An Integrodifferential Operator for Meromorphic Functions Associated with the Hurwitz-Lerch Zeta Function

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1 Filomat 30:7 (2016), DOI /FIL A Pulished y Faculty of Sciences Mathematics, University of Niš, Seria Availale at: An Integrodifferential Operator for Meromorphic Functions Associated with the Hurwitz-Lerch Zeta Function Adel A. Attiya a, Oh Sang Kwon, Park Ji Hyang c, Nak Eun Cho d a Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, Egypt Department of Mathematics, Kyungsung University, Busan , pulic of Korea c Department of Applied Mathematics, Pukyong National University, Busan , pulic of Korea d Corresponding Author, Department of Applied Mathematics, Pukyong National University, Busan , pulic of Korea Astract. In this paper, we introduce a new integrodifferential operator associated with the Hurwitz Lerch Zeta function in the puncture open disk of the meromorphic functions. We also otain some properties of the third-order differential suordination superordination for this integrodifferential operator, y using certain classes of admissile functions. 1. Introduction Let Σ denote the class of functions f of the form f = 1 z + a k z k k=0 (1.1) which are analytic in the punctured open unit disc U = U\{0} = {z C : 0 < z < 1}. The function f has a simple pole at z = 0. We egin y recalling that a general Hurwitz-Lerch Zeta function Φ(z, s, ) defined y (see, for example, [18, P. 121 et seq.] [19, P. 194 et seq.]) Φ(z, s, ) = k=0 z k (k + ) s, (1.2) 2010 Mathematics Suject Classification. 30C45 Keywords. Analytic functions; meromorphic functions; Hurwitz-Lerch Zeta function; Srivastava-Attiya operator. ceived: 17 Decemer 2014; Accepted: 13 March 2015 Communicated y Hari M. Srivastava The authors would like to express their thanks to Professor H.M. Srivastava, University of Victoria, for his valuale suggestions. This research was supported y the Basic Science search Program through the National search Foundation of Korea (NRF) funded y the Ministry of Education, Science Technology (No ). addresses: aattiy@mans.edu.eg (Adel A. Attiya), oskwon@ks.ac.kr (Oh Sang Kwon), jihyang1022@naver.com (Park Ji Hyang), necho@pknu.ac.kr (Nak Eun Cho)

2 A. A. Attiya et al. / Filomat 30:7 (2016), ( C \ Z 0, Z 0 = Z {0} = {0, 1, 2,... }, s C when z U, (s) > 1 when z = 1 ). Several properties of Φ(z, s, ) can e found in many papers, for example Attiya Hakami [3], Choi et al. [8], Cho et al. [7], Ferreira López [9], Gupta et al. [10] Luo Srivastava [14]. See, also Kuti Attiya( [11], [12]), Srivastava Attiya [17], Srivastava Gaoury [20], Srivastava et al. [21] Owa Attiya [16]. Analogous to the operator defined y Srivastava Attiya [17], we define the following operator associated with the Hurwitz-Lerch Zeta function, as follows: J s, : Σ Σ the operator defined y: J s, f = G(s, ; z) f (1.3) where the function G(s, ; z) defined y G(s, ; z) = s Φ(z, s, ) z (1.4) ( z U ; C\Z 0 ; s C) denotes the Hadamard product (or Convolution). Then we can see that J s, f = 1 ( ) s z + a k z k k k=0 ( z U ; f Σ; C\Z 0 ; s C). mark 1. We note that: 1. J f = f, 0, 2. J 1 2α f = 1, 1 α 2 α z α 1 1 z z 3. J 1, f = t f (t) dt, z J βα f = α,β Γ(α)z β+1 5. J s,1 f = 1 z + 0 k=0 0 z 6. J 1,1 f = z f, 0 t 1 α 2 f (t) dt (0 < α < 1 2 ), t β ( log z t ) α 1 f (t) dt (α > 0; β > 0), 1 (k+2) s a k z k, 7. J 1, 2 f = f z f, 2

3 8. J n, 1 f = 1 z + ( k) n a k z k (n N), k=0 9. J n,1 f = 1 z + (k + 2) n a k z k (n N), k=0 A. A. Attiya et al. / Filomat 30:7 (2016), where J the operator introduced y Cho et al. [6], 1, 1 α 2 J α,β the operator introduced y Lashin [13], J s,1 the operator introduced y Alhindi Darus [1], J the operator defined y Uralegaddi Somanatha 1, n [23] J is the operator analogous to the generalized Bernardi operator (for Bernardi operator see [5]), 1, when () > 0, the operator J introduced y Bajpai [4]. 1, We denote y H[a, n], the class of analytic functions in U in the form f = a + a k z k (a C; n N = {1, 2, }) k=n H = H[1, 1]. In our investigation we need the following definitions theorem: Definition 1.1. Let f F e analytic functions. The function f is said to e suordinate to F, written f F, if there exists a function w analytic in U with w(0) = 0 w < 1, such that f = F(w). If F is univalent, f F if only if f (0) = F(0) f (U) F(U). Definition 1.2. [2, P. 441] Let D e the set of analytic functions q univalent on U\E(q), where { } E(q) = ζ U : lim z ζ q =, is such that min q = ρ > 0 for ζ U\E(q). Further, let D(a) = { q D : q(0) = a } D1 = D(1). Definition 1.3. [2, P. 440] Let Ψ : C 4 U C h e univalent in U. If p is analytic in U satisfies the third-order differential suordination: ψ( p, z p, z 2 p, z 3 p ; z) h, (1.5) p is called a solution of the differential suordination. A univalent function q is called a dominant of the solutions of the differential suordination or more simply a dominant if p q for all p satisfying(1.5). A dominant q that satisfies q q for all dominants of (1.5) is called the est dominant of (1.5). Definition 1.4. [2, P. 440] Let Ω e a set in C, q D n N\{1}. The class of admissile functions Ψ n [Ω, q] consists of those functions ψ : C 4 U C that satisfy the admissiility condition : whenever ψ(r, s, t, u; z) Ω r = q, s = kζq, ( ) ( t ζq ) s + 1 k q + 1 ( ) ( u ζ 2 k 2 q ) s q, where z U; ζ U\E(q) k n.

4 A. A. Attiya et al. / Filomat 30:7 (2016), Analogous to the second order differential superordinations introduced y Miller Mocanu [15], Tang et al. [22] defined the differential superordinations as follows: Definition 1.5. [22, P. 3] Let ψ : C 4 U C the function h e analytic in U. If the functions p ψ( p, z p, z 2 p, z 3 p ) are univalent in U satisfy the following third-order differential superordination: h ψ( p, z p, z 2 p, z 3 p ), (1.6) p is called a solution of the differential superordination. An analytic function q is called a suordinant of the solutions of the differential superordination or more simply a suordinant if q p for p satisfying (1.6). A univalent suordinant q that satisfies q q for all supordinants q of (1.6) is said to e the est superordinant. Definition 1.6. [22, P. 4] Let Ω e a set in C, q H[a, n] q 0. The class of admissile functions Ψ n[ω, q] consists of those functions ψ : C 4 U C that satisfy the following admissiility condition : whenever ψ(r, s, t, u; ζ) Ω, r = q, s = zq m, ( ) t s ( zq ) m q + 1 ( ) u 1 ( z 2 s m q ) 2 q, where z U, ζ U m n 2. Also, we need the following theorems in our investigations: Theorem 1.1. [2, p. 449] Let p H[a, n] with n N\{1}. Also, let q D(a) satisfy the following conditions: ( ζq ) zp q > 0, q k, where z U; ζ U\E(q) k n. If Ω is a set in C, ψ Ψ n [Ω, q] ψ( p, z p, z 2 p, z 3 p ; z) Ω, p q.

5 A. A. Attiya et al. / Filomat 30:7 (2016), Theorem 1.2. [22, p. 4] Let q H[a, n] ψ Ψ n[ω, q].if ψ( p, z p, z 2 p, z 3 p ; z) is univalent in U p D(a) satisfy the following conditions: ( zq ) ζp q 0, q m, (z U; ζ U; m n 2) implies that Ω { ψ( p, z p, z 2 p, z 3 p ; z) : z U }, q p. In this paper, y using the third-order differential suordination superordination results y Antonino Miller [2] Tang et al. [22], we define certain classes of admissile functions investigate some suordination superordination properties of meromorphic functions associated with the integrodifferential operator J defined y (1.3). Furthermore, new differential swich-type theorems are s, otained. 2. Third Order Differential Suordination with J s, Definition 2.1. Let Ω e a set in C q D. The class of admissile functions Φ Γ [Ω, q] consists of those functions φ : C 4 U C that satisfy the admissiility condition: whenever φ(a 1, a 2, a 3, a 4 ; z) Ω, a 1 = q(ς), a 2 = k + q ζq, ( ) ( (a3 a 1 ) ζq ) (a 2 a 1 ) 2 k q + 1, ( ) ( 2 (a 4 a 1 ) 3 ( + 1) (a 3 a 1 ) ζ k 2 q ) (a 2 a 1 ) q, where z U, C\Z, s C, ζ U\E(q) k N\{1}. 0 Theorem 2.1. Let φ Φ Γ [Ω, q]. If f Σ q D 1 satisfy the following conditions: ( ζq ) z ( J s 1, q 0, f J s, f ) q k (2.1) { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U} Ω, (2.2) zj s, f q. (2.3)

6 Proof. Let us define the analytic function p as: A. A. Attiya et al. / Filomat 30:7 (2016), p = zj s, f (z U). (2.4) Using the definition of J f, we can prove that s, z ( J s, f ) = J s 1, f ( + 1)J s, f, (2.5) we get zj s 1, f = zp + p, which implies (2.6) zj s 2, f = z2 p + (2 + 1)zp + 2 p 2. (2.7) Also, we can see that zj s 3, f = z3 p + 3 ( + 1) z 2 p + ( )zp + 3 p 3. (2.8) Let us define the parameters a 1, a 2, a 3 a 4 as: a 1 = r, a 2 = s + r, a 3 = t + (1 + 2)s + 2 r 2 a 4 = u + 3 ( + 1) t + ( )s + 3 r 3. Now, we define the transformation ψ : C 4 U C ψ(r, s, t, u; z) = φ(a 1, a 2, a 3, a 4 ; z). (2.9) By using the relations from (2.4) to (2.8), we have ψ( p, z p, z 2 p, z 3 p ; z) (2.10) = φ ( zj s, f, zj s 1, f, zj s 2, f ; zj s 3, f ; z). Therefore, we can rewrite (2.2) as ψ( p, z p, z 2 p, z 3 p ; z) Ω. Then the proof is completed y showing that the admissiility condition for φ Φ Γ [Ω, q] is equivalent to the admissiility condition for ψ as given in Definition (1.3), since t s + 1 = (a 3 a 1 ) 2 a 2 a 1 (2.11) u s = 2 (a 4 a 1 ) 3 ( + 1) (a 3 a 1 ) (a 2 a 1 )

7 We also note that zp q = z ( J s 1, f J s, f ) q k. A. A. Attiya et al. / Filomat 30:7 (2016), Therefore, ψ Ψ 2 [Ω, q] hence y Theorem 1.1, p q. If Ω C is a simply connected domain, Ω = h(u) for some conformal mapping h of U onto Ω. In this case the class Φ Γ [h(u), q] is written as Φ Γ [h, q]. The following theorem is a directly consequence of Theorem 2.1. Theorem 2.2. Let φ Φ Γ [h, q]. If f Σ q D 1 satisfy the following conditions: ( ζq ) z ( J s 1, q 0, f J s, f ) q k (2.12) φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) h, (2.13) zj s, f q. The next corollary is an extension of Theorem 2.1 to the case where the ehavior of q on U is not known. Corollary 2.1. Let Ω C let q e univalent in U, q(0) = 1. Let φ Φ Γ [Ω, q ρ ] for some ρ (0, 1) where q ρ = q(ρz). If f Σ satisfies ( ζq ) ρ q 0, ρ z ( J s 1, f J s, f ) q ρ k (2.14) φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) Ω, (2.15) zj s, f q, where z U ζ U\E(q ρ ). Proof. By using Theorem 2.1, we have J s, f q ρ. Then we otain the result from q ρ q. Corollary 2.2. Let Ω C let q e univalent in U, q(0) = 1. Let φ Φ Γ [h, q ρ ] for some ρ (0, 1) where q ρ = q(ρz). If f Σ satisfies ( ζq ) ρ q 0, ρ z ( J s 1, f J s, f ) q ρ k (2.16)

8 A. A. Attiya et al. / Filomat 30:7 (2016), φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) h, (2.17) zj s, f q, where z U ζ U\E(q ρ ). Theorem 2.3. Let h e univalent in U. Let φ : C 4 U C. Suppose that the differential equation: ( φ q, zq + q, z2 q + (2 + 1)zq + 2 q, (2.18) 2 z 3 q + 3 ( + 1) z 2 q + ( )zq + 3 ) q ; z = h, 3 has a solution q with q(0) = 1 which satisfies (2.1). If f Σ satisfies (2.17) φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) is analytic in U, zj s, f q (2.19) q is the est dominant of (2.19). Proof. By using Theorem 2.1 that q is a dominant of (2.17). Since q satisfies (2.18), it is also a solution of (2.17) therefore q will e dominated y all dominants. Hence q is the est dominant. In the case q = 1 + Mz (M > 0) in view of the Definition 2.1, the class of admissile functions Φ Γ [Ω, q] denoted y Φ Γ [Ω, M] is defined elow. Definition 2.2. Let Ω e a set in C M > 0. The class of admissile functions Φ Γ [Ω, M] consists of those functions φ : C 4 U C that satisfy the admissiility condition φ 1 + ( + k) Meiθ Meiθ, 1 +, 1 + L + ( 2 + k(2 + 1) ) Me iθ, (2.20) N + 3( + 1)L + ( 3 + k ( )) Me iθ ; z Ω, 3 where z U, (Le iθ ) (k 1)kM (Ne iθ ) 0 for all real θ k N\{1}. Corollary 2.3. Let φ Φ Γ [Ω, M]. If f Σ satisfies the following conditions: z ( J s 1, f J s, f ) km (2.21) φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) Ω, (2.22) zj s, 1 < M.

9 A. A. Attiya et al. / Filomat 30:7 (2016), In the case Ω = q(u) = {ω : w 1 < M (M > 0)}, for simplification we denote y Φ Γ [M] to the class Φ Γ [Ω, M]. Corollary 2.4. Let φ Φ Γ [M]. If f Σ satisfies the condition (2.21) φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) 1 < M, (2.23) zj s, 1 < M. Putting φ( a 1, a 2, a 3, a 4 ; z) = a 2 = 1 + (+k)meiθ Corollary 2.5. Let M > 0 C\Z 0 zj s 1, f 1 < M, in Corollary 2.4, we have the following corollary: with () < k 2 (k N\{1}). If f Σ satisfies the condition (2.21) zj s, 1 < M. Corollary 2.6. Let k N\{1}, M > 0 C\Z. If f Σ satisfies the condition 0 z ( J s 1, f J s, f ) < km, (2.24) (2.21) zj s, 1 < M. Proof. Let φ( a 1, a 2, a 3, a 4 ; z) = a 2 a 1. Using Corollary 2.3 with Ω = h (U) h = km z (z U). Now we show that φ Φ Γ [Ω, M]. Since the condition (2.21) is satisfied from the condition (2.24) φ 1 + ( + k) Meiθ Meiθ, 1 +, 1 + L + ( 2 + k(2 + 1) ) Me iθ N + 3( + 1)L + ( 3 + k ( )) Me iθ ; z 3 = kme iθ = km, we have Corollary 2.6.

10 A. A. Attiya et al. / Filomat 30:7 (2016), Corollary 2.7. Let k N\{1}, M > 0 C\Z. If f Σ satisfies the condition (2.21) 0 z ( J s 3, f J s 2, f ) < 2 ( ) M, (2.25) 3 zj s, 1 < M. Proof. We define φ( a 1, a 2, a 3, a 4 ; z) = a 4 a 3. Using Corollary 2.3 with Ω = h (U) h = 2 ( ) M 3 z (z U). Now we show that φ Φ Γ [Ω, M]. Since φ 1 + ( + k) Meiθ Meiθ, 1 +, 1 + L + ( 2 + k(2 + 1) ) Me iθ N + 3( + 1)L + ( 3 + k ( )) Me iθ ; z 3 = N + (2 + 3) L + k ( + 1) 2 Me iθ 3 = Ne iθ + (2 + 3) Le iθ + k ( + 1) 2 M 3 e iθ ( Ne iθ) ( Le iθ) + k M 3 (k 1)kM k M 3 2 ( ) M 3, we completes the proof of Corollary Third Order Differential Superordination with J s, Definition 3.1. Let Ω e a set in C q H with q 0. The class of admissile functions Φ [Ω, q] consists Γ of those functions φ : C 4 U C that satisfy the admissiility condition: whenever φ(a 1, a 2, a 3, a 4 ; ζ) Ω, a 1 = q, a 2 = ζq + q, m

11 A. A. Attiya et al. / Filomat 30:7 (2016), ( ) (a3 a 1 ) (a 2 a 1 ) 2 1 ( ζq ) m q + 1, ( ) 2 (a 4 a 1 ) 3 ( + 1) (a 3 a 1 ) ( z 2 (a 2 a 1 ) m q ) 2 q, where z U, C\Z, s C ζ U m N\{1}. 0 Theorem 3.1. ( zq ) q 0, Let φ Φ Γ [Ω, q]. If f Σ zj s, f D 1 satisfy the following conditions: z ( J s 1, f J s, f ) q m, (3.1) { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U} is univalent, Ω { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U}, (3.2) q zj s, f. Proof. Let the functions p ψe defined y (2.4) (2.9). Since φ Φ [Ω, q].therefore (2.10) (3.2) imply Γ Ω ψ( p, z p, z 2 p, z 3 p ; z). The admissile condition for φ Φ [Ω, q] is equivalent to the admissile condition for ψ in Definition 1.6 with n = 2. Γ Therefore, ψ Ψ [Ω, q], y using (3.1) Theorem 1.2, we have 2 q p which yields q zj s, f. Therefore we completes the proof of Theorem 3.1. If Ω C is a simply connected domain, Ω = h(u) for some conformal mapping h of U onto Ω. In this case the class Φ Γ [h(u), q] is written as Φ [h, q]. Γ The following theorem is a directly consequence of Theorem 2.1. Theorem 3.2. Let φ Φ Γ [h, q]. Also, let h e analytic in U. If f Σ zj s, f D 1 satisfies the condition (3.1), { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U} is univalent in U, h φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z), (3.3) q zj s, f.

12 A. A. Attiya et al. / Filomat 30:7 (2016), Theorem 3.3. Let h e analytic in U, also, let φ : C 4 U C ψ e given y (2.9). Suppose that the differential equation (2.18) has a solution q D 1. If f Σ satisfies the condition (3.1), { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U} is univalent in U, h φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z), q zj s, f. (3.4) q is the est suordinant of (3.3). Proof. The proof is similar to that of Theorem 2.3 it is eing omitted here. By comining Theorem 2.2 Theorem 3.2 we otain the following swich type result. Corollary 3.1. Let h 1 q 1 e analytic in U.Also, let h 2 e univalent in U, q 2 D 1 with q 1 (0) = q 2 (0) = 1 φ Φ Γ [h, q] Φ Γ [h, q]. If f Σ, zj s, f D 1 H, { φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) : z U} is univalent in U, the conditions (2.12) (3.1) are satisfied, Also, let h 1 φ( zj s, f, zj s 1, f, z J s 2, f, z J s 3, f ; z) h 2, (3.5) q 1 zj s, f q 2. ferences [1] K. R. Alhindi M. Darus, A new class of meromorphic functions involving the polylogarithm function, J. Complex Anal. 2014, Art. ID , 5 pp. [2] J. A. Antonino S. S. Miller, Third-order differential inequalities suordinations in the complex plane, Complex Var. Theory Appl. (56) (5) (2011), [3] A.A. Attiya A. Hakami, Some suordination results associated with generalized Srivastava-Attiya operator, Adv. Difference Equ. 2013, 2013:105, 14 pp. [4] S. K. Bajpai, A note on a class of meromorphic univalent functions, v. Roum. Math. Pures Appl. 22 (1977), [5] S.D. Bernardi, Convex starlike univalent functions, Trans. Amer. Math. Soc. 135(1969), [6] Nak E. Cho, Y. S. Woo S. Owa. Argument estimates of certain meromorphic functions. New extension of historical theorems for univalent function theory (Japanese) (Kyoto, 1999), Sūrikaisekikenkyūsho Kōkyūroku No (2000), [7] N.E. Cho, I.H. Kim H.M. Srivastava, Swich-type theorems for multivalent functions associated with the Srivastava-Attiya operator, Appl. Math. Comput. 217 (2010), no. 2, [8] J. Choi, D.S. Jang H.M. Srivastava, A generalization of the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 19(2008), no. 1-2, [9] C. Ferreira J.L. López, Asymptotic expansions of the Hurwitz-Lerch zeta function, J. Math. Anal. Appl. 298(2004), [10] P.L. Gupta, R.C. Gupta, S. Ong H.M. Srivastava, A class of Hurwitz-Lerch zeta distriutions their applications in reliaility, Appl. Math. Comput. 196 (2008), no. 2, [11] M.A. Kuti A.A. Attiya, Differential suordination result with the Srivastava-Attiya integral operator, J. Inequal. Appl. 2010(2010), [12] M.A. Kuti A.A. Attiya, Differential suordination results for certain integrodifferential operator it s applications, Astr. Appl. Anal., 2012(2012), 13 pp. [13] A.Y. Lashin, On certain suclasses of meromorphic functions associated with certain integral operators. Comput. Math. Appl. 59 (2010), no. 1, [14] Q.M. Luo H.M. Srivastava, Some generalizations of the Apostol-Bernoulli Apostol-Euler polynomials, J. Math. Anal. Appl. 308(2005), [15] S.S. Miller P.T. Mocanu, Suordinants of differential superordinations, Complex Var. Theory Appl. 48 (2003), no. 10,

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