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International Journal of Differential Equations
Volume 2011 (2011), Article ID 902830, 19 pages
http://dx.doi.org/10.1155/2011/902830
Research Article

Differential Subordination and Superordination for Srivastava-Attiya Operator

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D. Ehsan, Malaysia

Received 28 March 2011; Accepted 1 June 2011

Academic Editor: Khalil Ezzinbi

Copyright © 2011 Maisarah Haji Mohd and Maslina Darus. 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.

Abstract

Due to the well-known Srivastava-Attiya operator, we investigate here some results relating the p-valent of the operator with differential subordination and subordination. Further, we obtain some interesting results on sandwich-type theorem for the same.

1. Introduction and Motivation

Let (𝑈) be the class of analytic functions in the open unit disc 𝑈 and let [𝑎,𝑛] be the subclass of (𝑈) consisting functions of the form 𝑓(𝑧)=𝑎+𝑎𝑛𝑧𝑛+𝑎𝑛+1𝑧𝑛+1+, with 0=[0,1] and =[1,1]. For two functions 𝑓1 and 𝑓2 analytic in 𝑈, the function 𝑓1 is subordinate to 𝑓2, or 𝑓2 superordinate to 𝑓1, written as 𝑓1𝑓2 if there exists a function 𝑤(𝑧), analytic in 𝑈 with 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑓1(𝑧)=𝑓2(𝑤(𝑧)). In particular, if the function 𝑓2 is univalent in 𝑈, then 𝑓1𝑓2 is equivalent to 𝑓1(0)=𝑓2(0) and 𝑓1(𝑈)𝑓2(𝑈).

Let 𝑓,(𝑈) and 𝜓3×𝑈. If 𝑓 and 𝜓(𝑓(𝑧),𝑧𝑓(𝑧),𝑧2𝑓(𝑧);𝑧) are univalent and 𝑓 satisfies the second-order differential subordination𝜓𝑓(𝑧),𝑧𝑓(𝑧),𝑧2𝑓(𝑧);𝑧(𝑧),(1.1) then 𝑓 is called a solution of the differential subordination. The univalent function 𝐹 is called a dominant if 𝑓𝐹 for all 𝑓 satisfying (1.1). Miller and Mocanu discussed many interesting results containing the above mentioned subordination and also many applications of the field of differential subordination in [1]. In that direction, many differential subordination and differential superordination problems for analytic functions defined by means of linear operators were investigated. See [211] for related results.

Let 𝒜𝑝 denote the class of functions of the form𝑓(𝑧)=𝑧𝑝+𝑛=1𝑎𝑛+𝑝𝑧𝑛+𝑝(𝑧𝑈,𝑝=1,2,3,),(1.2) which are analytic and p-valent in 𝑈. For 𝑓 satisfying (1.2), let the generalized Srivastava-Attiya operator [12] be denoted by𝐽𝑠,𝑏𝑓(𝑧)=𝐺𝑠,𝑏𝑓(𝑧)𝑏𝑍0=0,1,2,,(1.3) where𝐺𝑠,𝑏=(1+𝑏)𝑠𝜑(𝑧,𝑠,𝑏)𝑏𝑠,(1.4) with1𝜑(𝑧,𝑠,𝑏)=𝑏𝑠+𝑧𝑝(1+𝑏)𝑠+𝑧1+𝑝(2+𝑏)𝑠+,(1.5) and the symbol (*) denotes the usual Hadamard product (or convolution). From the equations, we can see that𝐽𝑠,𝑏𝑓(𝑧)=𝑧𝑝+𝑛=11+𝑏𝑛+1+𝑏𝑠𝑎𝑛+𝑝𝑧𝑛+𝑝.(1.6) Note that for 𝑝=1 in (1.6), 𝐽𝑠,𝑏𝑓(𝑧) coincides with the Srivastava-Attiya operator [13]. Further, observe that for proper choices of 𝑠 and 𝑏, the operator 𝐽𝑠,𝑏𝑓(𝑧) coincides with the following: (i)𝐽0,𝑏𝑓(𝑧)=𝑓(𝑧), (ii)𝐽1,0𝑓(𝑧)=𝐴(𝑓)(𝑧) [14],(iii)𝐽1,𝛾𝑓(𝑧)=𝛾(𝑓)(𝑧),(𝛾>1) [15, 16],(iv)𝐽𝜎,1𝑓(𝑧)=𝐼𝜎(𝑓)(𝑧),(𝜎>0) [17],(v)𝐽𝛼,𝛽𝑓(𝑧)=𝑃𝛼𝛽(𝑓)(𝑧),(𝛼1,𝛽>1) [18].

Since the above mentioned operator, the generalized Srivastava-Attiya operator, 𝐽𝑠,𝑏𝑓(𝑧) reduces to the well-known operators introduced and studied in the literature by suitably specializing the values of 𝑠 and 𝑏 and also in view of the several interesting properties and characteristics of well-known differential subordination results, we aim to associate these two motivating findings and obtain certain other related results. Further, we consider the differential superordination problems associated with the same operator. In addition, we also obtain interesting sandwich-type theorems.

The following definitions and theorems were discussed and will be needed to prove our results.

Definition 1.1 (see [1], Definition 2.2b, page 21). Denote by 𝑄 the set of all functions 𝑞 that are analytic and injective on 𝑈𝐸(𝑞) where 𝐸(𝑞)=𝜁𝜕𝑈lim𝑧𝜁=(1.7) and are such that 𝑞(𝜁)0 for 𝜁𝜕𝑈𝐸(𝑞). Further let the subclass of 𝑄 for which 𝑞(0)=𝑎 be denoted by 𝑄(𝑎), 𝑄(0)𝑄0, and 𝑄(1)𝑄1.

Definition 1.2 (see [1], Definition 2.3a, page 27). Let Ω be a set in ,𝑞𝑄, and let 𝑛 be a positive integer. The class of admissible functions Ψ𝑛[Ω,𝑞] consists of those functions 𝜓3×𝑈 that satisfy the admissibility condition 𝜓(𝑐,𝑑,𝑒;𝑧)Ω whenever 𝑐=𝑞(𝜁),𝑑=𝑘𝜁𝑞(𝜁), and 𝑒Re𝑑+1𝑘Re𝜁𝑞(𝜁)𝑞(𝜁)+1,(1.8)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘𝑛. Let Ψ1[Ω,𝑞]=Ψ[Ω,𝑞].

Definition 1.3 (see [19], Definition 3, page 817). Let Ω be a set in , 𝑞[𝑎,𝑛] with 𝑞(𝑧)0. The class of admissible functions Ψ𝑛[Ω,𝑞] consists of those functions 𝜓3×𝑈 that satisfy the admissibility condition 𝜓(𝑐,𝑑,𝑒;𝜁)Ω whenever 𝑐=𝑞(𝑧),𝑑=𝑧𝑞(𝑧)/𝑚, and 𝑒Re𝑑1+1𝑚Re𝜁𝑞(𝜁)𝑞(𝜁)+1,(1.9)𝑧𝑈,𝜁𝜕𝑈, and 𝑚𝑛1. Let Ψ1[Ω,𝑞]=Ψ[Ω,𝑞].

Theorem 1.4 (see [1], Theorem 2.3b, page 28). Let 𝜓Ψ𝑛[Ω,𝑞] with 𝑞(0)=𝑎. If the analytic function 𝑗(𝑧)[𝑎,𝑛] satisfies 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);𝑧Ω,(1.10) then 𝑗(𝑧)𝑞(𝑧).

Theorem 1.5 (see [19], Theorem 1, page 818). Let 𝜓Ψ𝑛[Ω,𝑞] with 𝑞(0)=𝑎. If 𝑗𝑄(𝑎) and 𝜓(𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);𝑧) is univalent in 𝑈, then 𝜓Ω𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);𝑧𝑧𝑈(1.11) implies 𝑞(𝑧)𝑗(𝑧).

2. Subordination Results Associated with Generalized Srivastava-Attiya Operator

Definition 2.1. Let Ω be a set in and 𝑞𝑄0[0,𝑝]. The class of admissible functions Φ𝐽[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝑧)Ω(2.1) whenever 𝑢=𝑞(𝜁),𝑣=𝑘𝜁𝑞[]𝑞(𝜁)𝑝(1+𝑏)(𝜁)1+𝑏𝑏𝑍0,=0,1,2,,𝑝Re(1+𝑏)2[]𝑤𝑝(1+𝑏)2𝑢[]𝑢[](1+𝑏)𝑣+𝑝(1+𝑏)+2𝑝(1+𝑏)𝑘Re𝜁𝑞(𝜁)𝑞,(𝜁)+1(2.2)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘𝑝.

Theorem 2.2. Let 𝜙Φ𝐽[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧𝑧𝑈Ω,(2.3) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧)(𝑧𝑈).(2.4)

Proof. The following relation obtained in [13] 𝑧𝐽𝑠+1,𝑏𝑓[]𝐽(𝑧)=𝑝(1+𝑏)𝑠+1,𝑏𝑓(𝑧)+(1+𝑏)𝐽𝑠,𝑏𝑓(𝑧)(2.5) is equivalent to 𝐽𝑠,𝑏𝑓(𝑧)=𝑧𝐽𝑠+1,𝑏𝑓[]𝐽(𝑧)𝑝(1+𝑏)𝑠+1,𝑏𝑓(𝑧),1+𝑏(2.6) and hence J𝑠+1,𝑏𝑓(𝑧)=𝑧𝐽𝑠+2,𝑏𝑓[]𝐽(𝑧)𝑝(1+𝑏)𝑠+2,𝑏𝑓(𝑧).1+𝑏(2.7) Define the analytic function 𝑗 in 𝑈 by 𝑗(𝑧)=𝐽𝑠+2,𝑏𝑓(𝑧),(2.8) and then we get 𝐽𝑠+1,𝑏𝑓(𝑧)=𝑧𝑗[]𝑗(𝑧)𝑝(1+𝑏)(𝑧),𝐽1+𝑏𝑠,𝑏𝑧𝑓(𝑧)=2𝑗[](𝑧)+(12𝑝(1+𝑏))𝑧𝑗[](𝑧)+𝑝(1+𝑏)2𝑗(𝑧)(1+𝑏)2.(2.9) Further, let us define the transformations from 3 to by []𝑐𝑢=𝑐,𝑣=𝑑𝑝(1+𝑏),[][]1+𝑏𝑤=𝑒+(12𝑝(1+𝑏))𝑑+𝑝(1+𝑏)2𝑐(1+𝑏)2.(2.10) Let []𝑐𝜓(𝑐,𝑑,𝑒;𝑧)=𝜙(𝑢,𝑣,𝑤;𝑧),(2.11)𝜙(𝑢,𝑣,𝑤;𝑧)=𝜙𝑐,𝑑𝑝(1+𝑏),[][]1+𝑏𝑒+(12𝑝(1+𝑏))𝑑+𝑝(1+𝑏)2𝑐(1+𝑏)2.;𝑧(2.12) The proof will make use of Theorem 1.4. Using (2.8) and (2.9), from (2.12) we obtain 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗𝐽(𝑧);𝑧=𝜙𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏.𝑓(𝑧);𝑧(2.13) Hence (2.3) becomes 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);𝑧Ω.(2.14) Note that 𝑒𝑑+1=(1+𝑏)2[]𝑤𝑝(1+𝑏)2𝑢[]𝑢[],(1+𝑏)𝑣+𝑝(1+𝑏)+2𝑝(1+𝑏)(2.15) and since the admissibility condition for 𝜙Φ𝐽[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as given in Definition 1.2, hence 𝜓Ψ𝑝[Ω,𝑞], and by Theorem 1.4, 𝑗(𝑧)𝑞(𝑧),(2.16) or 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧).(2.17)

In the case 𝜙(𝑢,𝑣,𝑤;𝑧)=𝑣, we have the following example.

Example 2.3. Let the class of admissible functions Φ𝐽𝑣[Ω,𝑞] consist of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝑣=𝑘𝜁𝑞[]𝑞(𝜁)𝑝(1+𝑏)(𝜁)1+𝑏Ω,(2.18)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘𝑝 and 𝜙Φ𝐽𝑣[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝐽𝑠+1,𝑏𝑓(𝑧)Ω,(2.19) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧)(𝑧𝑈).(2.20)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽[,𝑞]. The following result follows immediately from Theorem 2.2.

Theorem 2.4. Let 𝜙Φ𝐽[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(𝑧),(2.21) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧).(2.22)

The next result occurs when the behavior of 𝑞 on 𝜕𝑈 is not known.

Corollary 2.5. Let Ω, 𝑞 be univalent in 𝑈and 𝑞(0)=0. Let 𝜙Φ𝐽[Ω,𝑞𝜌] for some 𝜌(0,1) where 𝑞𝜌(𝑧)=𝑞(𝜌𝑧). If 𝑓𝒜𝑝 and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧Ω,(2.23) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧).(2.24)

Proof. From Theorem 2.2, we see that 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞𝜌(𝑧) and the proof is complete.

Theorem 2.6. Let and 𝑞 be univalent in 𝑈, with 𝑞(0)=0 and set 𝑞𝜌(𝑧)=𝑞(𝜌𝑧) and 𝜌(𝑧)=(𝜌𝑧). Let 𝜙3×𝑈 satisfy one of the following conditions: (1)𝜙Φ𝐽[,𝑞𝜌], for some 𝜌(0,1), or(2)there exists 𝜌0(0,1) such that 𝜙Φ𝐽[𝜌,𝑞𝜌], for all 𝜌0(0,1). If 𝑓𝒜𝑝 satisfies (2.21), then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧).(2.25)

Proof. The proof is similar to the one in [1] and therefore is omitted.

The next results give the best dominant of the differential subordination (2.21).

Theorem 2.7. Let be univalent in 𝑈. Let 𝜙3×𝑈. Suppose that the differential equation 𝜙𝑞(𝑧),𝑧𝑞(𝑧),𝑧2𝑞(𝑧);𝑧=(𝑧)(2.26) has a solution 𝑞 with 𝑞(0)=0 and satisfy one of the following conditions: (1)𝑞𝑄0 and 𝜙Φ𝐽[,𝑞],(2)𝑞 is univalent in 𝑈 and 𝜙Φ𝐽[,𝑞𝜌], for some 𝜌(0,1), or(3)𝑞 is univalent in 𝑈 and there exists 𝜌0(0,1) such that 𝜙Φ𝐽[𝜌,𝑞𝜌], for all 𝜌0(0,1). If 𝑓𝒜𝑝 satisfies (2.21), then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑞(𝑧),(2.27) and 𝑞 is the best dominant.

Proof. Following the same arguments in [1], we deduce that 𝑞 is a dominant from Theorem 2.4 and Theorem 2.6. Since 𝑞 satisfies (2.26), it is also a solution of (2.21) and therefore 𝑞 will be dominated by all dominants. Hence 𝑞 is the best dominant.

Definition 2.8. Let Ω be a set in and 𝑞𝑄00. The class of admissible functions Φ𝐽,1[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝑧)Ω(2.28) whenever 𝑢=𝑞(𝜁),𝑣=𝑘𝜁𝑞(𝜁)𝑏𝑞(𝜁)1+𝑏𝑏𝑍0,=0,1,2,,𝑝Re(1+𝑏)2𝑤𝑏2𝑢(1+𝑏)𝑣+𝑏𝑢2𝑏𝑘Re𝜁𝑞(𝜁)𝑞,(𝜁)+1(2.29)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘1.

Theorem 2.9. Let 𝜙Φ𝐽,1[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧𝑧𝑈Ω,(2.30) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1𝑞(𝑧)(𝑧𝑈).(2.31)

Proof. Define the analytic function 𝑗 in 𝑈 by 𝐽𝑗(𝑧)=𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1.(2.32) Using the relations (2.5) and (2.32), we get 𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1=𝑧𝑗(𝑧)𝑏𝑗(𝑧),𝐽1+𝑏𝑠,𝑏𝑓(𝑧)𝑧𝑝1=𝑧2𝑗(𝑧)+(2𝑏+1)𝑧𝑗(𝑧)+𝑏2𝑗(𝑧)(1+𝑏)2.(2.33) Further, let us define the transformations from 3 to by 𝑢=𝑐,𝑣=𝑑+𝑏𝑐,1+𝑏𝑤=𝑒+(2𝑏+1)𝑑+𝑏2𝑐(1+𝑏)2.(2.34) Let 𝜓(𝑐,𝑑,𝑒;𝑧)=𝜙(𝑢,𝑣,𝑤;𝑧)=𝜙𝑐,𝑑+𝑏𝑐,1+𝑏𝑒+(2𝑏+1)𝑑+𝑏2𝑐(1+𝑏)2.;𝑧(2.35) The proof will make use of Theorem 1.4. Using (2.32) and (2.33), from (2.35) we obtain 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗𝐽(𝑧);𝑧=𝜙𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1.;𝑧(2.36) Hence (2.30) becomes 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);𝑧Ω.(2.37) Note that 𝑒𝑑+1=(1+𝑏)2𝑤𝑏2𝑢(1+𝑏)𝑣+𝑏𝑢2𝑏,(2.38) and since the admissibility condition for 𝜙Φ𝐽,1[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as given in Definition 1.2, hence 𝜓Ψ[Ω,𝑞], and by Theorem 1.4, 𝑗(𝑧)𝑞(𝑧),(2.39) or 𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1𝑞(𝑧).(2.40)

In the case 𝜙(𝑢,𝑣,𝑤;𝑧)=𝑣𝑢, we have the following example.

Example 2.10. Let the class of admissible functions Φ𝐽𝑣,1[Ω,𝑞] consist of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝑣𝑢=𝑘𝜁𝑞(𝜁)𝑝𝑞(𝜁)1+𝑏Ω,(2.41)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘𝑝 and 𝜙Φ𝐽𝑣,1[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1Ω(𝑧𝑈),(2.42) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1𝑞(𝑧)(𝑧𝑈).(2.43)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽,1[,𝑞]. The following result follows immediately from Theorem 2.9.

Theorem 2.11. Let 𝜙Φ𝐽,1[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧(𝑧),(2.44) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1𝑞(𝑧).(2.45)

Definition 2.12. Let Ω be a set in and 𝑞𝑄1. The class of admissible functions Φ𝐽,2[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝑧)Ω(2.46) whenever 𝑢=𝑞(𝜁),𝑣=𝑞(𝜁)+𝑘𝜁𝑞(𝜁)(1+𝑏)𝑞(𝜁)𝑏𝑍0,=0,1,2,,𝑝Re(𝑤𝑢)(1+𝑏)𝑢𝑣𝑢+(1+𝑏)(𝑤3𝑢)𝑘Re𝜁𝑞(𝜁)𝑞,(𝜁)+1(2.47)𝑧𝑈,𝜁𝜕𝑈𝐸(𝑞), and 𝑘1.

Theorem 2.13. Let 𝜙Φ𝐽,2[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)s,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧𝑧𝑈Ω,(2.48) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)𝑞(𝑧)(𝑧𝑈).(2.49)

Proof. Define the analytic function 𝑗 in 𝑈 by 𝐽𝑗(𝑧)=𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(2.50) Differentiating (2.50) yields 𝑧𝑗(𝑧)=𝑗(𝑧)𝑧𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏𝐽𝑓(𝑧)𝑠+3,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(2.51) From the relation (2.5) we get 𝑧𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏=[]𝑓(𝑧)𝑝(1+𝑏)+(1+𝑏)𝑗+𝑧𝑗(𝑧),𝑗(𝑧)(2.52) and hence 𝐽𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧)=𝑗(𝑧)+𝑧𝑗(𝑧).(1+𝑏)𝑗(𝑧)(2.53) Further computations show that 𝐽𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏[]𝑓(𝑧)=𝑗(𝑧)+2(1+𝑏)𝑗(𝑧)+1𝑧𝑗(𝑧)+𝑧2𝑗(𝑧)(1+𝑏)2𝑗(𝑧)2+(1+𝑏)𝑧𝑗.(𝑧)(2.54) Let us define the transformations from 3 to by 𝑑𝑢=𝑐,𝑣=𝑐+,[](1+𝑏)𝑐𝑤=𝑐+2(𝑏+1)𝑐+1𝑑+𝑒(1+𝑏)2𝑐2.+(1+𝑏)𝑑(2.55) Let 𝑑𝜓(𝑐,𝑑,𝑒;𝑧)=𝜙(𝑢,𝑣,𝑤;𝑧)=𝜙𝑐,𝑐+[](1+𝑏)𝑐,𝑐+2(𝑏+1)𝑐+1𝑑+𝑒(1+𝑏)2𝑐2.+(1+𝑏)𝑑;𝑧(2.56) The proof will make use of Theorem 1.4. Using (2.50), (2.53) and (2.54), from (2.56) we obtain 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗𝐽(𝑧);𝑧=𝜙𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏.𝑓(𝑧);𝑧(2.57) Hence (2.48) becomes 𝜓𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗(𝑧);zΩ.(2.58) Note that 𝑒𝑑+1=(𝑤𝑢)(1+𝑏)𝑢𝑣𝑢+(1+𝑏)(𝑤3𝑢),(2.59) and since the admissibility condition for 𝜙Φ𝐽,2[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as given in Definition 1.2, hence 𝜓Ψ[Ω,𝑞] and by Theorem 1.4, 𝑗(𝑧)𝑞(𝑧),(2.60) or 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)𝑞(𝑧).(2.61)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽,2[,𝑞]. The following result follows immediately from Theorem 2.13.

Theorem 2.14. Let 𝜙Φ𝐽,2[Ω,𝑞]. If 𝑓𝒜𝑝 satisfies 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧(𝑧),(2.62) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)𝑞(𝑧).(2.63)

3. Superordination Results Associated with Generalized Srivastava-Attiya Operator

Definition 3.1. Let Ω be a set in and 𝑞[0,𝑝] with 𝑧𝑞(𝑧)0. The class of admissible functions Φ𝐽[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝜁)Ω(3.1) whenever 𝑢=𝑞(𝑧),𝑣=𝑧𝑞[]𝑞(𝑧)𝑚𝑝(1+𝑏)(𝑧)𝑚(1+𝑏)𝑏𝑍0,=0,1,2,,𝑝Re(1+𝑏)2[]𝑤𝑝(1+𝑏)2𝑢([]𝑢[]11+𝑏)𝑣+𝑝(1+𝑏)+2𝑝(1+𝑏)𝑚Re𝑧𝑞(𝑧)𝑞(,𝑧)+1(3.2)𝑧𝑈,𝜁𝜕𝑈, and 𝑚𝑝.

Theorem 3.2. Let 𝜙Φ𝐽[Ω,𝑞]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓𝑄0 and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.3) is univalent in 𝑈, then 𝜙𝐽Ω𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧𝑧𝑈(3.4) implies that 𝑞(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧).(3.5)

Proof. From (2.13) and (3.4), we have 𝜓Ω𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗.(𝑧);𝑧𝑧𝑈(3.6) From (2.10), we see that the admissibility condition for 𝜙Φ𝐽[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as given in Definition 1.3. Hence 𝜓Ψ𝑝[Ω,𝑞], and by Theorem 1.5, 𝑞(𝑧)𝑗(𝑧) or 𝑞(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧).(3.7)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽[,𝑞]. The next result follows immediately from Theorem 3.2.

Theorem 3.3. Let be analytic in 𝑈 and 𝜙Φ𝐽[Ω,𝑞]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)𝑄0 and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.8) is univalent in 𝑈, then 𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧,(3.9) and then 𝑞(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧).(3.10)

Theorems 3.2 and 3.3 can only be used to obtain subordinants for differential superordination of the form (3.4) and (3.9). The following theorems prove the existence of the best subordinant of (3.9) for certain 𝜙.

Theorem 3.4. Let be analytic in 𝑈 and 𝜙3×𝑈. Suppose that the differential equation 𝜙𝑞(𝑧),𝑧𝑞(𝑧),𝑧2𝑞(𝑧);𝑧=(𝑧)(3.11) has a solution 𝑞𝑄0. If 𝜙Φ𝐽[Ω,𝑞], 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)𝑄0, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.12) is univalent in 𝑈, then 𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.13) implies that 𝑞(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧),(3.14) and 𝑞(𝑧) is the best subordinant.

Proof. The result can be obtained by similar proof of Theorem 2.7.

The next result, the sandwich-type theorem follows from Theorems 2.4 and 3.3.

Corollary 3.5. Let 1  and 𝑞1  be analytic in 𝑈, and let 2  be univalent function in 𝑈, 𝑞2𝑄0  with 𝑞1(0)=𝑞2(0)=0 and 𝜙Φ𝐽[2,𝑞2]Φ𝐽[1,𝑞1]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)[0,𝑝]𝑄0, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.15) is univalent in 𝑈, then 1𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧2(𝑧)(3.16) implies that 𝑞1(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧)𝑞2(𝑧).(3.17)

Definition 3.6. Let Ω be a set in and 𝑞0 with 𝑧𝑞(𝑧)0. The class of admissible functions Φ𝐽,1[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝜁)Ω(3.18) whenever 𝑢=𝑞(𝑧),𝑣=𝑧𝑞(𝑧)𝑚𝑏𝑞(𝑧)𝑚(1+𝑏)𝑏𝑍0,=0,1,2,,𝑝Re(1+𝑏)2𝑤𝑏2𝑢1(1+𝑏)𝑣+𝑏𝑢2𝑏𝑚Re𝑧𝑞(𝑧)𝑞,(𝑧)+1(3.19)𝑧𝑈,𝜁𝜕𝑈, and 𝑚1.

The following result is associated with Theorem 2.9.

Theorem 3.7. Let 𝜙Φ𝐻,1[Ω,𝑞]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)/𝑧𝑝1𝑄0, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧(3.20) is univalent in 𝑈, then 𝜙𝐽Ω𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧𝑧𝑈(3.21) implies that 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1.(3.22)

Proof. From (2.36) and (3.21), we have 𝜙Ω𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗.(𝑧);𝑧𝑧𝑈(3.23) From (2.34), we see that the admissibility condition for 𝜙Φ𝐽,1[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as in Definition 1.3. Hence 𝜓Ψ[Ω,𝑞], and by Theorem 1.5, 𝑞(𝑧)𝑗(𝑧) or 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1.(3.24)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽,1[,𝑞]. The next result follows immediately from Theorem 3.7.

Theorem 3.8. Let 𝑞0, and let be analytic on 𝑈, and let 𝜙Φ𝐽,1[Ω,𝑞]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)/𝑧𝑝1𝑄0, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧(3.25) is univalent in 𝑈, then 𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧(3.26) implies that 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1.(3.27)

Combining Theorems 2.11 and 3.8, we obtain the following sandwich-type theorem.

Corollary 3.9. Let 1 and 𝑞1 be analytic in 𝑈, let 2 be univalent function in 𝑈, 𝑞2𝑄0 with 𝑞1(0)=𝑞2(0)=0, and 𝜙Φ𝐽,1[2,𝑞2]Φ𝐽,1[1,𝑞1]. If 𝑓𝒜𝑝, 𝐽𝑠+2,𝑏𝑓(𝑧)/𝑧𝑝10𝑄0, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧(3.28) is univalent in 𝑈, then 1𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠+1,𝑏𝑓(𝑧)𝑧𝑝1,𝐽𝑠,𝑏𝑓(𝑧)𝑧𝑝1;𝑧2(𝑧)(3.29) implies that 𝑞1𝐽(𝑧)𝑠+2,𝑏𝑓(𝑧)𝑧𝑝1𝑞2(𝑧).(3.30)

Definition 3.10. Let Ω be a set in and 𝑞(𝑧)0,𝑧𝑞(𝑧)0, and 𝑞. The class of admissible functions Φ𝐽,2[Ω,𝑞] consists of those functions 𝜙3×𝑈 that satisfy the admissibility condition: 𝜙(𝑢,𝑣,𝑤;𝜁)Ω(3.31) whenever 𝑢=𝑞(𝑧),𝑣=𝑞(𝑧)+𝑧𝑞(𝑧)𝑚(1+𝑏)𝑞(𝑧)𝑏𝑍0,=0,1,2,,𝑝Re(𝑤𝑢)(1+𝑏)𝑢1𝑣𝑢+(1+𝑏)(𝑤3𝑢)𝑚Re𝑧𝑞(𝑧)𝑞,(𝑧)+1(3.32)𝑧𝑈,𝜁𝜕𝑈, and 𝑚1.

The following result is associated with Theorem 2.13.

Theorem 3.11. Let 𝜙Φ𝐽,2[Ω,𝑞]. If 𝑓𝒜𝑝,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)𝑄1, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧(3.33) is univalent in 𝑈, then 𝜙𝐽Ω𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧𝑧𝑈(3.34) implies that 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(3.35)

Proof. From (2.57) and (3.34), we have 𝜙Ω𝑗(𝑧),𝑧𝑗(𝑧),𝑧2𝑗.(𝑧);𝑧𝑧𝑈(3.36) From (2.55), we see that the admissibility condition for 𝜙Φ𝐽,2[Ω,𝑞] is equivalent to the admissibility condition for 𝜓 as in Definition 1.3. Hence 𝜓Ψ[Ω,𝑞], and by Theorem 1.5, 𝑞(𝑧)𝑗(𝑧) or 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(3.37)

If Ω is a simply connected domain, then Ω(𝑈) for some conformal mapping (𝑧) of 𝑈 onto Ω and the class is written as Φ𝐽,2[,𝑞]. The next result follows immediately from Theorem 3.11 as in the previous section.

Theorem 3.12. Let 𝑞, let be analytic in 𝑈, and let 𝜙Φ𝐽,2[Ω,𝑞]. If 𝑓𝒜𝑝,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)𝑄1 and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧(3.38) is univalent in 𝑈, then 𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧(3.39) implies that 𝐽𝑞(𝑧)𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(3.40)

Combining Theorems 2.14 and 3.12, we obtain the following sandwich-type theorem.

Corollary 3.13. Let 1  and 𝑞1 be analytic in 𝑈, let 2 be univalent function in 𝑈, 𝑞2𝑄0 with 𝑞1(0)=𝑞2(0)=0, and 𝜙Φ𝐽,2[2,𝑞2]Φ𝐽,2[1,𝑞1]. If 𝑓𝒜𝑝,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)𝑄1, and 𝜙𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧(3.41) is univalent in 𝑈, then 1𝐽(𝑧)𝜙𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏,𝐽𝑓(𝑧)𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏,𝐽𝑓(𝑧)𝑠,𝑏𝑓(𝑧)𝐽𝑠+1,𝑏𝑓(𝑧);𝑧2(𝑧)(3.42) implies that 𝑞1𝐽(𝑧)𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)𝑞(𝑧).(3.43)

Other work related to certain operators concerning the subordination and superordination can be found in [2025].

Acknowledgment

The work presented here was partially supported by UKM-ST-FRGS-0244-2010.

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