Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID pot

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 401913, 16 pages doi:10.1155/2011/401913 Research Article On Certain Subclasses of Meromorphically p-Valent n Functions Associated by the Linear Operator Dλ Amin Saif and Adem Kılıcman ¸ Department of Mathematics, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia Correspondence should be addressed to Adem Kılıcman, akilicman@putra.upm.edu.my ¸ Received 26 July 2010; Accepted 28 February 2011 Academic Editor: Jong Kim Copyright q 2011 A Saif and A Kılıcman This is an open access article distributed under the ¸ Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The purpose of this paper is to introduce two novel subclasses Γλ n, α, β and Γ∗ n, α, β of λ n meromorphic p-valent functions by using the linear operator Dλ Then we prove the necessary and sufficient conditions for a function f in order to be in the new classes Further we study several important properties such as coefficients inequalities, inclusion properties, the growth and distortion theorems, the radii of meromorphically p-valent starlikeness, convexity, and subordination properties We also prove that the results are sharp for a certain subclass of functions Introduction Let Σp denote the class of functions of the form f z z−p ∞ ak zk ak ≥ 0; p ∈ N {1, 2, } , k p 1.1 which are meromorphic and p-valent in the punctured unit disc U∗ {z ∈ C : < |z| < 1} n U − {0} For the functions f in the class Σp , we define a linear operator Dλ by the following form: Dλ f z pλ f z Dλ f z Dλ f z , λ≥0 , f z , Dλ f λzf z , z Dλ f z Dλ Dλ f z , 1.2 Journal of Inequalities and Applications and in general for n n Dλ f z 0, 1, 2, , we can write ∞ zp pλ n kλ ak zk , n ∈ N0 N ∪ {0}; p ∈ N k p 1.3 Then we can observe easily that for f ∈ Σp , n zλ Dλ f z n Dλ f z − n pλ Dλ f z , p ∈ N; n ∈ N0 1.4 Recall 1, that a function f ∈ Σp is said to be meromorphically starlike of order α if it is satisfying the following condition: Re − zf z f z > α, z ∈ U∗ , 1.5 for some α α < Similarly recall a function f ∈ Σp is said to be meromorphically convex of order α if it is satisfying the following condition: Re −1 − zf z f z z ∈ U∗ for some α ≤ α < > α, 1.6 Let Σp α be a subclass of Σp consisting the functions which satisfy the following inequality: Re − n z Dλ f z n Dλ f z > pα, z ∈ U∗ ; α ≥ 1.7 In the following definitions, we will define subclasses Γλ n, α, β and Γ∗ n, α, β by using the λ n linear operator Dλ Definition 1.1 For fixed parameters α ≥ 0, ≤ β < 1, the meromorphically p-valent function f z ∈ Σp α will be in the class Γλ n, α, β if it satisfies the following inequality: Re − n z Dλ f z n p Dλ f z ≥α n z Dλ f z n p Dλ f z β, n ∈ N0 1.8 Definition 1.2 For fixed parameters α ≥ 1/ β ; ≤ β < 1, the meromorphically p-valent function f z ∈ Σp α will be in the class Γ∗ n, α, β if it satisfies the following inequality: λ n z Dλ f z n p Dλ f z α αβ ≤ Re − n z Dλ f z n p Dλ f z α − αβ, ∀ n ∈ N0 1.9 Journal of Inequalities and Applications Meromorphically multivalent functions have been extensively studied by several authors, see for example, Aouf 4–6 , Joshi and Srivastava , Mogra 8, , Owa et al 10 , Srivastava et al 11 , Raina and Srivastava 12 , Uralegaddi and Ganigi 13 , Uralegaddi and Somanatha 14 , and Yang 15 Similarly, in 16 , some new subclasses of meromorphic functions in the punctured unit disk was considered In 17 , similar results were proved by using the p-valent functions that satisfy the following differential subordinations: z Ip r, λ f z p−j j Ip r, λ f z j ≺ a aB A − B β z a Bz 1.10 and studied the related coefficients inequalities with β complex number This paper is organized as follows It consists of four sections Sections and investigate the various important properties and characteristics of the classes Γλ n, α, β and Γ∗ n, α, β by giving the necessary and sufficient conditions Further we study the growth λ and distortion theorems and determine the radii of meromorphically p-valent starlikeness of order μ ≤ μ < p and meromorphically p-valent convexity of order μ ≤ μ < p In Section we give some results related to the subordination properties Properties of the Class Γλ n, α, β We begin by giving the necessary and sufficient conditions for functions f in order to be in the class Γλ n, α, β Lemma 2.1 see Let Ra ⎧ ⎪ ⎪a − α β , ⎪ ⎪ ⎪ α ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1−a for a ≤ 1− α2 for a ≥ −2 1−β 1−a , 1−β , α α 1−β α α 2.1 Then {w : |w − a| ≤ Ra } ⊆ w : Re w ≥ α|w − 1| β 2.2 Theorem 2.2 Let f ∈ Σp Then f is in the class Γλ n, α, β if and only if ∞ p α β k α kλ pλ n ak ≤ p − β k p α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 2.3 Journal of Inequalities and Applications Proof Suppose that f ∈ Γλ n, α, β Then by the inequalities 1.3 and 1.8 , we get that Re − n z Dλ f z ≥α n p Dλ f z n z Dλ f z β n p Dλ f z 2.4 That is, 1− Re ≥α n ∞ pλ ak zk p k p k/p kλ n ∞ pλ ak zk p k p kλ ∞ k p k/p ∞ k p 1 ∞ k p ≥ Re α · Re kλ kλ ∞ k p ∞ k p n ak zk pλ k/p β kλ kλ ∞ k p β p n n ak zk β kλ p ak zk pλ pλ α k/p n ak zk pλ n ak zk pλ n β p pλ ak zk pβ kλ pλ kλ 2.5 p p p , that is, Re ∞ k p k p 1−β − ∞ k p kα pα kλ n ak zk pλ n ak zk p ≥ p 2.6 Taking z to be real and putting z → 1− through real values, then the inequality 2.6 yields ∞ k p k p 1−β − ∞ k p kα pα kλ pβ kλ pλ n ak n pλ ak ≥ 0, 2.7 which leads us at once to 2.3 In order to prove the converse, suppose that the inequality 2.3 holds true In Lemma 2.1, since ≤ 1 − β /α α , put a Then for p ∈ N and n ∈ N0 , let n n −z Dλ f z /p Dλ f z If we let z ∈ ∂U∗ {z ∈ C : |z| 1}, we get from the wnp inequalities 1.3 and 2.3 that |wnp − 1| ≤ R1 Thus by Lemma 2.1 above, we ge that Re − n z Dλ f z n p Dλ f z −1 Re wnp ≥ α wnp − α n z Dλ f z n p Dλ f z β, β α− n z Dλ f z n p Dλ f z −1 β α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 2.8 Therefore by the maximum modulus theorem, we obtain f ∈ Γλ n, α, β Journal of Inequalities and Applications Corollary 2.3 If f ∈ Γλ n, α, β , then p 1−β ak ≤ p α β k kλ α pλ n, α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 2.9 The result is sharp for the function f z given by f z ∞ z−p p 1−β k p p α β k α kλ pλ k nz , α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 2.10 Theorem 2.4 The class Γλ n, α, β is closed under convex linear combinations Proof Suppose the function f z ∞ z−p ak zk,j ak,j ≥ 0; j 1, 2; p ∈ N , k p 2.11 be in the class Γλ n, α, β It is sufficient to show that the function h z defined by hz − δ f1 z δf2 z 0≤δ≤1 , 2.12 is also in the class Γλ n, α, β Since ∞ z−p h z δak,2 zk,j , − δ ak,1 0≤δ≤1 , k p 2.13 and by Theorem 2.2, we get that ∞ p α β k α kλ pλ k n α − δ ak,1 δak,2 k p ∞ 1−δ p α β kλ n pλ ak,1 k p ∞ 2.14 δ p α β k α kλ pλ n ak,2 k p ≤ 1−δ p 1−β δp − β p 1−β , α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 Hence f ∈ Γλ n, α, β The following are the growth and distortion theorems for the class Γλ n, α, β Journal of Inequalities and Applications Theorem 2.5 If f ∈ Γλ n, α, β , then p m−1 ! p−1 ! p ≤ − 1−β β 2α p n−1 p!2−n · p−m ! 1−β m−1 ! p−1 ! β 2α p n−1 · r 2p r − p m ≤ f p!2−n 2p − p r r p−m ! m z 2.15 m r < 1; α ≥ 0; ≤ β < 1; p ∈ N; n, m ∈ N0 ; p > m < |z| The result is sharp for the function f given by ∞ z−p f z 1−β β 2α k p 1 2p p nz , n ∈ N0 ; p ∈ N 2.16 Proof From Theorem 2.2, we get that p 2α β 2p p n ! ∞ ∞ k!ak ≤ k p p α β k α kλ pλ n ak k p 2.17 ≤p 1−β , that is, ∞ k!ak ≤ k p p 1−β p p 2α β − β p!2−n ! 2p n 2α β p n−1 2.18 By the differentiating the function f in the form 1.1 m times with respect to z, we get that fm z −1 m p m−1 ! p−1 ! z− p ∞ m k p k! ak zk−m , k−m ! m ∈ N0 ; p ∈ N 2.19 and Theorem 2.5 follows easily from 2.18 and 2.19 Finally, it is easy to see that the bounds in 2.15 are attained for the function f given by 2.18 Next we determine the radii of meromorphically p-valent starlikeness of order μ ≤ μ < p and meromorphically p-valent convexity of order μ ≤ μ < p for the class Γλ n, α, β Theorem 2.6 If f ∈ Γλ n, α, β , then f is meromorphically p-valent starlike of order μ ≤ μ < in the disk |z| < r1 , that is, Re − zf z f z >μ ≤ μ < p; |z| < r1 ; p ∈ N , 2.20 Journal of Inequalities and Applications where r1 p−μ p α inf β k p k k≥p kλ α pλ 1/ k p n μ 1−β 2.21 Proof By the form 1.1 , we get that zf z /f z zf z /f z p −p p−μ 2μ ≤ ∞ p ak zk k p k ∞ z−p k p k−p ∞ k p p − μ ak |z|−p ∞ k p p |z|k ∞ k p k−p p ak |z|k k ∞ k p p−μ k k−p 2μ ak zk 2μ ak |z|k 2.22 p 2μ ak |z|k p Then the following incurability zf z /f z p −p zf z /f z 2μ ≤ 1, ≤ μ < p; p ∈ N 2.23 also holds if ∞ k μ ak |z|k p−μ k p p ≤ 1, ≤ μ < p; p ∈ N 2.24 Then by Corollary 2.3 the inequality 2.24 will be true if k μ p−μ |z|k p ≤ p α β k α kλ pλ kλ pλ n , p 1−β ≤ μ < p; p ∈ N , 2.25 , ≤ μ < p; p ∈ N 2.26 that is, |z|k p ≤ p−μ p α β p k k α μ 1−β n Therefore the inequality 2.26 leads us to the disc |z| < r1 , where r1 is given by the form 2.21 Theorem 2.7 If f ∈ Γλ n, α, β , then f is meromorphically p-valent convex of order μ ≤ μ < in the disk |z| < r2 , that is, Re −1 − zf z f z >μ ≤ μ < p; |z| < r2 ; p ∈ N , 2.27 Journal of Inequalities and Applications where r2 p−μ inf α β k k k k≥p α kλ pλ 1/ k p n μ 1−β 2.28 Proof By the form 1.1 , we get that −p zf z /f z ∞ k p p zf z /f z ∞ k p 2p p − μ z−p 2μ ≤ ∞ k p ∞ k p p |z|k ∞ k p ∞ k p k k−p p ak |z|k k k 2μ ak zk k k−p k k 2p p − μ ak |z|−p 2p p − μ p ak zk k k k k−p 2μ ak |z|k 2.29 p 2μ ak |z|k p Then the following incurability: p zf z /f z −p zf z /f z 2μ ≤ 1, ≤ μ < p; p ∈ N 2.30 will hold if ∞ k p k k μ 1p p − μ ak |z|k p ≤ 1, ≤ μ < p; p ∈ N 2.31 Then by Corollary 2.3 the inequality 2.31 will be true if k k μ p p−μ |z|k p ≤ p α β k kλ α pλ n , p 1−β ≤ μ < p; p ∈ N , 2.32 that is, |z| k p ≤ p−μ α β k k k α kλ μ 1−β pλ n , ≤ μ < p; p ∈ N 2.33 Therefore the inequality 2.33 leads us to the disc |z| < r2 , where r2 is given by the form 2.28 Properties of the Class Γ∗ n, α, β λ We first give the necessary and sufficient conditions for functions f in order to be in the class Γ∗ n, α, β λ Journal of Inequalities and Applications Lemma 3.1 see Let μ > δ and ⎧ ⎨a − δ, for a ≤ 2μ δ, ⎩2 μ a − μ − δ , Ra for a ≥ 2μ δ 3.1 Then {w : |w − a| ≤ Ra } ⊆ w : w − μ δ ≤ Re w μ−δ 3.2 Lemma 3.2 Let α ≥ and ≤ β < ⎧ ⎨a − αβ, for a ≤ 2α αβ, ⎩2 α a − α − αβ , for a ≥ 2α Ra αβ 3.3 Then {w : |w − a| ≤ Ra } ⊆ w : w − α ≤ Re w αβ α − αβ Proof Since α ≥ and ≤ β < 1, then α > αβ Then in Lemma 3.1, put μ α and δ 3.4 αβ Theorem 3.3 Let f ∈ Σp Then f is in the class Γ∗ n, α, β if and only if λ ∞ k pαβ kλ pλ n ak ≤ p − αβ α≥ k p 1 β ; ≤ β < 1; p ∈ N; n ∈ N0 3.5 Proof Suppose that f ∈ Γ∗ n, α, β Then by the inequality 1.9 , we get that λ n z Dλ f z n p Dλ f z α αβ ≤ Re − n z Dλ f z α − αβ n p Dλ f z 3.6 That is, Re n z Dλ f z n p Dλ f z α αβ ≤ n z Dλ f z n p Dλ f z ≤ Re − n z Dλ f z n p Dλ f z α αβ 3.7 α − αβ, 10 Journal of Inequalities and Applications that is, n 2z Dλ f z Re ≤ 2αβ n p Dλ f z 3.8 Hence by the inequality 1.3 , ∞ k p 12 ∞ k p 1p −2p − αβ Re p k pαβ kλ kλ n ak zk pλ n ak zk pλ p p ≤ 3.9 Taking z to be real and putting z → 1− through real values, then the inequality 3.9 yields ∞ k p 12 ∞ k p 1p −2p − αβ p k pαβ kλ kλ n pλ pλ ak n ak ≤ 0, 3.10 which leads us at once to 3.5 In order to prove the converse, consider that the inequality 3.5 holds true In Lemma 3.2 above, since α > αβ and α ≥ 1/ β , that is, ≤ 2α αβ, we can put n n −z Dλ f z /p Dλ f z Now, if we let a Then for p ∈ N and n ∈ N0 , let wnp ∗ z ∈ ∂U {z ∈ C : |z| 1}, we get from the inequalities 1.3 and 3.5 that |wnp − 1| ≤ R1 Thus by Lemma 3.2 above, we ge that n z Dλ f z α n p Dλ f z − αβ n z Dλ f z n p Dλ f z w− α ≤ Re w − − α 3.11 αβ α − αβ Re{w} n z Dλ f z p αβ n Dλ f α − αβ, z α − αβ α≥ β ; ≤ β < 1; p ∈ N; n ∈ N0 Therefore by the maximum modulus theorem, we obtain f ∈ Γ∗ n, α, β λ Corollary 3.4 If f ∈ Γ∗ n, α, β , then λ ak ≤ p − αβ k pαβ kλ pλ n α≥ β ; ≤ β < 1; p ∈ N; n ∈ N0 3.12 Journal of Inequalities and Applications 11 The result is sharp for the function f z given by f z ∞ z−p p − αβ k k p pαβ kλ pλ k nz α≥ β ; ≤ β < 1; p ∈ N; n ∈ N0 3.13 Theorem 3.5 The class Γ∗ n, α, β is closed under convex linear combinations λ Proof This proof is similar as the proof of Theorem 2.4 The following are the growth and distortion theorems for the class Γ∗ n, α, β λ Theorem 3.6 If f ∈ Γ∗ n, α, β , then λ p m−1 ! p−1 ! ≤ p − − αβ αβ p 1 m−1 ! p!2−n 2p − p r r p−m ! − αβ p−1 ! < |z| n−1 · r < 1; α ≥ αβ p β n−1 · m m ≤ f p!2−n 2p − p r r p−m ! z m 3.14 ; ≤ β < 1; p ∈ N; n, m ∈ N0 ; p > m The result is sharp for the function f given by ∞ z−p f z − αβ k p 1 αβ 2p p nz , n ∈ N0 ; p ∈ N 3.15 Next we determine the radii of meromorphically p-valent starlikeness of order μ ≤ μ < p and meromorphically p-valent convexity of order μ ≤ μ < p for the class Γ∗ n, α, β λ Theorem 3.7 If f ∈ Γ∗ n, α, β , then f is meromorphically p-valent starlike of order μ ≤ μ < λ in the disk |z| < r1 , that is, Re − zf z f z >μ ≤ μ < p; |z| < r1 ; p ∈ N , 3.16 where r1 inf k≥p p−μ k p k pαβ kλ μ − αβ Proof This proof is similar to the proof of Theorem 2.6 pλ n 1/ k p 3.17 12 Journal of Inequalities and Applications Theorem 3.8 If f ∈ Γ∗ n, α, β , then f is meromorphically p-valent convex of order μ ≤ μ < λ in the disk |z| < r2 , that is, Re −1 − zf z f z ≤ μ < p; |z| < r2 ; p ∈ N , >μ 3.18 where r2 p−μ k inf pαβ kλ k k k≥p pλ n 1/ k p μ − αβ 3.19 Proof This proof is similar to the proof of Theorem 2.7 Subordination Properties If f and g are analytic functions in U, we say that f is subordinate to g, written symbolically as follows: f ≺g in U or f z ≺g z z∈U 4.1 if there exists a function w which is analytic in U with w 0, |w z | < z∈U , 4.2 such that f z g w z z∈U 4.3 Indeed it is known that f z ≺g z z∈U ⇒f g , f U ⊂g U 4.4 In particular, if the function g is univalent in U we have the following equivalence see 18 : f z ≺g z z ∈ U ⇐⇒ f g 0, f U ⊂g U 4.5 Let φ : C2 → C be a function and let h be univalent in U If J is analytic function in U and satisfied the differential subordination φ J z , J z ≺ h z then J is called a solution of the differential subordination φ J z , J z ≺ h z The univalent function q is called a dominant of the solution of the differential subordination, J ≺ q Journal of Inequalities and Applications 13 Lemma 4.1 see 19 Let q z / be univalent in U Let θ and φ be analytic in a domain D containing q U with φ w / when w ∈ q U Set zq z φ q z , Q z hz θ q z Q z 4.6 zq z φ q z , 4.7 Suppose that i Q z is starlike univalent in U, ii Re{zh z /Q z } > for z ∈ U If J is analytic function in U and θ J z zJ z φ J z ≺θ q z then J z ≺ q z and q is the best dominant Lemma 4.2 see 20 Let w, γ ∈ C and φ is convex and univalent in U with φ Re{wφ z γ } > for all z ∈ U If q is analytic in U with q and q z zq z ≺φ z wq z γ z∈U , and 4.8 then q z ≺ φ z and φ is the best dominant Theorem 4.3 Let q z / be univalent in U such that zq z /q z is starlike univalent in U and Re γ q z zq z zq z − q z q z > 0, , γ ∈ C, γ / 4.9 If f ∈ Σp satisfies the subordination n z Dλ f z n Dλ f n then z Dλ f z z n / Dλ f z γ n z Dλ f z n Dλ f − z n z Dλ f z n Dλ f ≺ q z z γ zq z , q z 4.10 ≺ q z and q is the best dominant Proof Our aim is to apply Lemma 4.1 Setting J z n z Dλ f z n Dλ f z −p ∞ k p ∞ k p k kλ kλ pλ pλ n ak zk n ak zk p p , n ∈ N0 ; p ∈ N , 4.11 θ w w and φ w γ /w, γ / It can be easily observed that J is analytic in U, θ is analytic in C, φ is analytic in C/{0} and φ w / By computation shows that zJ z J z n z Dλ f z n Dλ f z − n z Dλ f z n Dλ f z 4.12 14 Journal of Inequalities and Applications which yields, by 4.10 , the following subordination: J z γ γ zq z , q z zJ z ≺ q z J z 4.13 that is, θ J z zJ z φ J z ≺θ q z zq z φ q z 4.14 Now by letting Q z h z γ zq z , q z zq z φ q z Q z θ q z q z 4.15 γ zq z q z We find Qi starlike univalent in U and that Re zh z Q z Re n Hence by Lemma 4.1, z Dλ f z n / Dλ f z γ zq z zq z − q z q z q z > 4.16 ≺ q z and q is the best dominant Corollary 4.4 If f ∈ Σp and assume that 4.9 holds, then n implies that z Dλ f z is the best dominant Proof By setting the result n z Dλ f z n Dλ f n / Dλ f z γ ≺ z ≺ 1 and q z Az Bz Az / 1 A−B z Az Bz Bz , −1 ≤ B < A ≤ and Az / 4.17 Az / Bz Bz in Theorem 4.3, then we can obtain Corollary 4.5 If f ∈ Σp and assume that 4.9 holds, then n implies that z Dλ f z Proof By setting n / Dλ f z γ n z Dλ f z n Dλ f z ≺ eαz αz ≺ eαz , |α| < π and eαz is the best dominant and q z eαz in Theorem 4.3, where |α| < π 4.18 Journal of Inequalities and Applications 15 Theorem 4.6 Let w, γ ∈ C, and φ be convex and univalent in U with φ 0 for all z ∈ U If f ∈ Σp satisfies the subordination γ n z Dλ f z n / Dλ f z − w/p n z Dλ f z n / Dλ f z n n w − γ p Dλ f z /z Dλ f z n then −z Dλ f z n /p Dλ f z and Re{wφ z γ} > ≺φ z , 4.19 n ∈ N0 ; p ∈ N 4.20 ≺ φ z and φ is the best dominant Proof Our aim is to apply Lemma 4.2 Setting q z n −z Dλ f z n p Dλ f z p p ∞ k p ∞ k p k kλ p kλ pλ pλ n p n p ak zk ak zk It can be easily observed that q is analytic in U and q zq z q z n z Dλ f z n Dλ f z − , Computation shows that n z Dλ f z n Dλ f z 4.21 which yields, by 4.19 , the following subordination: q z n Hence by Lemma 4.2, −z Dλ f z zq z ≺φ z , wq z γ n / pDλ f z z∈U 4.22 ≺ φ z and φ is the best dominant Acknowledgments The authors express their sincere thanks to the referees for their very constructive comments and suggestions The authors also acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme 05-01-090720RU References M K Aouf and H M Hossen, “New criteria for meromorphic p-valent starlike functions,” Tsukuba Journal of Mathematics, vol 17, no 2, pp 481–486, 1993 S S Kumar, V Ravichandran, and G Murugusundaramoorthy, “Classes of meromorphic p-valent parabolic starlike functions with positive coefficients,” The Australian Journal of Mathematical Analysis and Applications, vol 2, no 2, pp 1–9, 2005 M Nunokawa and O P Ahuja, “On meromorphic starlike and convex functions,” Indian Journal of Pure and Applied Mathematics, vol 32, no 7, pp 1027–1032, 2001 M K Aouf, “Certain subclasses of meromorphically p-valent functions with positive or negative coefficients,” Mathematical and Computer Modelling, vol 47, no 9-10, pp 997–1008, 2008 M K Aouf, “Certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric function,” Computers & Mathematics with Applications, vol 55, no 3, pp 494–509, 2008 16 Journal of Inequalities and Applications M K Aouf, “On a certain class of meromorphic univalent functions with positive coefficients,” Rendiconti di Matematica e delle sue Applicazioni Serie VII, vol 11, no 2, pp 209–219, 1991 S B Joshi and H M Srivastava, “A certain family of meromorphically multivalent functions,” Computers & Mathematics with Applications, vol 38, no 3-4, pp 201–211, 1999 M L Mogra, “Meromorphic multivalent functions with positive coefficients I,” Mathematica Japonica, vol 35, no 1, pp 1–11, 1990 M L Mogra, “Meromorphic multivalent functions with positive coefficients II,” Mathematica Japonica, vol 35, no 6, pp 1089–1098, 1990 10 S Owa, H E Darwish, and M K Aouf, “Meromorphic multivalent functions with positive and fixed second coefficients,” Mathematica Japonica, vol 46, no 2, pp 231–236, 1997 11 H M Srivastava, H M Hossen, and M K Aouf, “A unified presentation of some classes of meromorphically multivalent functions,” Computers & Mathematics with Applications, vol 38, no 11-12, pp 63–70, 1999 12 R K Raina and H M Srivastava, “A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions,” Mathematical and Computer Modelling, vol 43, no 3-4, pp 350–356, 2006 13 B A Uralegaddi and M D Ganigi, “Meromorphic multivalent functions with positive coefficients,” The Nepali Mathematical Sciences Report, vol 11, no 2, pp 95–102, 1986 14 B A Uralegaddi and C Somanatha, “New criteria for meromorphic starlike univalent functions,” Bulletin of the Australian Mathematical Society, vol 43, no 1, pp 137–140, 1991 15 D G Yang, “On new subclasses of meromorphic p-valent functions,” Journal of Mathematical Research and Exposition, vol 15, no 1, pp 7–13, 1995 16 I Faisal, M Darus, and A Kılıcman, “New subclasses of meromorphic functions associated with ¸ hadamard product,” in Proceedings of the International Conference on Mathematical Sciences (ICMS ’10), vol 1309 of AIP Conference Proceedings, pp 272–279, 2010 ¸ 17 A Tehranchi and A Kılıcman, “On certain classes of p-valent functions by using complex-order and differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol 2010, Article ID 275935, 12 pages, 2010 18 H M Srivastava and S S Eker, “Some applications of a subordination theorem for a class of analytic functions,” Applied Mathematics Letters, vol 21, no 4, pp 394–399, 2008 19 S S Miller and P T Mocanu, “Subordinants of differential superordinations,” Complex Variables, vol 48, no 10, pp 815–826, 2003 20 Z.-G Wang, Y.-P Jiang, and H M Srivastava, “Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function,” Computers & Mathematics with Applications, vol 57, no 4, pp 571–586, 2008  ... if and only if ∞ p α β k α kλ pλ n ak ≤ p − β k p α ≥ 0; ≤ β < 1; p ∈ N; n ∈ N0 2.3 Journal of Inequalities and Applications Proof Suppose that f ∈ Γλ n, α, β Then by the inequalities 1.3 and. .. is called a dominant of the solution of the differential subordination, J ≺ q Journal of Inequalities and Applications 13 Lemma 4.1 see 19 Let q z / be univalent in U Let θ and φ be analytic in... complex-order and differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol 2010, Article ID 275935, 12 pages, 2010 18 H M Srivastava and S S Eker, “Some applications