A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions

and Applied Analysis 3 and, for k 1, 2, 3, . . ., we can write I λf z z −w ( Ik−1f z )′ 2 z −w 1 z −w ∞ ∑ n 1 1 λ n − 1 an z −w , 1.7 where λ ≥ 1, k ≥ 0 and z −w ∈ U . The differential operator I 1 is studied extensively by Ghanim and Darus 5, 6 and Ghanim et al. 7 . The Hadamard product or convolution of the functions f given by 1.3 with the function g and h given, respectively, by g z 1 z −w ∞ ∑ n 1 bn z −w , h z 1 z −w ∞ ∑ n 1 cn z −w , 1.8 can be expressed as follows: ( f ∗ g z 1 z −w ∞ ∑ n 1 anbn z −w , ( f ∗ h z 1 z −w ∞ ∑ n 1 ancn z −w . 1.9 Suppose that f and g are two analytic functions in the unit disk U. Then, we say that the function g is subordinate to the function f , and we write g z ≺ f z z ∈ U , 1.10 if there exists a Schwarz function z with 0 0 and | z | < 1 such that g z f z z ∈ U . 1.11 By applying the above subordination definition, we introduce here a new class Σw A, B, k, α, λ of meromorphically functions, which is defined as follows: Definition 1.1. A function f ∈ Σw of the form 1.3 is said to be in the class Σw A,B, k, α, λ if it satisfies the following subordination property: α I λ ( f ∗ g z I λ ( f ∗ h z ≺ α − A − B z −w 1 B z −w z −w ∈ U , 1.12 where −1 ≤ B < A ≤ 1, k ≥ 0, α > 0, λ ≥ 1, with condition I λ f ∗ h z / 0. 4 Abstract and Applied Analysis The purpose of this paper is to investigate the coefficient estimates, distortion properties, and the radius of starlikeness for the class Σw A,B, k, α, λ . Some applications of the main results involving generalized hypergeometric functions are also considered. 2. Characterization and Other Related Properties In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function f ∈ Σw of the form 1.3 to belong to the class Σw A,B, k, α, λ of meromorphically analytic functions. Theorem 2.1. The function f ∈ Σw is said to be a member of the class Σw A,B, k, α, λ if it satisfies ∞ ∑ n 1 1 λ n − 1 k αbn 1 B − cn α 1 B A − B an ≤ A − B. 2.1 The equality is attained for the function fn z given by fn z 1 z −w A − B 1 λ n − 1 k αbn 1 B − cn α 1 B A − B z −w . 2.2 Proof. Let f ∈ Σw A,B, k, α, λ , and suppose that α I λ ( f ∗ g z I λ ( f ∗ h z α − A − B z −w 1 B z −w . 2.3 Then, in view of 2.2 , we have ∣∣∣∣ α ∑∞ n 1 1 λ n − 1 an bn − cn z −w n 1 A − B −∞n 1 1 λ n − 1 an αBbn { A − B − αB}cn z −w n 1 ∣∣∣∣ ≤ α ∑∞ n 1 1 λ n − 1 an bn − cn |z −w| 1 A − B −∞n 1 1 λ n − 1 an αBbn { A − B − αB}cn |z −w| 1 ≤ 1. 2.4 Letting z −w → 1, we get ∞ ∑ n 1 1 λ n − 1 k αbn 1 B − cn α 1 B A − B an ≤ A − B , 2.5 which is equivalent to our condition of the theorem, so that f ∈ Σw A,B, k, α, λ . Hence we have the theorem. Abstract and Applied Analysis 5 Theorem 2.1 immediately yields the following result. Corollary 2.2. If the function f ∈ Σw belongs to the class Σw A,B, k, α, λ , thenand Applied Analysis 5 Theorem 2.1 immediately yields the following result. Corollary 2.2. If the function f ∈ Σw belongs to the class Σw A,B, k, α, λ , then an ≤ A − B 1 λ n − 1 k αbn 1 B − cn α 1 B A − B , 2.6 n ≥ 1, where the equality holds true for the functions fn z given by 2.2 . We now state the following growth and distortion properties for the class Σw A, B, k, α, λ . Theorem 2.3. If the function f defined by 1.3 is in the class Σw A,B, k, α, λ , then for 0 < |z−w| r < 1, one has 1 r − A − B αb1 1 B − c1 α 1 B A − B r ≤ ∣f z ∣ ≤ 1 r A − B αb1 1 B − c1 α 1 B A − B r, 1 r2 − A − B αb1 1 B − c1 α 1 B A − B ≤ ∣f ′ z ∣ ≤ 1 r2 A − B αb1 1 B − c1 α 1 B A − B . 2.7 Proof. Since f ∈ Σw A,B, k, α, λ , Theorem 2.1 readily yields the inequality ∞ ∑ n 1 an ≤ A − B αb1 1 B − c1 α 1 B A − B . 2.8 Thus, for 0 < |z −w| r < 1 and utilizing 2.8 , we have 6 Abstract and Applied Analysis ∣ ∣f z ∣ ∣ ≤ 1 |z −w| m ∑ n 1 an| z −w | ≤ 1 r r m ∑ n 1 an ≤ 1 r A − B αb1 1 B − c1 α 1 B A − B r, ∣ ∣f z ∣ ∣ ≥ 1 |z −w| − m ∑ n 1 an| z −w | ≥ 1 r − r m ∑ n 1 an ≥ 1 r − A − B αb1 1 B − c1 α 1 B A − B r. 2.9 Also, from Theorem 2.1, we get ∞ ∑ n 1 nan ≤ A − B αb1 1 B − c1 α 1 B A − B . 2.10


Introduction
Let A be the class of functions f which are analytic in the open unit disk U {z ∈ C : |z| < 1}. 1.1 As usual, we denote by S the subclass of A, consisting of functions which are also univalent in U.
Let w be a fixed point in U and A w {f ∈ H D : f w f w − 1 0}.In 1 , Kanas and Ronning introduced the following classes 2 Abstract and Applied Analysis Later, Acu and Owa 2 studied the classes extensively.
The class ST w is defined by geometric property that the image of any circular arc centered at w is starlike with respect to f w , and the corresponding class S c w is defined by the property that the image of any circular arc centered at w is convex.We observed that the definitions are somewhat similar to the ones introduced by Goodman in 3, 4 for uniformly starlike and convex functions except that, in this case, the point w is fixed.
Let Σ w denote the subclass of A w consisting of the function of the form The functions f in Σ w are said to be starlike functions of order β if and only if for some β 0 ≤ β < 1 .We denote by S * w β the class of all starlike functions of order β.Similarly, a function f in S w is said to be convex of order β if and only if for some β 0 ≤ β < 1 .We denote by C w β the class of all convex functions of order β.
For the function f ∈ Σ w , we define Abstract and Applied Analysis 3 and, for k 1, 2, 3, . .., we can write The differential operator I k 1 is studied extensively by Ghanim and Darus 5, 6 and Ghanim et al. 7 .
The Hadamard product or convolution of the functions f given by 1.3 with the function g and h given, respectively, by 1.8 can be expressed as follows: 1.9 Suppose that f and g are two analytic functions in the unit disk U.Then, we say that the function g is subordinate to the function f, and we write if there exists a Schwarz function z with 0 0 and By applying the above subordination definition, we introduce here a new class Σ w A, B, k, α, λ of meromorphically functions, which is defined as follows: 3 is said to be in the class Σ w A, B, k, α, λ if it satisfies the following subordination property: where The purpose of this paper is to investigate the coefficient estimates, distortion properties, and the radius of starlikeness for the class Σ w A, B, k, α, λ .Some applications of the main results involving generalized hypergeometric functions are also considered.

Characterization and Other Related Properties
In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function f ∈ Σ w of the form 1.3 to belong to the class Σ w A, B, k, α, λ of meromorphically analytic functions.
Theorem 2.1.The function f ∈ Σ w is said to be a member of the class The equality is attained for the function f n z given by Then, in view of 2.2 , we have

2.4
Letting z − w → 1, we get which is equivalent to our condition of the theorem, so that f ∈ Σ w A, B, k, α, λ .Hence we have the theorem.
n ≥ 1, where the equality holds true for the functions f n z given by 2.2 .
We now state the following growth and distortion properties for the class Σ w A, B, k, α, λ .

2.9
Also, from Theorem 2.1, we get

2.11
This completes the proof of Theorem 2.3.

Abstract and Applied Analysis 7
We next determine the radius of meromorphically starlikeness of the class Σ w A, B, k, α, λ , which is given by Theorem 2.4.Theorem 2.4.If the function f defined by 1.3 is in the class Σ w A, B, k, α, λ , then f is meromorphically starlike of order δ in the disk |z − w| < r 1 , where

2.12
The equality is attained for the function f n z given by 2.2 .
Proof.It suffices to prove that

2.14
Hence 2.14 holds true for With the aid of 2.1 and 2.16 , it is true to say that for fixed n

8
Abstract and Applied Analysis Solving 2.17 for |z − w|, we obtain

2.18
This completes the proof of Theorem 2.4.

Applications Involving Generalized Hypergeometric Functions
Let us define the function φ a, c; z by for c / 0, −1, −2, . .., and a ∈ C/{0}, where λ n λ λ 1 n 1 is the Pochhammer symbol.We note that φ a, c; z Corresponding to the function φ a, c; z and using the Hadamard product which was defined earlier in the introduction section for f z ∈ Σ, we define here a new linear operator L * a, c on Σ by For a function f ∈ L * w a, c f z , we define and, for k 1, 2, 3, . ..,

3.6
We note I k L * w a, a f z studied by Ghanim and Darus 5, 6 and Ghanim et al. 7 , and also, I k L * 0 a, c f z studied by Ghanim and Darus 8, 9 and Ghanim et al. 10 .The subordination relation 1.12 in conjunction with 3.4 and 3.6 takes the following form: The equality is attained for the function f n z given by Proof.By using the same technique employed in the proof of Theorem 2.1 along with Definition 3.1, we can prove Theorem 3.2.
The following consequences of Theorem 3.2 can be deduced by applying 3.8 and 3.9 along with Definition 3.1.

3.11
The equality is attained for the function f n z given by 3.9 .
A slight background related to the formation of the present operator can be found in 11 , and other work can be tackled using this type of operator.Also, the meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava 12, 13 , Liu 14 , Liu and Srivastava 15 , and Cho and Kim 16 .

Abstract and Applied Analysis 5 Theorem 2 . 1 Corollary 2 . 2 .
immediately yields the following result.If the function f ∈ Σ w belongs to the class Σ w A, B, k, α, λ , then

Corollary 3 . 3 .. 10 n ≥ 1 , 9 . 3 . 4 . 3 inf n≥1 1 −1/ n 1 .
If the function f ∈ Σ w belongs to the class Σ w A, B, k, α, a, c , thena n ≤ A − B | c n 1 | n k αb n 1 B − c n α 1 B A − B | a n 1 | , 3where the equality holds true for the functions f n z given by 3.Corollary If the function f defined by 1.3 is in the class Σ w A, B, k, α, a, c , then f is meromorphically starlike of order δ in the disk |z − w| < r 3 , wherer δ αb n 1 B − c n α 1 B A − B | c n 1 | n 2 − δ A − B | a n 1 | the form 1.3 is said to be in the class Σ w A, B, k, α, a, c if it satisfies the subordination relation 3.7 above.
Theorem 3.2.The function f ∈ Σ w is said to be a member of the class Σ w A, B, k, α, a, c if it satisfies ∞ n 1