ON SUBCLASS OF MEROMORPHIC UNIVALENT FUNCTIONS DEFINED BY A LINEAR OPERATOR

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1 ITALIAN JOUNAL OF PUE AND APPLIED MATHEMATICS N ( ON SUBCLASS OF MEOMOPHIC UNIVALENT FUNCTIONS DEFINED BY A LINEA OPEATO ASSOCIATED WITH λ-genealized HUWITZ-LECH ZETA FUNCTION AND q-hypegeometic FUNCTION K. A. Challab M. Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600, Selangor, D. Ehsan Malaysia khalid math1363@yahoo.com maslina@ukm.edu.my F. Ghanim Department of Mathematics College of Sciences University of Sharjah Sharjah United Arab Emirates fgahmed@sharjah.ac.ae Abstract. In this article, a linear operator associated with the λ-generalized Hurwitz- Lerch zeta function and q-hypergeometric function by using the Hadamard product (or convolution is defined by the authors, a different interesting properties of certain subclass of meromorphic univalent functions related to a linear operator in the punctured unit disk are introduced and investigated. The authors also consider some closely related (known or new corollaries and consequences of the main results presented in this paper. Keywords: analytic functions, meromorphic functions, univalent functions, Hadamard product or (convolution, λ-generalized Hurwitz-Lerch zeta function, q-hypergeometric function, Srivastava-Attiya operator. 1. Introduction, definitions and preliminaries Normally, we are considering the class of meromorphic function f of the form (1.1 f(z = 1 z + a n z n, n=0 in the punctured open unit disc U = {z : z C, and 0 < z < 1} = U \ {0} and denoted by Σ.. Corresponding author

2 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS The set of complex numbers is, as usual C. By using ΣS (β and κ(β (β 0, we denote the subclasses of Σ that encompass all of the meromorphic functions, which are, starlike of the order β and convex of order β in U, respectively (see also the recent studies [33], [34]. In the case of the functions f j (z (j = 1, 2 which have been defined by f j (z = 1 z + a n,j z n, (j = 1, 2, n=1 the Hadamard product (or convolution of f 1 (z and f 2 (z can be dented by using (f 1 f 2 (z = 1 z + a n,1 a n,2 z n. n=1 For complex parameters β 1,..., β m (β j 0, 1,...; j = 1, 2,..., m and α 1,..., the q-hypergeometric function l Ψ m (z can be defined as (1.2 (α 1, q n...(, q n lψ m (α 1,..., ; β 1,..., β m ; q, z := (q, q n=0 n (β 1, q n...(β m, q n [ ( 1 n q (n 2 ] 1+m l z n, with ( n 2 = n(n 1/2 where q 0 when l > m+1 (l, m N0 = N {0}; z U. Also the q-shifted factorial can be defined for α, q C as a product of n factors by using { (1 α (1 αq... ( 1 αq n 1, (n N (1.3 (α; q n = 1, (n = 0, and in terms of basic analogue of the gamma function (1.4 (q α ; q n = Γ q(α + n(1 q n, n > 0. Γ q (α Interesting to note is that, lim q 1 ((q α ; q n /(1 q n = (α n = α(α (α + n 1 is the common Pochhammer symbol, and (1.5 (1.6 lψ m (α 1,..., ; β 1,..., β m ; z = n=0 (α 1 n...( n z n (β 1 n...(β m n n!. Now for z U, 0 < q < 1, and l = m + 1, the form of lψ m (α 1,..., ; β 1,..., β m ; q, z = n=0 (α 1, q n...(, q n (q, q n (β 1, q n...(β m, q n z n,

3 412 K.A. CHALLAB, M. DAUS and F. GHANIM is taken by the basic hypergeometric function which is defined in (1.2, and converges absolutely in the open unit disk U see [1]. Huda and Darus [1] and K.A Challab et al. [[2], [5]] introduced and studied a q-analogue of the Liu-Srivastava operator in correspondence with the function lψ m (α 1,..., ; β 1,..., β m ; q, z for the meromorphic functions of f Σ, which consist of the functions in the form of (1.1, and as presented below: lυ m (α 1,..., ; β 1,..., β m ; q, z f(z (1.7 = 1 z l Ψ m (α 1,..., ; β 1,..., β m ; q, z f(z = 1 z + l i=1 (α i, q n+1 (q, q m n+1 i=1 (β a n z n, i, q n+1 n=1 where s k=1 (α k, q n+1 = (α 1, q n+1 (α 2, q n+1...(α s, q n+1, where z U := {z C : 0 < z < 1}. ecently, Ghanim ([8]; see also [9] introduced the function G s,a which defined by (1.8 ] G s,a := (a + 1 [Φ(z, s s, a a s 1 + z(a + 1 s, G s,a = 1 z + ( a + 1 s z n, (z U. a + n n=1 Also, the function Φ(z, s, a be the well-known Hurwitz-Lerch zeta function as was defined by (see, e.g. [[28], p. 121 et seq.]; see also [[24], [23], p. 194 et seq.] Φ(z, s, a := n=0 z n (n + a s (a C \ Z 0 ; s C when z < 1; (s > 1 when z = 1. We recollect here that, Srivastava introduced and systematically investigated the following new group of λ-generalised Hurwitz-Lerch zeta functions (see for example, [3], [4], [17], [18], [19], [21], [22], [26], [27], [29], and [32] : Φ (ρ 1,...,ρ p,σ 1,...,σ r λ 1,...,λ p;µ 1,...,µ r (z, s, a; b, λ = 1 λγ(s. (1.9 n=0 p (λ j nρj (a + n s. r (µ H 2,0 0,2 j nσj ( [(a + nb 1λ (s, 1, 0, 1 ] z n λ n! (min{(a, (s} > 0; (b > 0; λ > 0,

4 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS where ( λ j C(j = 1,..., p and µ j C \ Z 0 (j = 1,..., r; ρ j > 0(j = 1,..., p; r p σ j > 0(j = 1,..., r; 1 + σ J ρ j 0 it was found that the equality in the convergence condition remained true for the suitably bounded values of z, which p r z < :=. had given. ρ ρ j j Definition 1.1. The H-function which was involved on the right-hand side of (1.9 was the well-known Fox s H-function [[12], Definition 1.1] (see also [16], [20] that [ ] Hp,r m,n (z = Hp,r m,n z (a 1,A 1,...,(a p,a p (b 1,B 1,...,(b r,b r = 1 Ξ(sz s ds (z C \ {0}; arg(z < π, 2πi l had defined, where m Ξ(s = Γ(b j + B j s. n Γ(1 a j A j s p j=n+1 Γ(a j + A j s. r j=m+1 Γ(1 b j B j s, a hollow product is also depicted as 1, m, n, p and r are integers such that 1 m r and 0 n p, σ σ j j A j > 0 (j = 1,..., p and B j > 0 (j = 1,..., r, a j C (j = 1,..., p and b j C (j = 1,..., r and l is a suitable Mellin-Barnes type contour separating the poles of the gamma functions {Γ(b j + B j s} m from the poles of the gamma functions lim b 0 H2,0 0,2 {Γ(1 a j + A j s} n. It is worth mentioning that, if we use the fact that [[32], p. 1496, emark 7] ( [(a + nb 1λ (s, 1, 0, λ ] 1 = λγ(s (λ > 0,

5 414 K.A. CHALLAB, M. DAUS and F. GHANIM equation (1.8 is reduced to the form of the following: Φ (ρ 1,...,ρ p,σ 1,...,σ r λ 1,...,λ p ;µ 1,...,µ r (z, s, a; b, λ = Φ (ρ 1,...,ρ p,σ 1,...,σ r λ 1,...,λ p ;µ 1,...,µ r (z, s, a (1.10 p (λ j nρj (a + n s. zn r (µ j nσj n!. n=0 Definition 1.2. The function Φ (ρ 1,...,ρ p,σ 1,...,σ r λ 1,...,λ p ;µ 1,...,µ r (z, s, a which was involved in (1.10 was a multiparameter extension and generalisation of the Hurwitz-Lerch zeta function Φ(z, s, a that Srivastava et al. introduced [17, p.503, Equation (6.2] and was defined by Φ (ρ 1,...,ρ p,σ 1,...,σ r λ 1,...,λ p;µ 1,...,µ r (z, s, a =: n=0 p (λ j nρj (a + n s. zn r (µ j nσj n! ( p, q N 0 ; λ j C(j = 1,..., p; a, µ j C \ Z 0 (j = 1,..., r; ρ j, σ k + (j = 1,..., p; k = 1,..., r; > 1 when s, z C; = 1 and s C when z < ; = 1 and (Ξ > 1 2 when z = with p := := ρ ρ j j r σ j r. σ σ j j, p ρ j and Ξ := s + r µ j p λ j + p r 2. Srivastava et al. [30, 31] presented and developed a new linear operator by applying new family of meromorphic λ-generalised Hurwitz-Lerch zeta functions. The convolution operator which was studied by Dziok-Srivastava [6], [7] is a generalisation of two other operators: they are the uscheweyh [15] operator and the Hohlov [10] operator. As a matter of fact, the Dziok-Srivastava convolution operator is in and of itself, a special case of the Srivastava-Wright operator (see, for more details[11], [25]. We have considered the new linear operator K f(z in this study such that, this has been defined by K f(z K s,a,λ,,β m (λ p,(µ r,b f(z : Σ Σ, (1.11 K f(z = G s,a,λ (λ p,(µ r,b (z l Υ m (α 1,..., ; β 1,..., β m ; q, z,

6 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS where the Hadamard product (or convolution of the analytical functions has been denoted by, and (1.12 G s,a,λ (λ p,(µ r,b (z [ :=(a + 1 s. = 1 z + p (λ j n n=1 Φ (1,...,1,1,...,1 λ 1,...,λ p ;µ 1,...,µ q (z, s, a; b, λ a s λγ(s s Λ(a + n, b, s, λ r (µ j n ( a + 1 a + n λγ(s Λ(a, b, s, λ+(a+1 s z z n n! ] gave the function G s,a,λ (λ p,(µ r,b(z, with Now Λ(a, b, s, λ := H 2,0 0,2 K f(z = 1 z + n=1 Λ(a + n, b, s, λ z n a n λγ(s n! K f(z = 1 z + Ω,β m,a,s n=1 [ab 1λ (s, 1, ( 0, 1 λ ]. l i=1 (α i, q n+1 (q, q n+1 m i=1 (β i, q n+1 λ p,µ r,q,b a z n n n! p (λ j n r (µ j n ( a + 1 a + n s (1.13 (z U ; α, λ j C (j = 1,..., p; β, µ j C\Z 0 (j = 1,..., r; p r+1 is obtained if (1.11 and (1.12 are combined, with and where min{(a, (s} > 0; λ > 0 if (b > 0 Ω,β m,a,s λ p,µ r,q,b = l i=1 (α i, q n+1 (q, q n+1 m i=1 (β i, q n+1 s C a C \ Z 0 if b = 0 p (λ j n r (µ j n ( a + 1 a + n s Λ(a + n, b, s, λ. λγ(s Let the class of all functions f(z Σ such that ( f(z µ +1 ( f(z K f(z µ 1 (1 ρ K α + ρ l g(z K αl+1. g(z K α > γ, l g(z (z U ; 0 γ < 1,

7 416 K.A. CHALLAB, M. DAUS and F. GHANIM be denoted by Σ s,a,λ,,β m (λ p,(µ r,b (γ, δ, µ, ρ, where the following inequality of: g(z (1.14 K αl+1 > δ (0 δ < 1; z U g(z is satisfied by g(z Σ. At this point and as follows, γ and µ are real numbers such that 0 γ < 1 and µ > 0 and ρ C with (ρ > 0. Different properties of certain subclass of the meromorphically analytical function class Σ in the punctured unit disk U have been investigated. One of these function class was introduced, first, and then the properties of the linear operator were investigated. K +1 K s,a,λ,+1,β m (λ p,(µ r,b f(z 2. Main results The following Lemmas were needed so that the main results could be proved: Lemma 2.1. (see[13]. Let Ω be a set in the complex plane C and let the function Ψ : C 2 C satisfy the following condition: Ψ(ir 2, a 1 / Ω for all real r 2, a (1 + r 2 2. If q(z is analytic in U with q(0 = 1 and Ψ(q(z, zq (z Ω (z U, then {q(z} > 0. Our first main result is now stated and proved as Theorem 2.1, which is presented below: Theorem 2.1. Let f(z Σ s,a,λ,,β m (λ p,(µ r,b (γ, δ, µ, ρ, α \ {0} and ρ 0. Then (2.1 f(z 2αγµ + δρ K α > l g(z 2αµ + δρ (0 γ < 1; µ > 0; z U, where the condition (1.14 is satisfied by the function g(z Σ. Proof. Let Now, let us suppose that (2.2 ξ = q(z = 1 1 ξ 2αγµ + δρ 2αµ + δρ [ f(z µ K α ξ] l g(z

8 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS defines the function q(z. In that case, the function q(z is analytical in U and q(0 = 1. If (2.3 h(z = Kg(z K +1 g(z, is put in, then, according to the hypothesis of Theorem 2.1, we get {h(z} > δ. Then, since f(z Σ s,a,λ,,β m (λ p,(µ r,b (γ, δ, µ, ρ (1 ρ f(z K g(z = [ξ + (1 ξq(z] + µ +1 f(z + ρ K αl+1. g(z ρ(1 ξ h(zzq (z, αµ is obtained by differentiating (2.2 with respect to z. Let us use ( ρ(1 ξ Ψ(r, s = ξ + (1 ξr + h(zs αµ f(z µ 1 K g(z to define the function Ψ(r, s. Then, if we use (2.3 and the fact that f(z Σ s,a,λ,,β m (λ p,(µ r,b (γ, δ, µ, ρ, {Ψ(q(z, zq (z; z U } Ω = {w C : (w > γ} is obtained. At this point, for all real numbers r 2, a (1 + r 2 2, we get ( ( ρ(1 ξ K g(z {Ψ(ir 2, a 1 } = ξ + αµ K αl+1 g(z ξ ρδ(1 ξ(1 + r 2 2 2αµ ξ ρδ(1 ξ αµ =: γ. Hence, for every z U, we have Ψ(ir 2, a 1 / Ω. Therefore, by using Lemma 2.1, we get that {q(z} > 0, which is, f(z µ K α > ξ (z U. l g(z The proof of Theorem 2.1 is obviously completed at this point. Corollary 2.1. If the functions of f(z and g(z are in the class Σ and also suppose that condition (1.14 is satisfied by the function of g(z, if the α \ {0}, ρ 0 and ( (1 ρ Kf(z K α + ρ Kαl+1 f(z (2.4 l g(z K αl+1 > γ (0 γ < 1; z U. g(z Then, +1 f(z K αl+1 > η := g(z γ(2α + δ + δ(ρ 1. 2α + ρδ

9 418 K.A. CHALLAB, M. DAUS and F. GHANIM Proof. We have λ K+1 f(z K +1 g(z = ( (1 ρ Kf(z K α + ρ Kαl+1 f(z l g(z K αl+1 + (ρ 1 Kf(z g(z K α. l g(z We can deduce the following desired inequality +1 f(z γ(2α + δ + δ(ρ 1 K αl+1 > η :=, g(z 2α + ρδ if we use of (2.4 and (2.1 (for µ = 1, and since ρ > 1. Corollary 2.2. Let α \ {0} and {ρ} > 0. If the following condition: {(1 ρ(zk f(z µ + ρ(zk +1 f(z(zk f(z µ 1 } > γ (0 γ < 1; µ > 0; z U, is satisfied by f(z Σ, then (2.5 {(zk f(z µ } > 2αµγ + {ρ} 2µα + {ρ}. In addition, if ρ 1 and α \ {0}, and if f(z Σ satisfies the following condition ((1 ρzk f(z + ρ(zk +1 f(z > γ, then (2.6 (zk +1 f(z > (2α + 1γ + ρ 1 2α + ρ (0 γ < 1; z U. Proof. The results of (2.5 and (2.6 are achieved by putting g(z = 1 z Theorem 2.1 and Corollary 2.1, respectively. emark 1. (i By putting ρ = 1 and for α i, β j > 0 (i = 1, 2,..., l and (j = 1, 2,..., m in Corollary 2.2, we get (zk +1 f(z.(zk f(z µ 1 > γ, this implies that {(zk f(z µ } > 2αγ + {ρ} 2α + {ρ} (z U. (ii For ρ C \ {0} with {ρ} > 0, µ = 1 and α i, β j > 0 (j = 1, 2,..., m in Corollary 2.2, we get {(1 ρzk f(z + ρ(zk +1 f(z} > γ (0 γ < 1; z U, this implies that (zk f(z > 2αγ + {ρ} 2α + {ρ} (0 γ < 1; z U. in (i = 1, 2,..., l and (iiifor ρ = 1, s = 0, α i = β j = 1 (i = 1, 2,..., l and (j = 1, 2,..., m, p 1 = r = 0 and λ 1 = 1, if we proceed to the limit as b 0 in Corollary 2.2, we have ( zf (z f(z (zf(zµ > γ (0 γ < 1; µ > 0; z U,

10 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS which implies that ([zf(z] µ > 2γµ + 1 2µ + 1 (0 γ < 1; µ > 0; z U. (ivfor ρ C \ {0} with {ρ} > 0, α i = β j = 1 (i = 1, 2,..., l and (j = 1, 2,..., m, µ = 1, s = 0, p 1 = r = 0, and λ 1 = 1, if we take the limit as b 0 in Corollary 2.2, we get ([(1 ρzf(z + ρ(z 2 f (z] > γ (0 γ < 1; µ > 0; z U, which implies that {zf(z} > 2γ + {ρ} 2 + {ρ} (0 γ < 1; µ > 0; z U. (v eplacing f(z by zf (z in emark 1 (ii above, we have which implies that {(1 ρz 2 f(z + ρ(z 3 f (z} > γ, {z 2 f (z} > 2γ + {ρ} 2 + {ρ} (z U. (vi For ρ with ρ 1, µ = 1, s = 0, α i = β j = 1 (i = 1, 2,..., l and (j = 1, 2,..., m, p 1 = r = 0, and λ 1 = 1, if we take the limit as b 0 in Corollary 2.2, we obtain {(1 ρzf(z + ρ(z 2 f (z} > γ, which implies that {zf(z} > 3γ + ρ ρ The following theorem gives a further extension of the previous result. Theorem 2.2. Let the functions f(z and g(z be in the class Σ. Let us also suppose that, if +1 f(z K αl+1 g(z Kαlf(z (1 γδ (2.7 K α >, l g(z 2α (0 γ < 1; α \ {0}; 0 δ < 1; z U, then the condition (1.14 is satisfied by the function g(z, hence f(z (2.8 K αl+1 > γ, (0 γ < 1; α \ {0}; 0 δ < 1; z U g(z and (2.9 { K +1 } f(z K αl+1 > g(z (2α δγ δ, 2α (0 γ < 1; α \ {0}; 0 δ < 1; z U.

11 420 K.A. CHALLAB, M. DAUS and F. GHANIM Proof. Let q(z = 1 f(z 1 γ K α γ. l g(z Then, the function q(z is analytic in U with q(0 = 1. At this point, by setting Φ(z = Kg(z K +1 g(z then, by using the hypothesis, we observe that {Φ(z} > δ, (z U. Now, (1 γzq (z{φ} α is shown by a simple computation, where Therefore, we get Ψ(r, a = Ψ(q(z, zq (z (z U Ω := = K+1 f(z K +1 g(z Kαlf(z K g(z = Ψ(q(z, zq (z, (1 γφ(za α (α \ {0}. { } δ(1 γ w : w C and (w > 2α if we use hypothesis (2.7. Now, for all real r 2, a 1 (1 + r 2 2 /2, or r 2, a 1 (1 + r 2 2 /2, we obtain {Ψ(ir 2, a 1 } = a 1(1 γ{φ} α δ(1 γ(1 r 2 2 2α δ(1 γ. 2α This shows that {Ψ(ir 2, a 1 } / Ω, (z U. Then, we get {q(z} > 0, (z U, with Lemma 2.1. The assertion of (2.8 is thus proved. The assertion of (2.9 is proved if (2.8 and (2.9 are used in the following identity, K αl+1 ( f(z K α K αl+1 g(z = l +1 f(z K αl+1 g(z Kαlf(z K α + Kf(z l g(z K α. l g(z The proof of Theorem 2.2 is now obviously completed. emark 2. Upon putting α i = β j = 1 (i = 1, 2,..., l and (j = 1, 2,..., m, s = 0, g(z = 1 z, p 1 = r = 0, and λ 1 = 1, if we take the limit as b 0 in Theorem 2.2, we obtain {zf(z + z 2 f δ(1 γ (z} > 2 (0 γ < 1; 0 δ < 1; z U, which implies that {zf(z} > γ (0 γ < 1; z U and {2zf(z + z 2 f (z} > γ(2 + δ δ 2 (0 γ < 1; 0 δ < 1; z U.

12 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS Concluding remarks and observations A remarkably general group of linear operators related to the λ generalised Hurwitz-Lerch zeta functions has been successfully applied in our present investigation. By using this general linear operator, we have a variety of properties of some new subclass of meromorphically univalent functions in the punctured unit disk U which were introduced and investigated. Several closely related (new or known consequences and corollaries of the main results (Theorem 2.1 and 2.2 presented in this paper have also been considered. Acknowledgements The work is supported by MOHE: FGS/1/2016/STG06/UKM/01/1. eferences [1] H. Aldweby and M. Darus, Integral operator defined by q-analogue of Liu-Srivastava operator, Studia Universitatis Babes-Bolyai, Mathematica, 58(4, [2] K.H. Challab, M. Darus, and F. Ghanim, A linear operator and associated families of meromorphically q-hypergeometric functions, AIP Conference Proceedings, 1830(1, , [3] K.H. Challab, M. Darus, and F. Ghanim, certain problems related to generalized Srivastava-Attiya operator, Asian-European Journal of Mathematics, , [4] K.H. Challab, M. Darus, and F. Ghanim, Further results related to generalized Hurwitz-Lerch zeta function and their applications, AIP Conference Proceedings, 1784(1, , [5] K.H. Challab, M. Darus, and F. Ghanim, ON q-hypegeometic FUNCTIONS, Far East Journal of Mathematical Sciences, 101(10, , [6] J. Dziok and H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Applied Mathematics and Computation, 103(1, 1 13, [7] J. Dziok and H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms and Special Functions, 14(1, 7 18, [8] F. Ghanim, A study of a certain subclass of Hurwitz-Lerch-Zeta function related to a linear operator, Abstract and Applied Analysis, 2013, 2013.

13 422 K.A. CHALLAB, M. DAUS and F. GHANIM [9] F. Ghanim, M. Darus, New result of analytic functions related to Hurwitz Zeta function, The Scientific World Journal, 2013, [10] Y.E. Hohlov, Operators and operations on the class of univalent functions, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, (10, 83 89, [11] V. Kiryakova, Criteria for univalence of the Dziok Srivastava and the Srivastava Wright operators in the class A, Applied Mathematics and Computation, 218(3, , [12] A.M. Mathai,.K. Saxena and H.J. Haubold, The H-function: theory and applications, Springer Science & Business Media, [13] S.S. Miller and P.T. Mocanu, Differential subordinations: theory and applications, CC Press, [14] S. Ponnusamy, Differential sobordination and Bazilevič functions, In Proceedings of the Indian Academy of Sciences-Mathematical Sciences, volume 105, pages , Springer, [15] S. uscheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society, pages , [16] H.M. Srivastava and H.L. Manocha, Treatise on generating functions, John Wiley & Sons, INC, 605 Third Ave, New York, NY 10158, USA, 1984, 500, [17] H.M. Srivastava, Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions, SpringerPlus, 2(1, 1 14, [18] H.M. Srivastava and S. Gaboury, New expansion formulas for a family of the-generalized Hurwitz-Lerch Zeta functions, International Journal of Mathematics and Mathematical Sciences, [19] H.M. Srivastava, S. Gaboury and A. Bayad, Expansion formulas for an extended Hurwitz-Lerch Zeta function obtained via fractional calculus, Advances in Difference Equations, 2014(1, 1 17, [20] H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-functions of one and two variables, with applications, South Asian Publishers, [21] H.M. Srivastava, D. Jankov, T.K. Pogány and.k. Saxena, Two-sided inequalities for the extended Hurwitz Lerch Zeta function, Computers & Mathematics with Applications, 62(1, , [22] H.M. Srivastava,.K. Saxena, T.K. Pogány and.k. Saxena, Integral and computational representations of the extended Hurwitz Lerch Zeta function, Integral Transforms and Special Functions, 22(7, , 2011.

14 ON SUBCLASS OF MEOMOPHICALLY UNIVALENT FUNCTIONS [23] H.M. Srivastava and C. Junesang, Zeta and q-zeta functions and associated series and integrals, Elsevier, [24] H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, In Mathematical Proceedings of the Cambridge Philosophical Society, volume 129, pages 77 84, Cambridge Univ Press, [25] H.M. Srivastava, Some fox-wright generalized hypergeometric functions and associated families of convolution operators, Applicable Analysis and Discrete Mathematics, 1(1, 56 71, [26] H.M. Srivastava, Some generalizations and basic (or q- extensions of the Bernoulli, Euler and genocchi polynomials, Appl. Math. Inf. Sci, 5(3, , [27] H.M. Srivastava, A new family of the λ-generalized Hurwitz-Lerch Zeta functions with applications, Appl. Math. Inf. Sci, 8, , [28] H.M. Srivastava and J. Choi, Series associated with Zeta and related functions, a new book, Journal of Applied Mathematics and Stochastic Analysis, 15(1, 97 97, [29] H.M. Srivastava, S. Gaboury, and B.J. Fugere, Further results involving a class of generalized Hurwitz-Lerch Zeta functions, ussian Journal of Mathematical Physics, 21(4, , [30] H. M. Srivastava, S. Gaboury and F. Ghanim, Certain subclasses of meromorphically univalent functions defined by a linear operator associated with the λ-generalized Hurwitz-Lerch zeta function, Integral Transforms Spec. Funct. 26(4 (2015, [31] H. M. Srivastava, S. Gaboury and F. Ghanim, Some Further Properties of a Linear Operator Associated with the λ-generalized Hurwitz-Lerch Zeta Function elated to the Class of Meromorphically Univalent Functions, Applied Mathematics and Computation 259 (2015, [32] H.M. Srivastava, S. Gaboury, and. Tremblay, New relations involving an extended multiparameter Hurwitz-Lerch Zeta function with applications. International Journal of Analysis, [33] H.M. Srivastava, A.Y. Lashin, and B.A. Frasin, Starlikeness and convexity of certain classes of meromorphically multivalent functions, Theory and Applications of Mathematics & Computer Science, 3(2, , [34] Z.G. Wang, H.M. Srivastava, and S.M. Yuan, Some basic properties of certain subclasses of meromorphically starlike functions, Journal of Inequalities and Applications, 2014(1, 1 13, Accepted: