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 China11301008Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province2013GGJS-146Foundation of Educational Committee of Henan Province17A1100141. Introduction

Denote by H(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 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)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 by Q the set of functions q(z) that are analytic and univalent on U¯∖E(q), where (8)Eq=ζ∈∂U:limz→ζqz=∞are such that minq′ζ=ε>0 for ζ∈∂U∖E(q). Furthermore, let (9)Qa=qz∈Q:q0=a,Q1=Q1.

Denote by A∗ the class of functions of the form(10)fz=1z+∑n=1∞anzn,which are analytic in the punctured unit disk (11)U∗=z∈C,0<z<1=U∖0.

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,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 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 [3], Garg et al. [4], Lin and Srivastava [5], and Srivastava et al. [6].

In 2007, by involving the general Hurwitz–Lerch Zeta function Φ(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 operator Js,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→0Ws,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∫0ztblogzts-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 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×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 [<xref ref-type="bibr" rid="B17">17</xref>, p. 3, Definition 5]).

Suppose that ψ: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 [<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 ω∈q(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 p∈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 [<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 p∈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 [<xref ref-type="bibr" rid="B8">16</xref>, p. 822]).

Suppose that q 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 operator Ws,bf(z).

Theorem 6.

Suppose that the function q∈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 function q∈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 function p(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 function q∈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 that 0≤ρ≤1, α,η∈C, the function q∈A∗ is univalent in U, and(65)R1+zq′′zq′z>max0,-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 function p(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 that 0≤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 function p(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 operator Ws,bf(z).

Theorem 11.

Suppose that functions q1,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 Theorems 9 and 10, we get the following result.

Corollary 12.

Suppose that 0≤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 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.

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