Abstract

New subclasses of analytic functions in the open unit disc are introduced which are defined using generalised multiplier transformations. Inclusion theorems are investigated for functions to be in the classes. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes.

1. Introduction

Let 𝐴 denote the class of functions 𝑓 normalised by βˆ‘π‘“(𝑧)=𝑧+βˆžπ‘›=2π‘Žπ‘›π‘§π‘› in the open unit disk πƒβˆΆ={π‘§βˆˆπ‚βˆΆ|𝑧|<1}. Also let 𝑆⋆,𝐢, and 𝐾 denote, respectively, the subclasses of 𝐴 consisting of functions which are starlike, convex, and close to convex in 𝐃. An analytic function 𝑓 is subordinate to an analytic function 𝑔, written 𝑓(𝑧)≺𝑔(𝑧)(π‘§βˆˆπƒ) if there exists an analytic function 𝑀 in 𝐃 such that 𝑀(0)=0 and |𝑀(𝑧)|<1 for |𝑧|<1 and 𝑓(𝑧)=𝑔(𝑀(𝑧)). In particular, if 𝑔 is univalent in 𝐃, then 𝑓(𝑧)≺𝑔(𝑧) is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝐃)βŠ‚π‘”(𝐃). The convolution of two analytic functions βˆ‘πœ‘(𝑧)=βˆžπ‘›=2π‘Žπ‘›π‘§π‘› and βˆ‘πœ“(𝑧)=βˆžπ‘›=0𝑏𝑛𝑧𝑛 is defined by βˆ‘πœ‘(𝑧)βˆ—πœ“(𝑧)=βˆžπ‘›=0π‘Žπ‘›π‘π‘›π‘§π‘›=πœ“(𝑧)βˆ—πœ‘(𝑧).

For any real numbers π‘˜ and πœ† where π‘˜β‰₯0, πœ†β‰₯0, 𝑐β‰₯0, Cǎtaş [1] defined the multiplier transformations 𝐼(π‘˜,πœ†,𝑐)𝑓(𝑧) by the following series:𝐼(π‘˜,πœ†,𝑐)𝑓(𝑧)=𝑧+βˆžξ“π‘›=2ξ‚Έ1+πœ†(π‘›βˆ’1)+𝑐1+π‘π‘˜π‘Žπ‘›π‘§π‘›.(1.1) Recently, some properties of functions using the multiplier transformations have been studied in [2–6]. Using the convolution, we extend the multiplier transformation in (1.1) to be a unified operator. The approach used is similar to Noor's [7], only we generalise and extend to include powers and uses the multiplier CΜ†ataş as basis instead of the Ruscheweyh operator.

Set the functionπ‘“π‘˜,𝑐(𝑧)=𝑧+βˆžξ“π‘›=2ξ‚Έ1+𝑐1+πœ†(π‘›βˆ’1)+π‘π‘˜π‘§π‘›(π‘˜,πœ†βˆˆπ‘,π‘˜β‰₯0,πœ†β‰₯0,𝑐β‰₯0),(1.2) and note that, for πœ†=1, π‘“π‘˜,𝑐(𝑧) is the generalised polylogarithm functions discussed in [8]. A new function π‘“πœ‡π‘˜,𝑐(𝑧) is defined in terms of the Hadamard product (or convolution) as follows:π‘“π‘˜,𝑐(𝑧)βˆ—π‘“πœ‡π‘˜,𝑐𝑧(𝑧)=(1βˆ’π‘§)πœ‡(πœ‡>0).(1.3) Motivated by [9–11] and analogous to (1.1), the following operator is introduced:πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)=π‘“πœ‡π‘˜,π‘βˆ—π‘“(𝑧)=𝑧+βˆžξ“π‘›=2(πœ‡)π‘›βˆ’1ξ‚Έ(π‘›βˆ’1)!1+πœ†(π‘›βˆ’1)+𝑐1+π‘π‘˜π‘Žπ‘›π‘§π‘›.(1.4) The operator πΌπ‘˜π‘(πœ†,πœ‡)𝑓 unifies other previously defined operators. For examples,(i)πΌπ‘˜π‘(πœ†,1)𝑓 is the 𝐼1(𝛿,πœ†,𝑙)𝑓 given in [1],(ii)πΌπ‘˜π‘(1,1)𝑓 is the πΌπ‘˜π‘π‘“ given in [12],

also, for any integer π‘˜,(iii)πΌπ‘˜0(πœ†,1)𝑓(𝑧)β‰‘π·π‘˜πœ†π‘“(𝑧) given in [13],(iv)πΌπ‘˜0(1,1)𝑓(𝑧)β‰‘π·π‘˜π‘“(𝑧) given in [14],(v)πΌπ‘˜1(1,1)𝑓(𝑧)β‰‘πΌπ‘˜π‘“(𝑧) given in [15].

The following relations are easily derived using the following definition:(1+𝑐)πΌπ‘π‘˜+1(πœ†,πœ‡)𝑓(𝑧)=(1βˆ’πœ†+𝑐)πΌπ‘˜π‘(ξ€ΊπΌπœ†,πœ‡)𝑓(𝑧)+πœ†π‘§π‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…ž,(1.5)πœ‡πΌπ‘˜π‘(ξ€ΊπΌπœ†,πœ‡+1)𝑓(𝑧)=π‘§π‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…ž+(πœ‡βˆ’1)πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧).(1.6) Let 𝑁 be the class of all analytic and univalent functions πœ™ in 𝐃 and for which πœ™(𝐃) is convex with πœ™(0)=1 and Re{πœ™(𝑧)}>0 for π‘§βˆˆπƒ. For πœ™,πœ“βˆˆπ‘, Ma and Minda [16] studied the subclasses 𝑆⋆(πœ™),𝐢(πœ™), and 𝐾(πœ™,πœ“) of the class 𝐴. These classes are defined using the principle of subordination as follows:𝑆⋆(πœ™)∢=π‘“βˆΆπ‘“βˆˆπ΄,π‘§π‘“ξ…ž(𝑧)ξ‚Ό,𝑓(𝑧)β‰Ίπœ™(𝑧)in𝐃𝐢(πœ™)∢=π‘“βˆΆπ‘“βˆˆπ΄,1+π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ό,ξ‚»(𝑧)β‰Ίπœ™(𝑧)in𝐃𝐾(πœ™,πœ“)∢=π‘“βˆΆπ‘“βˆˆπ΄,βˆƒπ‘”βˆˆπ‘†β‹†(πœ™)suchthatπ‘§π‘“ξ…ž(𝑧)ξ‚Ό.𝑔(𝑧)β‰Ίπœ“(𝑧)in𝐃(1.7) Obviously, we have the following relationships for special choices πœ™ and πœ“:𝑆⋆1+𝑧1βˆ’π‘§=𝑆⋆,𝐢1+𝑧1βˆ’π‘§=𝐢,𝐾1+𝑧,1βˆ’π‘§1+𝑧1βˆ’π‘§=𝐾.(1.8) Using the generalised multiplier transformations πΌπ‘˜π‘(πœ†,πœ‡)𝑓, new classes π‘†π‘˜π‘(πœ†,πœ‡;πœ™), πΆπ‘˜π‘(πœ†,πœ‡;πœ™) and πΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“) are introduced and defined belowπ‘†π‘˜π‘ξ€½(πœ†,πœ‡;πœ™)∢=π‘“βˆˆπ΄βˆΆπΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)βˆˆπ‘†β‹†ξ€Ύ,𝐢(πœ™)π‘˜π‘ξ€½(πœ†,πœ‡;πœ™)∢=π‘“βˆˆπ΄βˆΆπΌπ‘˜π‘ξ€Ύ,𝐾(πœ†,πœ‡)𝑓(𝑧)∈𝐢(πœ™)π‘˜π‘ξ€½(πœ†,πœ‡;πœ™,πœ“)∢=π‘“βˆˆπ΄βˆΆπΌπ‘˜π‘ξ€Ύ.(πœ†,πœ‡)𝑓(𝑧)∈𝐾(πœ™,πœ“)(1.9) It can be shown easily that𝑓(𝑧)βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™)βŸΊπ‘§π‘“ξ…ž(𝑧)βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™).(1.10) Janowski [17] introduced class 𝑆⋆[𝐴,𝐡]=𝑆⋆((1+𝐴𝑧)/(1+𝐡𝑧)) and in particular for πœ™(𝑧)=(1+𝐴𝑧)/(1+𝐡𝑧), we setπ‘†π‘˜π‘ξ‚€πœ†,πœ‡;1+𝐴𝑧1+𝐡𝑧=π‘†β‹†π‘˜,𝑐[]πœ‡;𝐴,𝐡(βˆ’1≀𝐡<𝐴≀1).(1.11) In [18], the authors studied the inclusion properties for classes defined using Dziok-Srivastava operator. This paper investigates the similar properties for analytic functions in the classes defined by the generalised multiplier transformations πΌπ‘˜π‘(πœ†,πœ‡)𝑓. Furthermore, applications of other families of integral operators are considered involving these classes.

2. Inclusion Properties Involving 𝐼𝐾𝑐(πœ†,πœ‡)𝑓

In proving our results, the following lemmas are needed.

Lemma 2.1 (see [19]). Let πœ™ be convex univalent in 𝐃, with πœ™(0)=1 and Re[πœ…πœ™(𝑧)+πœ‚]>0(πœ…,πœ‚βˆˆπ‚). If 𝑝 is analytic in 𝐃 with 𝑝(0)=1, then 𝑝(𝑧)+π‘§π‘ξ…ž(𝑧)πœ…π‘(𝑧)+πœ‚β‰Ίπœ™(𝑧)βŸΉπ‘(𝑧)β‰Ίπœ™(𝑧).(2.1)

Lemma 2.2 (see [20]). Let πœ™ be convex univalent in 𝐃 and πœ” be analytic in 𝐃 with Re{πœ”(𝑧)}β‰₯0. If 𝑝 is analytic in 𝐃 and 𝑝(0)=πœ™(0), then 𝑝(𝑧)+πœ”(𝑧)π‘§π‘ξ…ž(𝑧)β‰Ίπœ™(𝑧)βŸΉπ‘(𝑧)β‰Ίπœ™(𝑧).(2.2)

Theorem 2.3. For any real numbers π‘˜ and πœ† where π‘˜β‰₯0,πœ†β‰₯0 and 𝑐β‰₯0.
Let πœ™βˆˆπ‘ and Re{πœ™(𝑧)+(1βˆ’πœ†+𝑐)/πœ†}>0, then π‘†π‘π‘˜+1(πœ†,πœ‡;πœ™)βŠ‚π‘†π‘˜π‘(πœ†,πœ‡;πœ™)(πœ‡>0).

Proof. Let π‘“βˆˆπ‘†π‘π‘˜+1(πœ†,πœ‡;πœ™), and set 𝑝(𝑧)=(𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)) where 𝑝 is analytic in 𝐃 with 𝑝(0)=1. Rearranging (1.5), we have (1+𝑐)πΌπ‘π‘˜+1(πœ†,πœ‡)𝑓(𝑧)πΌπ‘˜π‘ξ€ΊπΌ(πœ†,πœ‡)𝑓(𝑧)=(1βˆ’πœ†+𝑐)+πœ†π‘§π‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘.(πœ†,πœ‡)𝑓(𝑧)(2.3) Next, differentiating (2.3) and multiplying by 𝑧 gives π‘§ξ€ΊπΌπ‘π‘˜+1ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘π‘˜+1=𝑧𝐼(πœ†,πœ‡)𝑓(𝑧)π‘˜π‘ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘+𝑧𝑧𝐼(πœ†,πœ‡)𝑓(𝑧)ξ‚€ξ‚€π‘˜π‘ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘ξ€Έξ‚(πœ†,πœ‡)𝑓(𝑧)ξ…žξ‚€π‘§ξ€ΊπΌπ‘˜π‘ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘ξ€Έ(πœ†,πœ‡)𝑓(𝑧)+(1βˆ’πœ†+𝑐)/πœ†=𝑝(𝑧)+π‘§π‘ξ…ž(𝑧).𝑝(𝑧)+(1βˆ’πœ†+𝑐)/πœ†(2.4) Since (𝑧[πΌπ‘π‘˜+1(πœ†,πœ‡)𝑓(𝑧)]ξ…ž)/(πΌπ‘π‘˜+1(πœ†,πœ‡)𝑓(𝑧))β‰Ίπœ™(𝑧) and applying Lemma 2.1, it follows that π‘β‰Ίπœ™. Thus π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™).

Theorem 2.4. Let π‘˜,πœ†βˆˆπ‘,π‘˜β‰₯0,πœ†β‰₯0, and πœ‡β‰₯1. Then π‘†π‘˜π‘(πœ†,πœ‡+1;πœ™)βŠ‚π‘†π‘˜π‘(πœ†,πœ‡;πœ™)(𝑐β‰₯0;πœ™βˆˆπ‘).

Proof. Let π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡+1;πœ™), and from (1.6), we obtain that πœ‡πΌπ‘˜π‘(πœ†,πœ‡+1)𝑓(𝑧)πΌπ‘˜π‘=𝑧𝐼(πœ†,πœ‡)π‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)+(πœ‡βˆ’1).(2.5) Making use of the differentiation on both sides in (2.5) and setting 𝑝(𝑧)=(𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)), we get the following: π‘§ξ€ΊπΌπ‘˜π‘(ξ€»πœ†,πœ‡+1)𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡+1)𝑓(𝑧)=𝑝(𝑧)+π‘§π‘ξ…ž(𝑧)𝑝(𝑧)+(πœ‡βˆ’1)β‰Ίπœ™(𝑧).(2.6) Since πœ‡β‰₯1 and Re{πœ™(𝑧)+(πœ‡βˆ’1)}>0, using Lemma 2.1, we conclude that π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™).

Corollary 2.5. Let πœ†β‰₯0,πœ‡β‰₯1, and βˆ’1≀𝐡<𝐴≀1. Then π‘†β‹†π‘˜+1,𝑐[πœ‡;𝐴,𝐡]βŠ‚π‘†β‹†π‘˜,𝑐[πœ‡;𝐴,𝐡] and π‘†β‹†π‘˜,𝑐[πœ‡+1;𝐴,𝐡]βŠ‚π‘†β‹†π‘˜,𝑐[πœ‡;𝐴,𝐡].

Theorem 2.6. Let πœ†β‰₯0 and πœ‡β‰₯1. Then πΆπ‘π‘˜+1(πœ†,πœ‡;πœ™)βŠ‚πΆπ‘˜π‘(πœ†,πœ‡;πœ™) and πΆπ‘˜π‘(πœ†,πœ‡+1;πœ™)βŠ‚πΆπ‘˜π‘(πœ†,πœ‡;πœ™).

Proof. Using (1.10) and Theorem 2.3, we observe that 𝑓(𝑧)βˆˆπΆπ‘π‘˜+1(πœ†,πœ‡;πœ™)βŸΊπ‘§π‘“ξ…ž(𝑧)βˆˆπ‘†π‘π‘˜+1(πœ†,πœ‡;πœ™)βŸΉπ‘§π‘“ξ…ž(𝑧)βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™)βŸΊπΌπ‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…ž(𝑧)βˆˆπ‘†β‹†ξ€ΊπΌ(πœ™)βŸΊπ‘§π‘˜π‘ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žβˆˆπ‘†β‹†(πœ™)βŸΊπΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)∈𝐢(πœ™)βŸΊπ‘“βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™).(2.7) To prove the second part of Theorem, using the similar manner and applying Theorem 2.4, the result is obtained.

Theorem 2.7. Let πœ†β‰₯0,𝑐β‰₯0 and Re{(1βˆ’πœ†+𝑐)/πœ†}>0.
Then πΎπ‘π‘˜+1(πœ†,πœ‡;πœ™,πœ“)βŠ‚πΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“) and πΎπ‘˜π‘(πœ†,πœ‡+1;πœ™,πœ“)βŠ‚πΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“)(πœ™,πœ“βˆˆπ‘).

Proof. Let π‘“βˆˆπΎπ‘π‘˜+1(πœ†,πœ‡;πœ™,πœ“). In view of the definition of the class πΎπ‘π‘˜+1(πœ†,πœ‡;πœ™,πœ“), there is a function π‘”βˆˆπ‘†π‘π‘˜+1(πœ†,πœ‡;πœ™) such that π‘§ξ€ΊπΌπ‘π‘˜+1(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘π‘˜+1(πœ†,πœ‡)𝑔(𝑧)β‰Ίπœ“(𝑧).(2.8) Applying Theorem 2.3, then π‘”βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™) and let π‘ž(𝑧)=(𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝑔(𝑧)]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝑔(𝑧))β‰Ίπœ™(𝑧).
Let the analytic function 𝑝 with 𝑝(0)=1 as 𝑧𝐼𝑝(𝑧)=π‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘.(πœ†,πœ‡)𝑔(𝑧)(2.9) Thus, rearranging and differentiating (2.9), we have ξ€ΊπΌπ‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…ž(𝑧)ξ…žπΌπ‘˜π‘=𝐼(πœ†,πœ‡)𝑔(𝑧)𝑝(𝑧)π‘˜π‘(ξ€»πœ†,πœ‡)𝑔(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝑔(𝑧)+π‘ξ…ž(𝑧).(2.10) Making use (1.5), (2.9), (2.10), and π‘ž(𝑧), we obtain that π‘§ξ€ΊπΌπ‘π‘˜+1(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘π‘˜+1=𝐼(πœ†,πœ‡)𝑔(𝑧)π‘π‘˜+1(πœ†,πœ‡)π‘§π‘“ξ…žξ€»(𝑧)πΌπ‘π‘˜+1=(πœ†,πœ‡)𝑔(𝑧)(1βˆ’πœ†+𝑐)πΌπ‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…žξ€ΊπΌ(𝑧)+πœ†π‘§π‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…žξ€»(𝑧)ξ…ž(1βˆ’πœ†+𝑐)πΌπ‘˜π‘ξ€ΊπΌ(πœ†,πœ‡)𝑔(𝑧)+πœ†π‘§π‘˜π‘ξ€»(πœ†,πœ‡)𝑔(𝑧)ξ…ž=ξ€·(1βˆ’πœ†+𝑐)πΌπ‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…žξ€Έ/𝐼(𝑧)π‘˜π‘ξ€Έ+𝐼(πœ†,πœ‡)𝑔(𝑧)πœ†π‘§π‘˜π‘(πœ†,πœ‡)π‘§π‘“ξ…žξ€»(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘ξ€Έ(πœ†,πœ‡)𝑔(𝑧)𝐼(1βˆ’πœ†+𝑐)+πœ†π‘§π‘˜π‘ξ€»(πœ†,πœ‡)𝑔(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘ξ€Έ=𝑝(πœ†,πœ‡)𝑔(𝑧)(1βˆ’πœ†+𝑐)𝑝(𝑧)+πœ†(𝑧)π‘ž(𝑧)+π‘ξ…žξ€»(𝑧)(1βˆ’πœ†+𝑐)+πœ†π‘ž(𝑧)=𝑝(𝑧)+π‘§π‘ξ…ž(𝑧)π‘ž(𝑧)+(1βˆ’πœ†+𝑐)/πœ†β‰Ίπœ“(𝑧).(2.11) Since π‘ž(𝑧)β‰Ίπœ™(𝑧) and Re{(1βˆ’πœ†+𝑐)/πœ†}>0, then Re{π‘ž(𝑧)+(1βˆ’πœ†+𝑐)/πœ†}>0. Using Lemma 2.2, we conclude that 𝑝(𝑧)β‰Ίπœ“(𝑧) and thus π‘“βˆˆπΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“). By using similar manner and (1.6), we obtain the second result.

In summary, using subordination technique inclusion properties has been established for certain analytic functions defined via the generalised multiplier transformation.

3. Inclusion Properties Involving 𝐹𝑐𝑓

In this section, we determine properties of generalised Bernardi-Libera-Livington integral operator defined by [21–24]𝐹𝑐[]=𝑓(𝑧)𝑐+1π‘§π‘ξ€œπ‘§0π‘‘π‘βˆ’1𝑓(𝑑)𝑑𝑑(𝑐>βˆ’1,Re𝑐β‰₯0)=𝑧+βˆžξ“π‘›=2𝑐+1π‘Žπ‘›+𝑐𝑛𝑧𝑛(3.1) and satisfies the following:π‘πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝐼𝑓(𝑧)+π‘§π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…ž=(𝑐+1)πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧).(3.2)

Theorem 3.1. If π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™), then πΉπ‘π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™).

Proof. Let π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™), then (𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧))β‰Ίπœ™(𝑧). Taking the differentiation on both sides of (3.2) and multiplying by 𝑧, we obtain π‘§ξ€ΊπΌπ‘˜π‘ξ€»(πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘=𝑧𝐼(πœ†,πœ‡)𝑓(𝑧)π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[𝑓]+𝑧𝑧𝐼(𝑧)ξ‚€ξ‚€π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…žξ‚€π‘§ξ€ΊπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…žξ‚/ξ€·πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]ξ€Έ.𝑓(𝑧)+𝑐(3.3) Setting 𝑝(𝑧)=(𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[𝑓(𝑧)]]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[𝑓(𝑧)]), we have π‘§ξ€ΊπΌπ‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)=𝑝(𝑧)+π‘§π‘ξ…ž(𝑧).𝑝(𝑧)+𝑐(3.4) Lemma 2.1 implies (𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[𝑓(𝑧)]]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[𝑓(𝑧)])β‰Ίπœ™(𝑧). Hence πΉπ‘π‘“βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™).

Theorem 3.2. Let π‘“βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™), then πΉπ‘π‘“βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™).

Proof. By using (1.10) and Theorem 3.1, it follows that π‘“βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™)βŸΊπ‘§π‘“ξ…ž(𝑧)βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™)βŸΉπΉπ‘ξ€Ίπ‘§π‘“ξ…žξ€»(𝑧)βˆˆπ‘†π‘˜π‘ξ€ΊπΉ(πœ†,πœ‡;πœ™)βŸΊπ‘§π‘[]𝑓(𝑧)ξ…žβˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™)βŸΊπΉπ‘[]𝑓(𝑧)βˆˆπΆπ‘˜π‘(πœ†,πœ‡;πœ™).(3.5)

Theorem 3.3. Let πœ™,πœ“βˆˆπ‘, and π‘“βˆˆπΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“), then πΉπ‘π‘“βˆˆπΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“).

Proof. Let π‘“βˆˆπΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“), then there exists function π‘”βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™) such that (𝑧[πΌπ‘˜π‘(πœ†,πœ‡)𝑓(𝑧)]ξ…ž)/(πΌπ‘˜π‘(πœ†,πœ‡)𝑔(𝑧))β‰Ίπœ“(𝑧). Since π‘”βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™) therefore from Theorem 3.1, 𝐹𝑐[𝑓(𝑧)]βˆˆπ‘†π‘˜π‘(πœ†,πœ‡;πœ™). Then let π‘§ξ€ΊπΌπ‘ž(𝑧)=π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑔(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑔(𝑧)β‰Ίπœ™(𝑧).(3.6) Set 𝑧𝐼𝑝(𝑧)=π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[].𝑔(𝑧)(3.7) By rearranging and differentiating (3.7), we obtain that ξ€ΊπΌπ‘˜π‘(πœ†,πœ‡)πΉπ‘ξ€Ίπ‘§π‘“ξ…ž(𝑧)ξ€»ξ€»ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]=𝐼𝑔(𝑧)𝑝(𝑧)π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑔(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[]+𝐼𝑔(𝑧)π‘˜π‘(πœ†,πœ‡)𝐹𝑐[]𝑝𝑔(𝑧)ξ…ž(𝑧)πΌπ‘˜π‘(πœ†,πœ‡)𝐹𝑐[].𝑔(𝑧)(3.8) Making use (3.2), (3.7), and (3.6), it can be derived that π‘§ξ€ΊπΌπ‘˜π‘(ξ€»πœ†,πœ‡)𝑓(𝑧)ξ…žπΌπ‘˜π‘(πœ†,πœ‡)𝑔(𝑧)=𝑝(𝑧)+π‘§π‘ξ…ž(𝑧).𝑐+π‘ž(𝑧)(3.9) Hence, applying Lemma 2.2, we conclude that 𝑝(𝑧)β‰Ίπœ“(𝑧), and it follows that 𝐹𝑐[𝑓(𝑧)]βˆˆπΎπ‘˜π‘(πœ†,πœ‡;πœ™,πœ“).

For analytic functions in the classes defined by generalised multiplier transformations, the generalised Bernardi-Libera-Livington integral operator has been shown to be preserved in these classes.

4. Conclusion

Results involving functions defined using the generalised multiplier transformation, namely, inclusion properties and the Bernardi-Libera-Livington integral operator were obtained using subordination principles. In [18], similar results were discussed for functions defined using the Dziok-Srivastava operator.

Acknowledgment

This research was supported by IPPP/UPGP/geran(RU/PPP)/PS207/2009A University Malaya Grants 2009.