International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 902830 | 19 pages | https://doi.org/10.1155/2011/902830

Differential Subordination and Superordination for Srivastava-Attiya Operator

Academic Editor: Khalil Ezzinbi
Received28 Mar 2011
Accepted01 Jun 2011
Published11 Aug 2011

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 β„‹(π‘ˆ) be the class of analytic functions in the open unit disc π‘ˆ and let β„‹[π‘Ž,𝑛] be the subclass of β„‹(π‘ˆ) consisting functions of the form 𝑓(𝑧)=π‘Ž+π‘Žπ‘›π‘§π‘›+π‘Žπ‘›+1𝑧𝑛+1+β‹―, with β„‹0=β„‹[0,1] and β„‹=β„‹[1,1]. For two functions 𝑓1 and 𝑓2 analytic in π‘ˆ, the function 𝑓1 is subordinate to 𝑓2, or 𝑓2 superordinate to 𝑓1, written as 𝑓1≺𝑓2 if there exists a function 𝑀(𝑧), analytic in π‘ˆ with 𝑀(0)=0 and |𝑀(𝑧)|<1 such that 𝑓1(𝑧)=𝑓2(𝑀(𝑧)). In particular, if the function 𝑓2 is univalent in π‘ˆ, then 𝑓1≺𝑓2 is equivalent to 𝑓1(0)=𝑓2(0) and 𝑓1(π‘ˆ)βŠ‚π‘“2(π‘ˆ).

Let 𝑓,β„Žβˆˆβ„‹(π‘ˆ) and πœ“βˆΆβ„‚3Γ—π‘ˆβ†’β„‚. If 𝑓 and πœ“(𝑓(𝑧),𝑧𝑓′(𝑧),𝑧2π‘“ξ…žξ…ž(𝑧);𝑧) are univalent and 𝑓 satisfies the second-order differential subordinationπœ“ξ€·π‘“(𝑧),π‘§π‘“ξ…ž(𝑧),𝑧2π‘“ξ…žξ…žξ€Έ(𝑧);π‘§β‰Ίβ„Ž(𝑧),(1.1) then 𝑓 is called a solution of the differential subordination. The univalent function 𝐹 is called a dominant if 𝑓≺𝐹 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 [2–11] for related results.

Let π’œπ‘ denote the class of functions of the form𝑓(𝑧)=𝑧𝑝+βˆžξ“π‘›=1π‘Žπ‘›+𝑝𝑧𝑛+𝑝(π‘§βˆˆπ‘ˆ,π‘βˆˆβ„•=1,2,3,…),(1.2) which are analytic and p-valent in π‘ˆ. For 𝑓 satisfying (1.2), let the generalized Srivastava-Attiya operator [12] be denoted by𝐽𝑠,𝑏𝑓(𝑧)=𝐺𝑠,π‘ξ‚€βˆ—π‘“(𝑧)π‘βˆˆβ„‚β§΅π‘0=0,βˆ’1,βˆ’2,…,(1.3) where𝐺𝑠,𝑏=(1+𝑏)π‘ ξ€Ίπœ‘(𝑧,𝑠,𝑏)βˆ’π‘βˆ’π‘ ξ€»,(1.4) with1πœ‘(𝑧,𝑠,𝑏)=𝑏𝑠+𝑧𝑝(1+𝑏)𝑠+𝑧1+𝑝(2+𝑏)𝑠+β‹―,(1.5) and the symbol (*) denotes the usual Hadamard product (or convolution). From the equations, we can see that𝐽𝑠,𝑏𝑓(𝑧)=𝑧𝑝+βˆžξ“π‘›=1ξ‚€1+𝑏𝑛+1+π‘π‘ π‘Žπ‘›+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)=ℐ𝛾(𝑓)(𝑧),(𝛾>βˆ’1) [15, 16],(iv)𝐽𝜎,1𝑓(𝑧)=𝐼𝜎(𝑓)(𝑧),(𝜎>0) [17],(v)𝐽𝛼,𝛽𝑓(𝑧)=𝑃𝛼𝛽(𝑓)(𝑧),(𝛼β‰₯1,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 π‘ž that are analytic and injective on π‘ˆβ§΅πΈ(π‘ž) where 𝐸(π‘ž)=πœβˆˆπœ•π‘ˆβˆΆlimπ‘§β†’πœξ‚Ό=∞(1.7) and are such that π‘žβ€²(𝜁)β‰ 0 for πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž). Further let the subclass of 𝑄 for which π‘ž(0)=π‘Ž be denoted by 𝑄(π‘Ž), 𝑄(0)≑𝑄0, and 𝑄(1)≑𝑄1.

Definition 1.2 (see [1], Definition 2.3a, page 27). Let Ξ© be a set in β„‚,π‘žβˆˆπ‘„, and let 𝑛 be a positive integer. The class of admissible functions Ψ𝑛[Ξ©,π‘ž] consists of those functions πœ“βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition πœ“(𝑐,𝑑,𝑒;𝑧)βˆ‰Ξ© whenever 𝑐=π‘ž(𝜁),𝑑=π‘˜πœπ‘žξ…ž(𝜁), and 𝑒Re𝑑+1β‰₯π‘˜Reπœπ‘žξ…žξ…ž(𝜁)π‘žξ…žξ‚Ό(𝜁)+1,(1.8)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯𝑛. Let Ξ¨1[Ξ©,π‘ž]=Ξ¨[Ξ©,π‘ž].

Definition 1.3 (see [19], Definition 3, page 817). Let Ξ© be a set in β„‚, π‘žβˆˆβ„‹[π‘Ž,𝑛] with π‘žξ…ž(𝑧)β‰ 0. The class of admissible functions Ξ¨ξ…žπ‘›[Ξ©,π‘ž] consists of those functions πœ“βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition πœ“(𝑐,𝑑,𝑒;𝜁)βˆ‰Ξ©β€‰whenever 𝑐=π‘ž(𝑧),𝑑=π‘§π‘žξ…ž(𝑧)/π‘š, and 𝑒Re𝑑β‰₯1+1π‘šξ‚»Reπœπ‘žξ…žξ…ž(𝜁)π‘žξ…žξ‚Ό(𝜁)+1,(1.9)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆ, and π‘šβ‰₯𝑛β‰₯1. Let Ξ¨ξ…ž1[Ξ©,π‘ž]=Ξ¨β€²[Ξ©,π‘ž].

Theorem 1.4 (see [1], Theorem 2.3b, page 28). Let πœ“βˆˆΞ¨π‘›[Ξ©,π‘ž] with π‘ž(0)=π‘Ž. If the analytic function 𝑗(𝑧)βˆˆβ„‹[π‘Ž,𝑛] satisfies πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έ(𝑧);π‘§βˆˆΞ©,(1.10) then 𝑗(𝑧)β‰Ίπ‘ž(𝑧).

Theorem 1.5 (see [19], Theorem 1, page 818). Let πœ“βˆˆΞ¨ξ…žπ‘›[Ξ©,π‘ž] with π‘ž(0)=π‘Ž. If π‘—βˆˆπ‘„(π‘Ž) and πœ“(𝑗(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…ž(𝑧);𝑧) is univalent in π‘ˆ, then ξ€½πœ“ξ€·Ξ©βŠ‚π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ€Ύ(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆ(1.11) implies π‘ž(𝑧)≺𝑗(𝑧).

2. Subordination Results Associated with Generalized Srivastava-Attiya Operator

Definition 2.1. Let Ξ© be a set in β„‚ and π‘žβˆˆπ‘„0βˆ©β„‹[0,𝑝]. The class of admissible functions Φ𝐽[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝑧)βˆ‰Ξ©(2.1) whenever 𝑒=π‘ž(𝜁),𝑣=π‘˜πœπ‘žξ…ž[]π‘ž(𝜁)βˆ’π‘βˆ’(1+𝑏)(𝜁)ξ‚€1+π‘π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(1+𝑏)2[]π‘€βˆ’π‘βˆ’(1+𝑏)2𝑒[]𝑒[]ξ‚Όξ‚»(1+𝑏)𝑣+π‘βˆ’(1+𝑏)+2π‘βˆ’(1+𝑏)β‰₯π‘˜Reπœπ‘žξ…žξ…ž(𝜁)π‘žξ…žξ‚Ό,(𝜁)+1(2.2)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯𝑝.

Theorem 2.2. Let πœ™βˆˆΞ¦π½[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies ξ€½πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆβŠ‚Ξ©,(2.3) then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧)(π‘§βˆˆπ‘ˆ).(2.4)

Proof. The following relation obtained in [13] π‘§π½ξ…žπ‘ +1,𝑏𝑓[]𝐽(𝑧)=π‘βˆ’(1+𝑏)𝑠+1,𝑏𝑓(𝑧)+(1+𝑏)𝐽𝑠,𝑏𝑓(𝑧)(2.5) is equivalent to 𝐽𝑠,𝑏𝑓(𝑧)=π‘§π½ξ…žπ‘ +1,𝑏𝑓[]𝐽(𝑧)βˆ’π‘βˆ’(1+𝑏)𝑠+1,𝑏𝑓(𝑧),1+𝑏(2.6) and hence J𝑠+1,𝑏𝑓(𝑧)=π‘§π½ξ…žπ‘ +2,𝑏𝑓[]𝐽(𝑧)βˆ’π‘βˆ’(1+𝑏)𝑠+2,𝑏𝑓(𝑧).1+𝑏(2.7) Define the analytic function 𝑗 in π‘ˆ by 𝑗(𝑧)=𝐽𝑠+2,𝑏𝑓(𝑧),(2.8) and then we get 𝐽𝑠+1,𝑏𝑓(𝑧)=π‘§π‘—ξ…ž[]𝑗(𝑧)βˆ’π‘βˆ’(1+𝑏)(𝑧),𝐽1+𝑏𝑠,𝑏𝑧𝑓(𝑧)=2π‘—ξ…žξ…ž[](𝑧)+(1βˆ’2π‘βˆ’(1+𝑏))π‘§π‘—ξ…ž[](𝑧)+π‘βˆ’(1+𝑏)2𝑗(𝑧)(1+𝑏)2.(2.9) Further, let us define the transformations from β„‚3 to β„‚ by []𝑐𝑒=𝑐,𝑣=π‘‘βˆ’π‘βˆ’(1+𝑏),[][]1+𝑏𝑀=𝑒+(1βˆ’2π‘βˆ’(1+𝑏))𝑑+π‘βˆ’(1+𝑏)2𝑐(1+𝑏)2.(2.10) Let ξ‚΅[]π‘πœ“(𝑐,𝑑,𝑒;𝑧)=πœ™(𝑒,𝑣,𝑀;𝑧),(2.11)πœ™(𝑒,𝑣,𝑀;𝑧)=πœ™π‘,π‘‘βˆ’π‘βˆ’(1+𝑏),[][]1+𝑏𝑒+(1βˆ’2π‘βˆ’(1+𝑏))𝑑+π‘βˆ’(1+𝑏)2𝑐(1+𝑏)2ξ‚Ά.;𝑧(2.12) The proof will make use of Theorem 1.4. Using (2.8) and (2.9), from (2.12) we obtain πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ€·π½(𝑧);𝑧=πœ™π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏.𝑓(𝑧);𝑧(2.13) Hence (2.3) becomes πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έ(𝑧);π‘§βˆˆΞ©.(2.14) Note that 𝑒𝑑+1=(1+𝑏)2[]π‘€βˆ’π‘βˆ’(1+𝑏)2𝑒[]𝑒[],(1+𝑏)𝑣+π‘βˆ’(1+𝑏)+2π‘βˆ’(1+𝑏)(2.15) and since the admissibility condition for πœ™βˆˆΞ¦π½[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as given in Definition 1.2, hence πœ“βˆˆΞ¨π‘[Ξ©,π‘ž], and by Theorem 1.4, 𝑗(𝑧)β‰Ίπ‘ž(𝑧),(2.16) or 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(2.17)

In the case πœ™(𝑒,𝑣,𝑀;𝑧)=𝑣, we have the following example.

Example 2.3. Let the class of admissible functions Φ𝐽𝑣[Ξ©,π‘ž] consist of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: 𝑣=π‘˜πœπ‘žξ…ž[]π‘ž(𝜁)βˆ’π‘βˆ’(1+𝑏)(𝜁)1+π‘βˆ‰Ξ©,(2.18)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯𝑝 and πœ™βˆˆΞ¦π½π‘£[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies 𝐽𝑠+1,𝑏𝑓(𝑧)βŠ‚Ξ©,(2.19) then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧)(π‘§βˆˆπ‘ˆ).(2.20)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Φ𝐽[β„Ž,π‘ž]. The following result follows immediately from Theorem 2.2.

Theorem 2.4. Let πœ™βˆˆΞ¦π½[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);π‘§β‰Ίβ„Ž(𝑧),(2.21) then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(2.22)

The next result occurs when the behavior of π‘ž on πœ•π‘ˆ is not known.

Corollary 2.5. Let Ξ©βŠ‚β„‚, π‘ž be univalent in π‘ˆand π‘ž(0)=0. Let πœ™βˆˆΞ¦π½[Ξ©,π‘žπœŒ] for some 𝜌∈(0,1) where π‘žπœŒ(𝑧)=π‘ž(πœŒπ‘§). If π‘“βˆˆπ’œπ‘ and πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);π‘§βˆˆΞ©,(2.23) then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(2.24)

Proof. From Theorem 2.2, we see that 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘žπœŒ(𝑧) and the proof is complete.

Theorem 2.6. Let β„Ž and π‘ž be univalent in π‘ˆ, with π‘ž(0)=0 and set π‘žπœŒ(𝑧)=π‘ž(πœŒπ‘§) and β„ŽπœŒ(𝑧)=β„Ž(πœŒπ‘§). Let πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ satisfy one of the following conditions: (1)πœ™βˆˆΞ¦π½[β„Ž,π‘žπœŒ], for some 𝜌∈(0,1), or(2)there exists 𝜌0∈(0,1) such that πœ™βˆˆΞ¦π½[β„ŽπœŒ,π‘žπœŒ], for all 𝜌0∈(0,1). If π‘“βˆˆπ’œπ‘ satisfies (2.21), then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(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 β„Ž be univalent in π‘ˆ. Let πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚. Suppose that the differential equation πœ™ξ€·π‘ž(𝑧),π‘§π‘žξ…ž(𝑧),𝑧2π‘žξ…žξ…žξ€Έ(𝑧);𝑧=β„Ž(𝑧)(2.26) has a solution π‘ž with π‘ž(0)=0 and satisfy one of the following conditions: (1)π‘žβˆˆπ‘„0 and πœ™βˆˆΞ¦π½[β„Ž,π‘ž],(2)π‘ž is univalent in π‘ˆ and πœ™βˆˆΞ¦π½[β„Ž,π‘žπœŒ], for some 𝜌∈(0,1), or(3)π‘ž is univalent in π‘ˆ and there exists 𝜌0∈(0,1) such that πœ™βˆˆΞ¦π½[β„ŽπœŒ,π‘žπœŒ], for all 𝜌0∈(0,1). If π‘“βˆˆπ’œπ‘ satisfies (2.21), then 𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧),(2.27) and π‘ž is the best dominant.

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

Definition 2.8. Let Ξ© be a set in β„‚ and π‘žβˆˆπ‘„0βˆ©β„‹0. The class of admissible functions Φ𝐽,1[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝑧)βˆ‰Ξ©(2.28) whenever 𝑒=π‘ž(𝜁),𝑣=π‘˜πœπ‘žξ…ž(𝜁)βˆ’π‘π‘ž(𝜁)ξ‚€1+π‘π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(1+𝑏)2π‘€βˆ’π‘2𝑒(1+𝑏)𝑣+π‘π‘’βˆ’2𝑏β‰₯π‘˜Reπœπ‘žξ…žξ…ž(𝜁)π‘žξ…žξ‚Ό,(𝜁)+1(2.29)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯1.

Theorem 2.9. Let πœ™βˆˆΞ¦π½,1[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies ξ‚»πœ™ξ‚΅π½π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Άξ‚Ό;π‘§βˆΆπ‘§βˆˆπ‘ˆβŠ‚Ξ©,(2.30) then 𝐽𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1β‰Ίπ‘ž(𝑧)(π‘§βˆˆπ‘ˆ).(2.31)

Proof. Define the analytic function 𝑗 in π‘ˆ by 𝐽𝑗(𝑧)=𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1.(2.32) Using the relations (2.5) and (2.32), we get 𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1=π‘§π‘—ξ…ž(𝑧)βˆ’π‘π‘—(𝑧),𝐽1+𝑏𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1=𝑧2π‘—ξ…žξ…ž(𝑧)+(2𝑏+1)π‘§π‘—ξ…ž(𝑧)+𝑏2𝑗(𝑧)(1+𝑏)2.(2.33) Further, let us define the transformations from β„‚3 to β„‚ by 𝑒=𝑐,𝑣=𝑑+𝑏𝑐,1+𝑏𝑀=𝑒+(2𝑏+1)𝑑+𝑏2𝑐(1+𝑏)2.(2.34) Let ξ‚΅πœ“(𝑐,𝑑,𝑒;𝑧)=πœ™(𝑒,𝑣,𝑀;𝑧)=πœ™π‘,𝑑+𝑏𝑐,1+𝑏𝑒+(2𝑏+1)𝑑+𝑏2𝑐(1+𝑏)2ξ‚Ά.;𝑧(2.35) The proof will make use of Theorem 1.4. Using (2.32) and (2.33), from (2.35) we obtain πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ‚΅π½(𝑧);𝑧=πœ™π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά.;𝑧(2.36) Hence (2.30) becomes πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έ(𝑧);π‘§βˆˆΞ©.(2.37) Note that 𝑒𝑑+1=(1+𝑏)2π‘€βˆ’π‘2𝑒(1+𝑏)𝑣+π‘π‘’βˆ’2𝑏,(2.38) and since the admissibility condition for πœ™βˆˆΞ¦π½,1[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as given in Definition 1.2, hence πœ“βˆˆΞ¨[Ξ©,π‘ž], and by Theorem 1.4, 𝑗(𝑧)β‰Ίπ‘ž(𝑧),(2.39) or 𝐽𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1β‰Ίπ‘ž(𝑧).(2.40)

In the case πœ™(𝑒,𝑣,𝑀;𝑧)=π‘£βˆ’π‘’, we have the following example.

Example 2.10. Let the class of admissible functions Φ𝐽𝑣,1[Ξ©,π‘ž] consist of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: π‘£βˆ’π‘’=π‘˜πœπ‘žξ…ž(𝜁)βˆ’π‘π‘ž(𝜁)1+π‘βˆ‰Ξ©,(2.41)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯𝑝 and πœ™βˆˆΞ¦π½π‘£,1[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies 𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1βˆ’π½π‘ ,𝑏𝑓(𝑧)π‘§π‘βˆ’1βŠ‚Ξ©(π‘§βˆˆπ‘ˆ),(2.42) then 𝐽𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1β‰Ίπ‘ž(𝑧)(π‘§βˆˆπ‘ˆ).(2.43)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Φ𝐽,1[β„Ž,π‘ž]. The following result follows immediately from Theorem 2.9.

Theorem 2.11. Let πœ™βˆˆΞ¦π½,1[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies πœ™ξ‚΅π½π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;π‘§β‰Ίβ„Ž(𝑧),(2.44) then 𝐽𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1β‰Ίπ‘ž(𝑧).(2.45)

Definition 2.12. Let Ξ© be a set in β„‚ and π‘žβˆˆπ‘„1βˆ©β„‹. The class of admissible functions Φ𝐽,2[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝑧)βˆ‰Ξ©(2.46) whenever 𝑒=π‘ž(𝜁),𝑣=π‘ž(𝜁)+π‘˜πœπ‘žξ…ž(𝜁)(ξ‚€1+𝑏)π‘ž(𝜁)π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(π‘€βˆ’π‘’)(1+𝑏)π‘’ξ‚Όξ‚»π‘£βˆ’π‘’+(1+𝑏)(π‘€βˆ’3𝑒)β‰₯π‘˜Reπœπ‘žξ…žξ…ž(𝜁)π‘žξ…žξ‚Ό,(𝜁)+1(2.47)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆβ§΅πΈ(π‘ž), and π‘˜β‰₯1.

Theorem 2.13. Let πœ™βˆˆΞ¦π½,2[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies ξ‚»πœ™ξ‚΅π½π‘ +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);π‘§βˆΆπ‘§βˆˆπ‘ˆβŠ‚Ξ©,(2.48) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧)(π‘§βˆˆπ‘ˆ).(2.49)

Proof. Define the analytic function 𝑗 in π‘ˆ by 𝐽𝑗(𝑧)=𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(2.50) Differentiating (2.50) yields π‘§π‘—ξ…ž(𝑧)=𝑗(𝑧)π‘§π½ξ…žπ‘ +2,𝑏𝑓(𝑧)𝐽𝑠+2,π‘βˆ’π½π‘“(𝑧)ξ…žπ‘ +3,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(2.51) From the relation (2.5) we get π‘§π½ξ…žπ‘ +2,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏=[]𝑓(𝑧)π‘βˆ’(1+𝑏)+(1+𝑏)𝑗+π‘§π‘—ξ…ž(𝑧),𝑗(𝑧)(2.52) and hence 𝐽𝑠+1,𝑏𝑓(𝑧)𝐽𝑠+2,𝑏𝑓(𝑧)=𝑗(𝑧)+π‘§π‘—ξ…ž(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)+1π‘§π‘—ξ…ž(𝑧)+𝑧2π‘—ξ…žξ…ž(𝑧)(1+𝑏)2𝑗(𝑧)2+(1+𝑏)π‘§π‘—ξ…ž.(𝑧)(2.54) Let us define the transformations from β„‚3 to β„‚ by 𝑑𝑒=𝑐,𝑣=𝑐+,[](1+𝑏)𝑐𝑀=𝑐+2(𝑏+1)𝑐+1𝑑+𝑒(1+𝑏)2𝑐2.+(1+𝑏)𝑑(2.55) Let ξ‚΅π‘‘πœ“(𝑐,𝑑,𝑒;𝑧)=πœ™(𝑒,𝑣,𝑀;𝑧)=πœ™π‘,𝑐+[](1+𝑏)𝑐,𝑐+2(𝑏+1)𝑐+1𝑑+𝑒(1+𝑏)2𝑐2ξ‚Ά.+(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 πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ‚΅π½(𝑧);𝑧=πœ™π‘ +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 πœ“ξ€·π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έ(𝑧);z∈Ω.(2.58) Note that 𝑒𝑑+1=(π‘€βˆ’π‘’)(1+𝑏)π‘’π‘£βˆ’π‘’+(1+𝑏)(π‘€βˆ’3𝑒),(2.59) and since the admissibility condition for πœ™βˆˆΞ¦π½,2[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as given in Definition 1.2, hence πœ“βˆˆΞ¨[Ξ©,π‘ž] and by Theorem 1.4, 𝑗(𝑧)β‰Ίπ‘ž(𝑧),(2.60) or 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(2.61)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Φ𝐽,2[β„Ž,π‘ž]. The following result follows immediately from Theorem 2.13.

Theorem 2.14. Let πœ™βˆˆΞ¦π½,2[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘ satisfies πœ™ξ‚΅π½π‘ +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);π‘§β‰Ίβ„Ž(𝑧),(2.62) then 𝐽𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(2.63)

3. Superordination Results Associated with Generalized Srivastava-Attiya Operator

Definition 3.1. Let Ξ© be a set in β„‚ and π‘žβˆˆβ„‹[0,𝑝] with π‘§π‘žξ…ž(𝑧)β‰ 0. The class of admissible functions Ξ¦ξ…žπ½[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝜁)βˆ‰Ξ©(3.1) whenever 𝑒=π‘ž(𝑧),𝑣=π‘§π‘žξ…ž[]π‘ž(𝑧)βˆ’π‘šπ‘βˆ’(1+𝑏)(𝑧)ξ‚€π‘š(1+𝑏)π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(1+𝑏)2[]π‘€βˆ’π‘βˆ’(1+𝑏)2𝑒([]𝑒[]ξ‚Όβ‰₯11+𝑏)𝑣+π‘βˆ’(1+𝑏)+2π‘βˆ’(1+𝑏)π‘šξ‚»Reπ‘§π‘žξ…žξ…ž(𝑧)π‘žξ…ž(ξ‚Ό,𝑧)+1(3.2)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆ, and π‘šβ‰₯𝑝.

Theorem 3.2. Let πœ™βˆˆΞ¦ξ…žπ½[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,π‘π‘“βˆˆπ‘„0 and πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.3) is univalent in π‘ˆ, then ξ€½πœ™ξ€·π½Ξ©βŠ‚π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.4) implies that π‘ž(𝑧)≺𝐽𝑠+2,𝑏𝑓(𝑧).(3.5)

Proof. From (2.13) and (3.4), we have ξ€½πœ“ξ€·Ξ©βŠ‚π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ€Ύ.(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.6) From (2.10), we see that the admissibility condition for πœ™βˆˆΞ¦ξ…žπ½[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as given in Definition 1.3. Hence πœ“βˆˆΞ¨ξ…žπ‘[Ξ©,π‘ž], and by Theorem 1.5, π‘ž(𝑧)≺𝑗(𝑧) or π‘ž(𝑧)≺𝐽𝑠+2,𝑏𝑓(𝑧).(3.7)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Ξ¦ξ…žπ½[β„Ž,π‘ž]. The next result follows immediately from Theorem 3.2.

Theorem 3.3. Let β„Ž be analytic in π‘ˆ and πœ™βˆˆΞ¦ξ…žπ½[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)βˆˆπ‘„0 and πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.8) is univalent in π‘ˆ, then β„Žξ€·π½(𝑧)β‰Ίπœ™π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧,(3.9) and then π‘ž(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 πœ™.

Theorem 3.4. Let β„Ž be analytic in π‘ˆ and πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚. Suppose that the differential equation πœ™ξ€·π‘ž(𝑧),π‘§π‘žξ…ž(𝑧),𝑧2π‘žξ…žξ…žξ€Έ(𝑧);𝑧=β„Ž(𝑧)(3.11) has a solution π‘žβˆˆπ‘„0. If πœ™βˆˆΞ¦ξ…žπ½[Ξ©,π‘ž], π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)βˆˆπ‘„0, and πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.12) is univalent in π‘ˆ, then β„Žξ€·π½(𝑧)β‰Ίπœ™π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.13) implies that π‘ž(𝑧)≺𝐽𝑠+2,𝑏𝑓(𝑧),(3.14) and π‘ž(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 β„Ž1  and π‘ž1  be analytic in π‘ˆ, and let β„Ž2  be univalent function in π‘ˆ, π‘ž2βˆˆπ‘„0  with π‘ž1(0)=π‘ž2(0)=0 and πœ™βˆˆΞ¦π½[β„Ž2,π‘ž2]βˆ©Ξ¦ξ…žπ½[β„Ž1,π‘ž1]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)βˆˆβ„‹[0,𝑝]βˆ©π‘„0, and πœ™ξ€·π½π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);𝑧(3.15) is univalent in π‘ˆ, then β„Ž1𝐽(𝑧)β‰Ίπœ™π‘ +2,𝑏𝑓(𝑧),𝐽𝑠+1,𝑏𝑓(𝑧),𝐽𝑠,𝑏𝑓(𝑧);π‘§β‰Ίβ„Ž2(𝑧)(3.16) implies that π‘ž1(𝑧)≺𝐽𝑠+2,𝑏𝑓(𝑧)β‰Ίπ‘ž2(𝑧).(3.17)

Definition 3.6. Let Ξ© be a set in β„‚ and π‘žβˆˆβ„‹0 with π‘§π‘žβ€²(𝑧)β‰ 0. The class of admissible functions Ξ¦ξ…žπ½,1[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝜁)∈Ω(3.18) whenever 𝑒=π‘ž(𝑧),𝑣=π‘§π‘žξ…ž(𝑧)βˆ’π‘šπ‘π‘ž(𝑧)ξ‚€π‘š(1+𝑏)π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(1+𝑏)2π‘€βˆ’π‘2𝑒β‰₯1(1+𝑏)𝑣+π‘π‘’βˆ’2π‘π‘šξ‚»Reπ‘§π‘žξ…žξ…ž(𝑧)π‘žξ…žξ‚Ό,(𝑧)+1(3.19)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆ, and π‘šβ‰₯1.

The following result is associated with Theorem 2.9.

Theorem 3.7. Let πœ™βˆˆΞ¦ξ…žπ»,1[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)/π‘§π‘βˆ’1βˆˆπ‘„0, and πœ™ξ‚΅π½π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;𝑧(3.20) is univalent in π‘ˆ, then ξ‚»πœ™ξ‚΅π½Ξ©βŠ‚π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Άξ‚Ό;π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.21) implies that π½π‘ž(𝑧)≺𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1.(3.22)

Proof. From (2.36) and (3.21), we have ξ€½πœ™ξ€·Ξ©βŠ‚π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ€Ύ.(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.23) From (2.34), we see that the admissibility condition for πœ™βˆˆΞ¦ξ…žπ½,1[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as in Definition 1.3. Hence πœ“βˆˆΞ¨ξ…ž[Ξ©,π‘ž], and by Theorem 1.5, π‘ž(𝑧)≺𝑗(𝑧) or π½π‘ž(𝑧)≺𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1.(3.24)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Ξ¦ξ…žπ½,1[β„Ž,π‘ž]. The next result follows immediately from Theorem 3.7.

Theorem 3.8. Let π‘žβˆˆβ„‹0, and let β„Ž be analytic on π‘ˆ, and let πœ™βˆˆΞ¦ξ…žπ½,1[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)/π‘§π‘βˆ’1βˆˆπ‘„0, and πœ™ξ‚΅π½π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;𝑧(3.25) is univalent in π‘ˆ, then ξ‚΅π½β„Ž(𝑧)β‰Ίπœ™π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;𝑧(3.26) implies that π½π‘ž(𝑧)≺𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1.(3.27)

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

Corollary 3.9. Let β„Ž1 and π‘ž1 be analytic in π‘ˆ, let β„Ž2 be univalent function in π‘ˆ, π‘ž2βˆˆπ‘„0 with π‘ž1(0)=π‘ž2(0)=0, and πœ™βˆˆΞ¦π½,1[β„Ž2,π‘ž2]βˆ©Ξ¦ξ…žπ½,1[β„Ž1,π‘ž1]. If π‘“βˆˆπ’œπ‘, 𝐽𝑠+2,𝑏𝑓(𝑧)/π‘§π‘βˆ’1βˆˆβ„‹0βˆ©π‘„0, and πœ™ξ‚΅π½π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;𝑧(3.28) is univalent in π‘ˆ, then β„Ž1𝐽(𝑧)β‰Ίπœ™π‘ +2,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠+1,𝑏𝑓(𝑧)π‘§π‘βˆ’1,𝐽𝑠,𝑏𝑓(𝑧)π‘§π‘βˆ’1ξ‚Ά;π‘§β‰Ίβ„Ž2(𝑧)(3.29) implies that π‘ž1𝐽(𝑧)≺𝑠+2,𝑏𝑓(𝑧)π‘§π‘βˆ’1β‰Ίπ‘ž2(𝑧).(3.30)

Definition 3.10. Let Ξ© be a set in β„‚ and π‘ž(𝑧)β‰ 0,π‘§π‘žξ…ž(𝑧)β‰ 0, and π‘žβˆˆβ„‹. The class of admissible functions Ξ¦ξ…žπ½,2[Ξ©,π‘ž] consists of those functions πœ™βˆΆβ„‚3Γ—π‘ˆβ†’β„‚ that satisfy the admissibility condition: πœ™(𝑒,𝑣,𝑀;𝜁)βˆ‰Ξ©(3.31) whenever 𝑒=π‘ž(𝑧),𝑣=π‘ž(𝑧)+π‘§π‘žξ…ž(𝑧)ξ‚€π‘š(1+𝑏)π‘ž(𝑧)π‘βˆˆβ„‚β§΅π‘0,ξ‚»=0,βˆ’1,βˆ’2,…,π‘βˆˆβ„•Re(π‘€βˆ’π‘’)(1+𝑏)𝑒β‰₯1π‘£βˆ’π‘’+(1+𝑏)(π‘€βˆ’3𝑒)π‘šξ‚»Reπ‘§π‘žξ…žξ…ž(𝑧)π‘žξ…žξ‚Ό,(𝑧)+1(3.32)π‘§βˆˆπ‘ˆ,πœβˆˆπœ•π‘ˆ, and π‘šβ‰₯1.

The following result is associated with Theorem 2.13.

Theorem 3.11. Let πœ™βˆˆΞ¦ξ…žπ½,2[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)βˆˆπ‘„1, and πœ™ξ‚΅π½π‘ +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(3.33) is univalent in π‘ˆ, then ξ‚»πœ™ξ‚΅π½Ξ©βŠ‚π‘ +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);π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.34) implies that π½π‘ž(𝑧)≺𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(3.35)

Proof. From (2.57) and (3.34), we have ξ€½πœ™ξ€·Ξ©βŠ‚π‘—(𝑧),π‘§π‘—ξ…ž(𝑧),𝑧2π‘—ξ…žξ…žξ€Έξ€Ύ.(𝑧);π‘§βˆΆπ‘§βˆˆπ‘ˆ(3.36) From (2.55), we see that the admissibility condition for πœ™βˆˆΞ¦ξ…žπ½,2[Ξ©,π‘ž] is equivalent to the admissibility condition for πœ“ as in Definition 1.3. Hence πœ“βˆˆΞ¨β€²[Ξ©,π‘ž], and by Theorem 1.5, π‘ž(𝑧)≺𝑗(𝑧) or π½π‘ž(𝑧)≺𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏.𝑓(𝑧)(3.37)

If Ξ©β‰ β„‚ is a simply connected domain, then Ξ©βˆˆβ„Ž(π‘ˆ) for some conformal mapping β„Ž(𝑧) of π‘ˆ onto Ξ© and the class is written as Ξ¦ξ…žπ½,2[β„Ž,π‘ž]. The next result follows immediately from Theorem 3.11 as in the previous section.

Theorem 3.12. Let π‘žβˆˆβ„‹, let β„Ž be analytic in π‘ˆ, and let πœ™βˆˆΞ¦ξ…žπ½,2[Ξ©,π‘ž]. If π‘“βˆˆπ’œπ‘,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)βˆˆπ‘„1 and πœ™ξ‚΅π½π‘ +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(3.38) is univalent in π‘ˆ, then ξ‚΅π½β„Ž(𝑧)β‰Ίπœ™π‘ +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(3.39) implies that π½π‘ž(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 β„Ž1  and π‘ž1 be analytic in π‘ˆ, let β„Ž2 be univalent function in π‘ˆ, π‘ž2βˆˆπ‘„0 with π‘ž1(0)=π‘ž2(0)=0, and πœ™βˆˆΞ¦π½,2[β„Ž2,π‘ž2]βˆ©Ξ¦ξ…žπ½,2[β„Ž1,π‘ž1]. If π‘“βˆˆπ’œπ‘,𝐽𝑠+2,𝑏𝑓(𝑧)/𝐽𝑠+3,𝑏𝑓(𝑧)βˆˆβ„‹βˆ©π‘„1, and πœ™ξ‚΅π½π‘ +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(3.41) is univalent in π‘ˆ, then β„Ž1𝐽(𝑧)β‰Ίπœ™π‘ +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);π‘§β‰Ίβ„Ž2(𝑧)(3.42) implies that π‘ž1𝐽(𝑧)≺𝑠+2,𝑏𝑓(𝑧)𝐽𝑠+3,𝑏𝑓(𝑧)β‰Ίπ‘ž(𝑧).(3.43)

Other work related to certain operators concerning the subordination and superordination can be found in [20–25].

Acknowledgment

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

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Copyright Β© 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.


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