and p ∈ P γ implies Re p z > γ in E; ii when k 0, γ 0, we have the class P m introduced in 5 .
For k 0, m 2, we obtain the class * γ of meromorphic starlike functions of order γ.
We can define the class k − MV m γ by the following relation: When k 0, m 2, γ 0, we obtain c of meromorphic convex functions.
We note that the classes k − MR α m γ and k − MV α m γ are related by relation 1.14 .For k 0, γ 0, we have 0 − MV m 0 MV m , the class of meromorphic functions of bounded boundary rotation which was studied in 6 .The functions f ∈ MV m have integral representation of the form where μ t is a real-valued function of bounded variation on 0, 2π satisfying the conditions With simple computations, it can easily be seen that the third of conditions 1.16 guarantees that the singularity of f z at z 0 is a simple pole with no logarithm term.
Also it is known that f ∈ MV m if and only if f E is a domain containing infinity with boundary rotation at most mπ, see 6 .The class V m is wellknown 1 and consists of analytic functions with boundary rotation at most mπ.Noonan 6 established the relation between the classes V m and MV m as follows.
A function f ∈ MV m if and only if there exists g ∈ V m of the form g z z

1.18
We note that φ i ∈ * and therefore 1/φ i ∈ S * of analytic functions, i 1, 2 in 1.18 .This give us by distortion results and subordination for the class S * .
We can easily extend the relations 1.17 and 1.18 by noting that f ∈ V m γ implies that there exists Throughout this paper, we will assume k ∈ 0, 1 , m ≥ 2, γ ∈ 0, 1 and α > 1 unless otherwise stated.
We also note that all the results proved in this paper hold for k ≥ 0 in general.

Preliminary Results
The following lemma is a generalized version of a result proved in 3 .
Lemma 2.1 see 4 .Let 0 ≤ k < ∞ 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 2.1 .
Lemma 2.2 see 8 .Let u u 1 iu 2 , v v 1 iv 2 and let ψ u, v be complex-valued function satisfying the following conditions: If h z 1 n 1 c n z n is a function analytic in E such that h z , zh z ∈ D, and Re ψ h z , zh z > 0 for z ∈ E, then Re h z > 0 for z ∈ E. Lemma 2.3 see 9 .Let p and q be analytic in E and Re p z ≥ 0, q 0 1 and Re q z > 0 for z ∈ E. Further let A / 0 and B be complex constants such that A B / 0. Then where

2.5
This result is sharp for A real and nonnegative constant.

Main Results
Theorem 3.1.One has This result is best possible and sharpness follows from the best dominant property.
Proof.Let f ∈ k − MR α 1 m γ , and set Then Using identity 1.9 , it follows that is analytic in |z| < r 1 and p 0 1.

Abstract and Applied Analysis 7
Now, from 1.9 and 3.4 , we obtain

3.9
Then using convolution technique, we have

3.10
Thus, from 3.7 and 3.10 , we obtain Abstract and Applied Analysis It can easily be seen that Re βk/ k 1 δ > γ, so we apply Lemma 2.1 to have from 3.11 where q k,γ z is the best dominant and is given as

3.15
Proof.From 3.6 , we have

3.16
Proceeding as in Theorem 3.1, it follows that

3.18
We construct a functional ψ u, v by taking u H i z , v zH i z .Then The first two conditions of Lemma 2.2 are easily verified.For condition iii , we proceed as follows: Re

3.20
By putting where

3.22
From A 1 ≤ 0, we obtain γ 1 as given by 3.15 and Applying Lemma 2.2, we now have H i ∈ P , i 1, 2 and therefore h i ∈ P γ 1 in E, consequently h ∈ P m γ 1 in E and the proof is complete.
We note that, for α 0, we have

3.23
Also, for α 1, γ 1/3 and γ 1 ≈ 6/21.We will now investigate the rate of growth of coefficients for f ∈ k − MV α m γ and the corresponding result for the class k − MR α m γ will follow from the relation 1.14 .
and O 1 depends only on m and σ.
The exponent {β 1 − α 2 } in 3.24 is best possible for the class MV α m σ as can be seen from the function f 0 ∈ MV α m σ given by A n z n and z re iθ , 0 < r < 1, we have 3.27 Pommerenke 10 has shown that 2π 0 1

3.28
Thus we use 1.19 , 3.26 , and 3.28 to have from 3.27

3.29
We take r 1 − 1/n , A n Γ n α 2 / Γ α 1 n 1 !a n , Γ denotes gamma function, and have where β 1 is as given in 3.24 and O 1 is a constant depending only on m and σ.This completes the proof.
Next we will show that the class k − MR α m γ is preserved under an integral operator.For Re c > 0, the generalized Bernardi operator for the class is defined in 11 as below.
Let f ∈ Σ and be given by 1.1 .Then the integral transform F c is defined as a n z n .

3.32
It easily follows from 3.31 that and as in 1.9 ,

3.34
From 3.33 and 3.34 , we have

3.35
We now prove the following.Proof.We put Then F 1 z is single valued and analytic in |z| < r 1 and H z defined by is analytic in |z| < r 1 , H 0 1.Form 3.33 , 3.34 , and 3.37 , we obtain and with the convolution technique used before, we have, for i 1, 2

3.40
Since Re βk/ k 1 δ > γ, we apply Lemma 2.1 to have H i z ≺ q k,γ z ≺ p k,γ z , where q k,γ z is the best dominant.The required result now follows from 3.37 .

3.41
The proof follows on the similar lines of Corollary 3.2.

3.42
Proof.We write

3.43
Since F ∈ MR α m σ , p ∈ P m in E with p 0 1, and p i ∈ P , i 1, 2. Proceeding on the similar lines as before, we obtain form 3.42 and 3.43 and with convolution technique as previously used, we get from 3.44

Theorem 3 . 4 .
Let f ∈ k − MR α m γ .ThenF z , defined by 3.31 , also belong to the same class in E * .

Theorem 3 . 7 .
Let F z be defined by 3.31 and let, for Re c > 0, F ∈ MR α m σ , σ k γ / 1 k .Then f ∈ MR α m σ for |z| < ρ, where ρ ρ A, B |A B| is given by 3.42 .Now, from 3.44 and 3.46 , we have the required result that f ∈ MR α m σ in |z| < ρ A, B .From Lemma 2.3 it follows that this result is sharp for c > 0.