# Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function

Authors: Xiao-Yuan Wang; Lei Shi; Zhi-Ren Wang
Journal: Journal of Complex Analysis
 (2018)

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## Abstract

The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operatorWs,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.


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## Body

## 1. Introduction

Denote byH(U) the class of functions analytic in the unite disk(1)U=z:z∈C,z<1of the form (2)Ha,n=f:f∈HU,fz=a+∑k=n∞akzka∈C;n∈N=1,2,… and let H=H[1,1].For two functionsf(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)fz≺gzz∈U,if there exists a Schwarz function ω(z), which is analytic in U, with(4)ω0=0,ωz<1,such that (5)fz=gωzz∈U.It is generally known that (6)fz≺gzz∈U⟹f0=g0,fU⊂gU.Furthermore, if the function g(z) is univalent in U, then (7)fz≺gzz∈U⇔f0=g0,fU⊂gU.Denote byQ the set of functions q(z) that are analytic and univalent on U¯∖E(q), where (8)Eq=ζ∈∂U:limz→ζ⁡qz=∞are such that min⁡q′ζ=ε>0 for ζ∈∂U∖E(q). Furthermore, let (9)Qa=qz∈Q:q0=a,Q1=Q1.Denote byA∗ the class of functions of the form(10)fz=1z+∑n=1∞anzn,which areanalytic in thepunctured unit disk (11)U∗=z∈C,0<z<1=U∖0.We recall the general Hurwitz–Lerch Zeta functionΦ(z,s,a) (see, e.g., [1, p. 121] and [2, p. 194]) defined by(12)Φz,s,a≔∑k=0∞zkk+asa∈C∖Z0-;s∈Cwhenz<1;Rs>1whenz=1,where (13)Z0-≔Z-∪0=0,-1,-2,….In recent years, the general Hurwitz–Lerch Zeta functionΦ(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 [3], Garg et al. [4], Lin and Srivastava [5], and Srivastava et al. [6].In 2007, by involving the general Hurwitz–Lerch Zeta functionΦ(z,s,a), Srivastava and Attiya [7] (also see [8–11]) introduced the integral operator(14)Js,bfz=z+∑k=2∞1+bk+bsckzkb∈C∖Z-;s∈C;z∈U.Analogous to abovementioned operatorJs,bf, Wang and Shi [12] introduced a new integral operator(15)Ws,b:Σ⟶Σdefined by(16)Ws,bfz≔Θs,bz∗fzb∈C∖Z0-∪1;s∈C;f∈Σ;z∈U∗,where(17)Θs,bz≔b-1sΦz,s,b-b-s+1zb-1sz∈U∗,and “∗” denotes the Hadamard product.From (10), (12), (16), and (17), we easily find that(18)Ws,bfz=1z+∑k=1∞b-1b+ksakzk.It is true that b∈C∖{Z-∪{1}}, the integral operator Ws,b defined as (19)Ws,0fz≔limb→0⁡Ws,bfz.We can deduce that(20)W0,bfz=fz,(21)W-1,0fz=-zf′z,(22)W-1,-1fz=fz-zf′z2,(23)Ws,2fz=1z+∑k=1∞1k+2sakzk,(24)W1,b+1fz=1z+∑k=1∞bk+b+1akzk=bzb+1∫0ztbftdtb>0,(25)Wα,β+1fz=βαΓszβ+1∫0ztblog⁡zts-1ftdtα>0;β>0.We also see that(26)W1,γfz=γ-1zγ∫0ztγ-1ftdtRγ>1.Furthermore, by (18), we observe that(27)Ws+1,bfz=b-1zb∫0ztb-1Ws,bfzdtRb>1.Operator (23) was introduced and studied by Alhindi and Darus [13]; operators (24) and (25) were introduced by Lashin [14].The main purpose of this paper is to derive some third-order differential subordination, differential superordination properties, and sandwich-type theorems of the integral operatorWs,bf(z).


## 2. Preliminary Results

We will investigate our main results by using following definitions and lemmas.Definition 1 (see [15, p. 440, Definition  1]).
Suppose thatΨ:C4×U→C, 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,zp′z,z2p′′z,z3p′′′z;z≺hz,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~z≺qz,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 [16], Tang et al. [17] introduced the following third-order differential superordinations.Definition 2 (see [17, p. 3, Definition 5]).
Suppose thatψ:C4×U→C and the function h(z) is analytic in U. If the functions p(z) and (30)ψpz,zp′z,z2p′′z,z3p′′′z;zare univalent in U and satisfy the third-order differential superordination(31)hz≺ψpz,zp′z,z2p′′z,z3p′′′z;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)qz≺q~zfor all superordinants q(z) of (31) is said to be the best superordinant.Lemma 3 (see [18, p. 132], [19, p. 190]).
Suppose thatq 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 ω∈q(U). Set Φ(z)=zq′(z)ϕ(q(z)) and h(z)=θ(q(z))+Φ(z). Suppose that(1)
Φ is star-like in U;(2)
Rzh′(z)/Φ(z)>0.
Ifp∈H[q(0),n] for some n∈N with p(U)⊂D and(33)θpz+zp′zϕpz≺θqz+zq′zϕqz,then p≺q and q is the best dominant.Lemma 4 (see [20, p. 332]).
Suppose thatq 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 (1)
Φ is star-like in U;(2)
Rθ′(q(z))/ϕ(q(z))>0.
Ifp∈H[q(0),1]∩Q, with p(U)⊆D, θ(p(z))+zp′(z)ϕ(p(z)) is univalent in U, and (34)θqz+zq′zϕqz≺θpz+zp′zϕpz,then q≺p and q is the best dominant.Lemma 5 (see [16, p. 822]).
Suppose thatq is univalent complex in the open unit disk U and γ∈C, with R(γ)>0. If p∈H[q(0),1]∩Q, p(z)+γzp′(z) is univalent in U, and (35)qz+γzq′z≺pz+γzp′zz∈U,then q≺p 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 operatorWs,bf(z).Theorem 6.
Suppose that the functionq∈A∗ is nonzero univalent in U with q(0)=1 and(36)R1+zq′′zq′z-zq′zqz>0z∈U.Let 0≤ρ≤1 and η∈C. If f∈H[0,p] satisfies (37)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U,(38)η1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1≺zq′zqz,then(39)1-ρzWs,bfz+ρzWs+1,bfzη≺qzand 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)pz≔1-ρzWs,bfz+ρzWs+1,bfzη.Then p is analytic in U. Logarithmically differentiating both sides of (40) with respect to z, we have(41)zp′zpz=η1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1.To apply Lemma 3, we set (42)θω≔1,ϕω≔1ωω∈C∖0,Φz=zq′zϕqz=zq′zqzz∈U,hz=θqz+Φz=1+zq′zqz.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)Rzh′zΦz>0.By a routine calculation using (40) and (41) we find that (44)θpz+zp′zϕpz=1+η1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1.Therefore, hypothesis (38) is equivalently written as (45)θpz+zp′zϕpz≺1+zq′zqz=θqz+zq′zϕqz.We know that condition (33) is also satisfied. From an application of Lemma 3, we have (46)pz≺qz.Thus, we get the assertions in (39). Thus, the proof of Theorem 6 is completed.Theorem 7.
Suppose that the functionq∈A∗ is a univalent mapping of U into the right half plane with q(0)=1 and(47)R1+zq′′zq′z-zq′zqz>0z∈U.Let 0≤ρ≤1 and η∈C, f∈H[0,p] satisfy (48)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U.If(49)Δz≺qz+zq′zqz,where(50)Δz=1-ρzWs,bfz+ρzWs+1,bfzη+η1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1,then(51)1-ρzWs,bfz+ρzWs+1,bfzη≺qzand 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 functionp(z) is defined by (40). If set (52)θω≔ω,ϕω≔1ωω∈C∖0,Φz=zq′zϕqz=zq′zqzz∈U,hz=θqz+Φz=qz+Φzwe easily get (53)Rzh′zΦz=Rqz+1+zq′′zq′z-zq′zqz>0z∈U.By virtue of (41), hypothesis (49) can be rewritten as (54)θpz+zp′zϕpz≺θqz+zq′zϕqz.Therefore, by making use of Lemma 3, we derive that (55)pz≺qzz∈U.Thus, the assertion in (49) follows. The proof of Theorem 7 is completed.Theorem 8.
Suppose that the functionq∈A∗ is a univalent mapping of U into the right half plane with q(0)=1 and satisfies condition(56)R1+zq′′zq′z-zq′zqz>0z∈U.Let 0≤ρ≤1, η∈C, and f∈H[0,p] satisfy (57)1-ρzWs,bfz+ρzWs+1,bfzη∈H1,1∩Q.Let function Δ(z) be univalent in U, where Δ(z) is defined by (50). If(58)qz+zq′zqz≺Δz,then(59)qz≺1-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)θω≔ω,ϕω≔1ωω∈C∖0,Φz=zq′zϕqz=zq′zqzz∈U,obviously, Φ is star-like in U and (61)Rθ′qzϕqz=Rqzz∈U.Suppose that function p is defined by (40). By simple calculation, from (41), we know that (62)θpz+zp′zϕpz=Δz.Hence, condition (58) can be equivalently written as (63)θqz+zq′zϕqz≺θpz+zp′zϕpz.Therefore, by Lemma 4, we have (64)qz≺pzz∈Uand q is the best subordinant. The proof of Theorem 8 is completed.Theorem 9.
Suppose that0≤ρ≤1, α,η∈C, the function q∈A∗ is univalent in U, and(65)R1+zq′′zq′z>max⁡0,-Rα.Let f∈H[0,p] satisfy (66)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U.Denote by(67)Ξz=1-ρzWs,bfz+ρzWs+1,bfzη×α+η1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1z∈U.If(68)Ξz≺αqz+zq′z,then(69)1-ρzWs,bfz+ρzWs+1,bfzη≤qzand q is the best dominant in (69). When η=0, the left side hand expressions of (69) are interpreted as 1.Proof.
Suppose that functionp(z) is defined by (40). Making using of (41), we have(70)zp′z=ηpz1-ρzWs,bfz′+ρzWs+1,bfz′1-ρWs,bfz+ρWs+1,bfz-1.Therefore, by putting (71)θω≔αω,ϕω≔1ω∈C,Φz=zq′zϕqz=zq′zz∈U,hz=θqz+Φz=αqz+zq′z,obviously, Φ is star-like in U and (72)Rzh′zΦz=Rα+1+zq′′zq′z>0.Furthermore, by substituting the expression for p(z),zp′(z) from (40) and (70), respectively, we get (73)θpz+zp′zϕpz=αpz+zp′zϕpz=Ξz,where Ξ(z) is given by (67). Hypothesis (68) can be equivalently written as (74)θpz+zp′zϕpz≺θqz+zq′zϕqz.From Lemma 3, we get (75)pz≺qz.Thus, we get assertion (69) of Theorem 9.Theorem 10.
Suppose that0≤ρ≤1,η∈C,α∈C∖{0},R(α)>0; function q∈A∗ is univalent in U with q(0)=1. Let function f∈H[0,p] satisfy (76)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U,1-ρzWs,bfz+ρzWs+1,bfzη∈H1,1∩Q.If Ξ(z) defined by (67) is univalent and satisfies(77)αqz+zq′z≺Ξz,then(78)1-ρzWs,bfz+ρzWs+1,bfzη≺qzand q is the best subordinant in (78). When η=0, the left hand side expressions of (78) are interpreted as 1.Proof.
Suppose that functionp(z) is defined by (40). From (41), we get (79)αpz+zp′zϕpz=Ξz.Hypothesis (77) can be rewritten as (80)qz+1αzq′z≺pz+1αzp′z.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 operatorWs,bf(z).Theorem 11.
Suppose that functionsq1,q2∈A∗ are univalent mapping of U into the right half plane and satisfy conditions (81)q10=q20=1,R1+zqj′′zqj′z-zqj′zqjz>0j=1,2;z∈U.Let 0≤ρ≤1, α,η∈C, and f∈H[0,p] satisfy (82)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U,1-ρzWs,bfz+ρzWs+1,bfzη∈H1,1∩Q.If function Δ(z) is given by (50) and satisfies (83)q1z+zq1′zq1z≺Δz≺q2z+zq2′zq2z,then(84)q1z≺1-ρzWs,bfz+ρzWs+1,bfzη≺q2z,where q1 and q2 are, respectively, the best subordinant and the best dominant in (84).Combining Theorems9 and 10, we get the following result.Corollary 12.
Suppose that0≤ρ≤1, η∈C, and α∈C∖{0} with R(α)>0. Functions q1 and q2 are univalent convex in U with q1(0)=q2(0)=1. Let f∈H[0,p] satisfy (85)1-ρzWs,bfz+ρzWs+1,bfz≠0z∈U,1-ρzWs,bfz+ρzWs+1,bfzη∈H1,1∩Q.If function Ξ(z) is given by (67) and satisfies (86)q1z+zq1′z≺Ξz≺αq2z+zq2′z,then(87)q1z≺1-ρzWs,bfz+ρzWs+1,bfzη≺q2z,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 operatorWs,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.



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*Source: 5915864-2018-04-08.xml*