JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi 10.1155/2018/5915864 5915864 Research Article Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function http://orcid.org/0000-0002-6550-8046 Wang Xiao-Yuan 1 http://orcid.org/0000-0003-3079-9944 Shi Lei 2 Wang Zhi-Ren 1 Goswami Pranay 1 College of Science Yanshan University Hebei Qinhuangdao 066004 China ysu.edu.cn 2 School of Mathematics and Statistics Anyang Normal University Henan Anyang 455000 China aynu.edu.cn 2018 842018 2018 16 08 2017 06 03 2018 842018 2018 Copyright © 2018 Xiao-Yuan Wang et al. 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 aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.

National Natural Science Foundation of China 11301008 Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province 2013GGJS-146 Foundation of Educational Committee of Henan Province 17A110014
1. Introduction

Denote by H(U) the class of functions analytic in the unite disk(1)U=z:zC,z<1of the form (2)Ha,n=f:fHU,fz=a+k=nakzkaC;nN=1,2, and let H=H[1,1].

For two functions f(z) and g(z) to be analytic in U, f(z) is said to be subordinate to g(z) in U and written by (3)fzgzzU,if there exists a Schwarz function ω(z), which is analytic in U, with(4)ω0=0,ωz<1,such that (5)fz=gωzzU.It is generally known that (6)fzgzzUf0=g0,fUgU.Furthermore, if the function g(z) is univalent in U, then (7)fzgzzUf0=g0,fUgU.

Denote by Q the set of functions q(z) that are analytic and univalent on U¯E(q), where (8)Eq=ζU:limzζqz=are such that minqζ=ε>0 for ζUE(q). Furthermore, let (9)Qa=qzQ:q0=a,Q1=Q1.

Denote by A the class of functions of the form(10)fz=1z+n=1anzn,which are analytic in the punctured unit disk (11)U=zC,0<z<1=U0.

We recall the general Hurwitz–Lerch Zeta function Φ(z,s,a) (see, e.g., [1, p. 121] and [2, p. 194]) defined by(12)Φz,s,ak=0zkk+asaCZ0-;sCwhenz<1;Rs>1whenz=1,where (13)Z0-Z-0=0,-1,-2,.

In recent years, the general Hurwitz–Lerch Zeta function Φ(z,s,a) was investigated by many researchers. A huge amount of interesting properties and consequences can be found in, for example, Choi and Srivastava , Garg et al. , Lin and Srivastava , and Srivastava et al. .

In 2007, by involving the general Hurwitz–Lerch Zeta function Φ(z,s,a), Srivastava and Attiya  (also see ) introduced the integral operator(14)Js,bfz=z+k=21+bk+bsckzkbCZ-;sC;zU.

Analogous to abovementioned operator Js,bf, Wang and Shi  introduced a new integral operator(15)Ws,b:ΣΣdefined by(16)Ws,bfzΘs,bzfzbCZ0-1;sC;fΣ;zU,where(17)Θs,bzb-1sΦz,s,b-b-s+1zb-1szU,and “” denotes the Hadamard product.

From (10), (12), (16), and (17), we easily find that(18)Ws,bfz=1z+k=1b-1b+ksakzk.It is true that bC{Z-{1}}, the integral operator Ws,b defined as (19)Ws,0fzlimb0Ws,bfz.

We can deduce that(20)W0,bfz=fz,(21)W-1,0fz=-zfz,(22)W-1,-1fz=fz-zfz2,(23)Ws,2fz=1z+k=11k+2sakzk,(24)W1,b+1fz=1z+k=1bk+b+1akzk=bzb+10ztbftdtb>0,(25)Wα,β+1fz=βαΓszβ+10ztblogzts-1ftdtα>0;β>0.

We also see that(26)W1,γfz=γ-1zγ0ztγ-1ftdtRγ>1.Furthermore, by (18), we observe that(27)Ws+1,bfz=b-1zb0ztb-1Ws,bfzdtRb>1.Operator (23) was introduced and studied by Alhindi and Darus ; operators (24) and (25) were introduced by Lashin .

The main purpose of this paper is to derive some third-order differential subordination, differential superordination properties, and sandwich-type theorems of the integral operator Ws,bf(z).

2. Preliminary Results

We will investigate our main results by using following definitions and lemmas.

Definition 1 (see [<xref ref-type="bibr" rid="B2">15</xref>, p. 440, Definition  1]).

Suppose that Ψ:C4×UC, q(z), and h(z) are univalent in U. If p(z) is analytic in U and satisfies the third-order differential subordination(28)ψpz,zpz,z2pz,z3pz;zhz,then p(z) is called a solution of the differential subordination. q(z) is called a dominant of the solutions of the differential subordination or more simply a dominant if p(z)q(z) for all p(z) satisfying (28). A dominant q~(z) that satisfies (29)q~zqz,for all dominants of (28), is called the best dominant of (28).

As the second-order differential superordinations were introduced and investigated by Miller and Mocanu , Tang et al.  introduced the following third-order differential superordinations.

Definition 2 (see [<xref ref-type="bibr" rid="B17">17</xref>, p. 3, Definition 5]).

Suppose that ψ:C4×UC and the function h(z) is analytic in U. If the functions p(z) and (30)ψpz,zpz,z2pz,z3pz;zare univalent in U and satisfy the third-order differential superordination(31)hzψpz,zpz,z2pz,z3pz;z,then p(z) is called a solution of the differential superordination. An analytic function q(z) is called a subordinant of the solutions of the differential superordination or more simply a subordinant if q(z)p(z) satisfies (31) for p(z) satisfying (31). A univalent subordinant q~(z) that satisfies (32)qzq~zfor all superordinants q(z) of (31) is said to be the best superordinant.

Lemma 3 (see [<xref ref-type="bibr" rid="B9">18</xref>, p. 132], [<xref ref-type="bibr" rid="B11">19</xref>, p. 190]).

Suppose that q is univalent in the open unit disk U and θ and ϕ are analytic in a domain D containing q(U) with ϕ(ω)0 when ωq(U). Set Φ(z)=zq(z)ϕ(q(z)) and h(z)=θ(q(z))+Φ(z). Suppose that

Φ is star-like in U;

Rzh(z)/Φ(z)>0.

If pH[q(0),n] for some nN with p(U)D and(33)θpz+zpzϕpzθqz+zqzϕqz,then pq and q is the best dominant.

Lemma 4 (see [<xref ref-type="bibr" rid="B10">20</xref>, p. 332]).

Suppose that q is univalent in the open unit disk U and θ and ϕ are analytic in a domain D containing q(U). Set Φ(z)=zq(z)ϕ(q(z)). Suppose that

Φ is star-like in U;

Rθ(q(z))/ϕ(q(z))>0.

If pH[q(0),1]Q, with p(U)D, θ(p(z))+zp(z)ϕ(p(z)) is univalent in U, and (34)θqz+zqzϕqzθpz+zpzϕpz,then qp and q is the best dominant.

Lemma 5 (see [<xref ref-type="bibr" rid="B8">16</xref>, p. 822]).

Suppose that q is univalent complex in the open unit disk U and γC, with R(γ)>0. If pH[q(0),1]Q, p(z)+γzp(z) is univalent in U, and (35)qz+γzqzpz+γzpzzU,then qp and q is the best dominant.

3. Main Results

In this section, we state several third-order differential subordination and differential superordination results associated with the operator Ws,bf(z).

Theorem 6.

Suppose that the function qA is nonzero univalent in U with q(0)=1 and(36)R1+zqzqz-zqzqz>0zU.Let 0ρ1 and ηC. If fH[0,p] satisfies (37)1-ρzWs,bfz+ρzWs+1,bfz0zU,(38)η1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1zqzqz,then(39)1-ρzWs,bfz+ρzWs+1,bfzηqzand q is the best dominant in (39). When η=0 the left hand side expressions in (39) are interpreted as 1.

Proof.

Suppose that(40)pz1-ρzWs,bfz+ρzWs+1,bfzη.Then p is analytic in U. Logarithmically differentiating both sides of (40) with respect to z, we have(41)zpzpz=η1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1.To apply Lemma 3, we set (42)θω1,ϕω1ωωC0,Φz=zqzϕqz=zqzqzzU,hz=θqz+Φz=1+zqzqz.By means of (36) we see that Φ(z) is univalent star-like in U. Since h(z)=1+Φ(z), we furthermore get that(43)RzhzΦz>0.By a routine calculation using (40) and (41) we find that (44)θpz+zpzϕpz=1+η1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1.Therefore, hypothesis (38) is equivalently written as (45)θpz+zpzϕpz1+zqzqz=θqz+zqzϕqz.We know that condition (33) is also satisfied. From an application of Lemma 3, we have (46)pzqz.Thus, we get the assertions in (39). Thus, the proof of Theorem 6 is completed.

Theorem 7.

Suppose that the function qA is a univalent mapping of U into the right half plane with q(0)=1 and(47)R1+zqzqz-zqzqz>0zU.Let 0ρ1 and ηC, fH[0,p] satisfy (48)1-ρzWs,bfz+ρzWs+1,bfz0zU.If(49)Δzqz+zqzqz,where(50)Δz=1-ρzWs,bfz+ρzWs+1,bfzη+η1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1,then(51)1-ρzWs,bfz+ρzWs+1,bfzηqzand q is the best dominant in (51). When η=0, the left hand side expression of (51) is interpreted as 1.

Proof.

Suppose that the function p(z) is defined by (40). If set (52)θωω,ϕω1ωωC0,Φz=zqzϕqz=zqzqzzU,hz=θqz+Φz=qz+Φzwe easily get (53)RzhzΦz=Rqz+1+zqzqz-zqzqz>0zU.By virtue of (41), hypothesis (49) can be rewritten as (54)θpz+zpzϕpzθqz+zqzϕqz.Therefore, by making use of Lemma 3, we derive that (55)pzqzzU.Thus, the assertion in (49) follows. The proof of Theorem 7 is completed.

Theorem 8.

Suppose that the function qA is a univalent mapping of U into the right half plane with q(0)=1 and satisfies condition(56)R1+zqzqz-zqzqz>0zU.Let 0ρ1, ηC, and fH[0,p] satisfy (57)1-ρzWs,bfz+ρzWs+1,bfzηH1,1Q.Let function Δ(z) be univalent in U, where Δ(z) is defined by (50). If(58)qz+zqzqzΔz,then(59)qz1-ρzWs,bfz+ρzWs+1,bfzηand q is the best subordinant in (59). When η=0, the left hand side expressions of (59) are interpreted as 1.

Proof.

By putting (60)θωω,ϕω1ωωC0,Φz=zqzϕqz=zqzqzzU,obviously, Φ is star-like in U and (61)Rθqzϕqz=RqzzU.Suppose that function p is defined by (40). By simple calculation, from (41), we know that (62)θpz+zpzϕpz=Δz.Hence, condition (58) can be equivalently written as (63)θqz+zqzϕqzθpz+zpzϕpz.Therefore, by Lemma 4, we have (64)qzpzzUand q is the best subordinant. The proof of Theorem 8 is completed.

Theorem 9.

Suppose that 0ρ1, α,ηC, the function qA is univalent in U, and(65)R1+zqzqz>max0,-Rα.Let fH[0,p] satisfy (66)1-ρzWs,bfz+ρzWs+1,bfz0zU.Denote by(67)Ξz=1-ρzWs,bfz+ρzWs+1,bfzη×α+η1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1zU.If(68)Ξzαqz+zqz,then(69)1-ρzWs,bfz+ρzWs+1,bfzηqzand q is the best dominant in (69). When η=0, the left side hand expressions of (69) are interpreted as 1.

Proof.

Suppose that function p(z) is defined by (40). Making using of (41), we have(70)zpz=ηpz1-ρzWs,bfz+ρzWs+1,bfz1-ρWs,bfz+ρWs+1,bfz-1.Therefore, by putting (71)θωαω,ϕω1ωC,Φz=zqzϕqz=zqzzU,hz=θqz+Φz=αqz+zqz,obviously, Φ is star-like in U and (72)RzhzΦz=Rα+1+zqzqz>0.Furthermore, by substituting the expression for p(z),zp(z) from (40) and (70), respectively, we get (73)θpz+zpzϕpz=αpz+zpzϕpz=Ξz,where Ξ(z) is given by (67). Hypothesis (68) can be equivalently written as (74)θpz+zpzϕpzθqz+zqzϕqz.From Lemma 3, we get (75)pzqz.Thus, we get assertion (69) of Theorem 9.

Theorem 10.

Suppose that 0ρ1,ηC,αC{0},R(α)>0; function qA is univalent in U with q(0)=1. Let function fH[0,p] satisfy (76)1-ρzWs,bfz+ρzWs+1,bfz0zU,1-ρzWs,bfz+ρzWs+1,bfzηH1,1Q.If Ξ(z) defined by (67) is univalent and satisfies(77)αqz+zqzΞz,then(78)1-ρzWs,bfz+ρzWs+1,bfzηqzand q is the best subordinant in (78). When η=0, the left hand side expressions of (78) are interpreted as 1.

Proof.

Suppose that function p(z) is defined by (40). From (41), we get (79)αpz+zpzϕpz=Ξz.Hypothesis (77) can be rewritten as (80)qz+1αzqzpz+1αzpz.Then, combining Lemma 5 with γ=1/α, we have (78). Theorem 10 follows immediately.

Following that, we display some sandwich-type theorems associated with the operator Ws,bf(z).

Theorem 11.

Suppose that functions q1,q2A are univalent mapping of U into the right half plane and satisfy conditions (81)q10=q20=1,R1+zqjzqjz-zqjzqjz>0j=1,2;zU.Let 0ρ1, α,ηC, and fH[0,p] satisfy (82)1-ρzWs,bfz+ρzWs+1,bfz0zU,1-ρzWs,bfz+ρzWs+1,bfzηH1,1Q.If function Δ(z) is given by (50) and satisfies (83)q1z+zq1zq1zΔzq2z+zq2zq2z,then(84)q1z1-ρzWs,bfz+ρzWs+1,bfzηq2z,where q1 and q2 are, respectively, the best subordinant and the best dominant in (84).

Combining Theorems 9 and 10, we get the following result.

Corollary 12.

Suppose that 0ρ1, ηC, and αC{0} with R(α)>0. Functions q1 and q2 are univalent convex in U with q1(0)=q2(0)=1. Let fH[0,p] satisfy (85)1-ρzWs,bfz+ρzWs+1,bfz0zU,1-ρzWs,bfz+ρzWs+1,bfzηH1,1Q.If function Ξ(z) is given by (67) and satisfies (86)q1z+zq1zΞzαq2z+zq2z,then(87)q1z1-ρzWs,bfz+ρzWs+1,bfzηq2z,where q1 and q2 are, respectively, the best subordinant and the best dominant in (87).

4. Conclusions

In the present paper, making use of the integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function, we have derived several third-order differential subordination and differential superordination consequences of meromorphic functions in the punctured unit disk. Furthermore, the sandwich-type theorems are considered. These subordinate relationships have shown the upper and lower bounds of the operator in the punctured unit disk.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of the paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant no. 11301008, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Foundation of Educational Committee of Henan Province under Grant no. 17A110014.

Srivastava H. M. Choi J. Series Associated with the Zeta and Related Functions 2001 London, UK Kluwer Academic 10.1007/978-94-015-9672-5 MR1849375 Zbl1014.33001 Srivastava H. M. Choi J. Zeta and q-Zeta Functions and Associated Series and Integrals 2012 Amsterdam, Netherlands Elsevier Zbl1239.33002 Choi J. Srivastava H. M. Certain families of series associated with the Hurwitz-Lerch zeta function Applied Mathematics and Computation 2005 170 1 399 409 MR2177231 10.1016/j.amc.2004.12.004 Zbl1082.11052 2-s2.0-26844443518 Garg M. Jain K. Srivastava H. M. Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions Integral Transforms and Special Functions 2006 17 11 803 815 MR2263956 10.1080/10652460600926907 Zbl1184.11005 2-s2.0-33749589040 Lin S.-D. Srivastava H. M. Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations Applied Mathematics and Computation 2004 154 3 725 733 MR2072816 10.1016/S0096-3003(03)00746-X Zbl1078.11054 2-s2.0-3042752343 M. Srivastava H. Luo M. K. Raina R. New Results Involving a Class of Generalized Hurwitz-Lerch Zeta Functions and Their Applications Turkish Journal of Analysis and Number Theory 2013 1 1 26 35 10.12691/tjant-1-1-7 Srivastava H. M. Attiya A. A. An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination Integral Transforms and Special Functions 2007 18 3-4 207 216 MR2319582 10.1080/10652460701208577 Zbl1112.30007 2-s2.0-33947384551 Liu J.-L. Sufficient conditions for strongly star-like functions involving the generalized Srivastava-Attiya operator Integral Transforms and Special Functions 2011 22 2 79 90 MR2749387 10.1080/10652469.2010.498110 Zbl1207.30017 2-s2.0-78650271031 Wang Z.-G. Liu Z.-H. Sun Y. Some properties of the generalized Srivastava-Attiya operator Integral Transforms and Special Functions 2012 23 3 223 236 MR2891465 10.1080/10652469.2011.585425 Zbl1246.30033 2-s2.0-84863372262 Sun Y. Kuang W.-P. Wang Z.-G. Properties for uniformly starlike and related functions under the Srivastava-Attiya operator Applied Mathematics and Computation 2011 218 7 3615 3623 MR2851461 10.1016/j.amc.2011.09.002 Zbl1244.30030 2-s2.0-80054973101 Yuan S.-M. Liu Z.-M. Some properties of two subclasses of k-fold symmetric functions associated with Srivastava-Attiya operator Applied Mathematics and Computation 2011 218 3 1136 1141 10.1016/j.amc.2011.03.080 MR2831369 Wang Z.-G. Shi L. Some subclasses of meromorphic functions involving the Hurwitz-Lerch zeta function Hacettepe Journal of Mathematics and Statistics 2016 45 5 1449 1460 MR3699555 Zbl1368.30009 2-s2.0-84996497472 10.15672/HJMS.20164514284 Alhindi K. R. Darus M. A new class of meromorphic functions involving the polylogarithm function Journal of Complex Analysis 2014 2014 5 864805 10.1155/2014/864805 MR3251596 Lashin A. Y. On certain subclasses of meromorphic functions associated with certain integral operators Computers & Mathematics with Applications 2010 59 1 524 531 10.1016/j.camwa.2009.06.015 MR2575541 Antonino J. A. Miller S. S. Third-order differential inequalities and subordinations in the complex plane Complex Variables and Elliptic Equations. An International Journal 2011 56 5 439 454 MR2795467 10.1080/17476931003728404 Zbl1220.30035 2-s2.0-79956095971 Miller S. S. Mocanu P. T. Subordinants of differential superordinations Complex Variables, Theory and Application 2003 48 10 815 826 10.1080/02781070310001599322 MR2014390 Zbl1039.30011 Tang H. Srivastava H. M. Li S.-H. Ma L.-N. Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator Abstract and Applied Analysis 2014 2014 2-s2.0-84904664868 10.1155/2014/792175 792175 Miller S. S. Mocanu P. T. Differential Subordinations: Theory and Applications 2000 225 New York, NY, USA Marcel Dekker Monographs and Textbooks in Pure and Applied Mathematics MR1760285 Zbl0954.34003 Miller S. S. Mocanu P. T. On some classes of first-order differential subordinations Michigan Mathematical Journal 1985 32 2 185 195 10.1307/mmj/1029003185 MR783572 Zbl0575.30019 Miller S. S. Mocanu P. T. Briot-Bouquet differential superordinations and sandwich theorems Journal of Mathematical Analysis and Applications 2007 329 1 327 335 MR2306804 10.1016/j.jmaa.2006.05.080 Zbl1138.30009 2-s2.0-33846308876