About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 632429, 9 pages
http://dx.doi.org/10.5402/2012/632429
Research Article

Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations

1Faculty of Computer and Mathematical Sciences, MARA University of Technology, 40450 Shah Alam, Selangor, Malaysia
2Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 20 January 2012; Accepted 25 March 2012

Academic Editors: O. Miyagaki and W. Yu

Copyright © 2012 Rashidah Omar and Suzeini Abdul Halim. 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

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 𝑓(𝑧)=𝑧+𝑛=2𝑎𝑛𝑧𝑛 in the open unit disk 𝐃={𝑧𝐂|𝑧|<1}. Also let 𝑆,𝐶, 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)(𝑧𝐃) if there exists an analytic function 𝑤 in 𝐃 such that 𝑤(0)=0 and |𝑤(𝑧)|<1 for |𝑧|<1 and 𝑓(𝑧)=𝑔(𝑤(𝑧)). In particular, if 𝑔 is univalent in 𝐃, then 𝑓(𝑧)𝑔(𝑧) is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝐃)𝑔(𝐃). The convolution of two analytic functions 𝜑(𝑧)=𝑛=2𝑎𝑛𝑧𝑛 and 𝜓(𝑧)=𝑛=0𝑏𝑛𝑧𝑛 is defined by 𝜑(𝑧)𝜓(𝑧)=𝑛=0𝑎𝑛𝑏𝑛𝑧𝑛=𝜓(𝑧)𝜑(𝑧).

For any real numbers 𝑘 and 𝜆 where 𝑘0, 𝜆0, 𝑐0, Cǎtaş [1] defined the multiplier transformations 𝐼(𝑘,𝜆,𝑐)𝑓(𝑧) by the following series:𝐼(𝑘,𝜆,𝑐)𝑓(𝑧)=𝑧+𝑛=21+𝜆(𝑛1)+𝑐1+𝑐𝑘𝑎𝑛𝑧𝑛.(1.1) Recently, some properties of functions using the multiplier transformations have been studied in [26]. 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 C̆ataş as basis instead of the Ruscheweyh operator.

Set the function𝑓𝑘,𝑐(𝑧)=𝑧+𝑛=21+𝑐1+𝜆(𝑛1)+𝑐𝑘𝑧𝑛(𝑘,𝜆𝐑,𝑘0,𝜆0,𝑐0),(1.2) and note that, for 𝜆=1, 𝑓𝑘,𝑐(𝑧) is the generalised polylogarithm functions discussed in [8]. A new function 𝑓𝜇𝑘,𝑐(𝑧) is defined in terms of the Hadamard product (or convolution) as follows:𝑓𝑘,𝑐(𝑧)𝑓𝜇𝑘,𝑐𝑧(𝑧)=(1𝑧)𝜇(𝜇>0).(1.3) Motivated by [911] and analogous to (1.1), the following operator is introduced:𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)=𝑓𝜇𝑘,𝑐𝑓(𝑧)=𝑧+𝑛=2(𝜇)𝑛1(𝑛1)!1+𝜆(𝑛1)+𝑐1+𝑐𝑘𝑎𝑛𝑧𝑛.(1.4) The operator 𝐼𝑘𝑐(𝜆,𝜇)𝑓 unifies other previously defined operators. For examples,(i)𝐼𝑘𝑐(𝜆,1)𝑓 is the 𝐼1(𝛿,𝜆,𝑙)𝑓 given in [1],(ii)𝐼𝑘𝑐(1,1)𝑓 is the 𝐼𝑘𝑐𝑓 given in [12],

also, for any integer 𝑘,(iii)𝐼𝑘0(𝜆,1)𝑓(𝑧)𝐷𝑘𝜆𝑓(𝑧) given in [13],(iv)𝐼𝑘0(1,1)𝑓(𝑧)𝐷𝑘𝑓(𝑧) given in [14],(v)𝐼𝑘1(1,1)𝑓(𝑧)𝐼𝑘𝑓(𝑧) given in [15].

The following relations are easily derived using the following definition:(1+𝑐)𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)=(1𝜆+𝑐)𝐼𝑘𝑐(𝐼𝜆,𝜇)𝑓(𝑧)+𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑓(𝑧),(1.5)𝜇𝐼𝑘𝑐(𝐼𝜆,𝜇+1)𝑓(𝑧)=𝑧𝑘𝑐(𝜆,𝜇)𝑓(𝑧)+(𝜇1)𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧).(1.6) Let 𝑁 be the class of all analytic and univalent functions 𝜙 in 𝐃 and for which 𝜙(𝐃) is convex with 𝜙(0)=1 and Re{𝜙(𝑧)}>0 for 𝑧𝐃. For 𝜙,𝜓𝑁, Ma and Minda [16] studied the subclasses 𝑆(𝜙),𝐶(𝜙), and 𝐾(𝜙,𝜓) of the class 𝐴. These classes are defined using the principle of subordination as follows:𝑆(𝜙)=𝑓𝑓𝐴,𝑧𝑓(𝑧),𝑓(𝑧)𝜙(𝑧)in𝐃𝐶(𝜙)=𝑓𝑓𝐴,1+𝑧𝑓(𝑧)𝑓,(𝑧)𝜙(𝑧)in𝐃𝐾(𝜙,𝜓)=𝑓𝑓𝐴,𝑔𝑆(𝜙)suchthat𝑧𝑓(𝑧).𝑔(𝑧)𝜓(𝑧)in𝐃(1.7) Obviously, we have the following relationships for special choices 𝜙 and 𝜓:𝑆1+𝑧1𝑧=𝑆,𝐶1+𝑧1𝑧=𝐶,𝐾1+𝑧,1𝑧1+𝑧1𝑧=𝐾.(1.8) Using the generalised multiplier transformations 𝐼𝑘𝑐(𝜆,𝜇)𝑓, new classes 𝑆𝑘𝑐(𝜆,𝜇;𝜙), 𝐶𝑘𝑐(𝜆,𝜇;𝜙) and 𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓) are introduced and defined below𝑆𝑘𝑐(𝜆,𝜇;𝜙)=𝑓𝐴𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝑆,𝐶(𝜙)𝑘𝑐(𝜆,𝜇;𝜙)=𝑓𝐴𝐼𝑘𝑐,𝐾(𝜆,𝜇)𝑓(𝑧)𝐶(𝜙)𝑘𝑐(𝜆,𝜇;𝜙,𝜓)=𝑓𝐴𝐼𝑘𝑐.(𝜆,𝜇)𝑓(𝑧)𝐾(𝜙,𝜓)(1.9) It can be shown easily that𝑓(𝑧)𝐶𝑘𝑐(𝜆,𝜇;𝜙)𝑧𝑓(𝑧)𝑆𝑘𝑐(𝜆,𝜇;𝜙).(1.10) Janowski [17] introduced class 𝑆[𝐴,𝐵]=𝑆((1+𝐴𝑧)/(1+𝐵𝑧)) and in particular for 𝜙(𝑧)=(1+𝐴𝑧)/(1+𝐵𝑧), we set𝑆𝑘𝑐𝜆,𝜇;1+𝐴𝑧1+𝐵𝑧=𝑆𝑘,𝑐[]𝜇;𝐴,𝐵(1𝐵<𝐴1).(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 𝐼𝑘𝑐(𝜆,𝜇)𝑓. Furthermore, applications of other families of integral operators are considered involving these classes.

2. Inclusion Properties Involving 𝐼𝐾𝑐(𝜆,𝜇)𝑓

In proving our results, the following lemmas are needed.

Lemma 2.1 (see [19]). Let 𝜙 be convex univalent in 𝐃, with 𝜙(0)=1 and Re[𝜅𝜙(𝑧)+𝜂]>0(𝜅,𝜂𝐂). If 𝑝 is analytic in 𝐃 with 𝑝(0)=1, then 𝑝(𝑧)+𝑧𝑝(𝑧)𝜅𝑝(𝑧)+𝜂𝜙(𝑧)𝑝(𝑧)𝜙(𝑧).(2.1)

Lemma 2.2 (see [20]). Let 𝜙 be convex univalent in 𝐃 and 𝜔 be analytic in 𝐃 with Re{𝜔(𝑧)}0. If 𝑝 is analytic in 𝐃 and 𝑝(0)=𝜙(0), then 𝑝(𝑧)+𝜔(𝑧)𝑧𝑝(𝑧)𝜙(𝑧)𝑝(𝑧)𝜙(𝑧).(2.2)

Theorem 2.3. For any real numbers 𝑘 and 𝜆 where 𝑘0,𝜆0 and 𝑐0.
Let 𝜙𝑁 and Re{𝜙(𝑧)+(1𝜆+𝑐)/𝜆}>0, then 𝑆𝑐𝑘+1(𝜆,𝜇;𝜙)𝑆𝑘𝑐(𝜆,𝜇;𝜙)(𝜇>0).

Proof. Let 𝑓𝑆𝑐𝑘+1(𝜆,𝜇;𝜙), and set 𝑝(𝑧)=(𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)])/(𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)) where 𝑝 is analytic in 𝐃 with 𝑝(0)=1. Rearranging (1.5), we have (1+𝑐)𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐𝐼(𝜆,𝜇)𝑓(𝑧)=(1𝜆+𝑐)+𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐.(𝜆,𝜇)𝑓(𝑧)(2.3) Next, differentiating (2.3) and multiplying by 𝑧 gives 𝑧𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)𝐼𝑐𝑘+1=𝑧𝐼(𝜆,𝜇)𝑓(𝑧)𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐+𝑧𝑧𝐼(𝜆,𝜇)𝑓(𝑧)𝑘𝑐(𝜆,𝜇)𝑓(𝑧)/𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝑧𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)/𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)+(1𝜆+𝑐)/𝜆=𝑝(𝑧)+𝑧𝑝(𝑧).𝑝(𝑧)+(1𝜆+𝑐)/𝜆(2.4) Since (𝑧[𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)])/(𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧))𝜙(𝑧) and applying Lemma 2.1, it follows that 𝑝𝜙. Thus 𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙).

Theorem 2.4. Let 𝑘,𝜆𝐑,𝑘0,𝜆0, and 𝜇1. Then 𝑆𝑘𝑐(𝜆,𝜇+1;𝜙)𝑆𝑘𝑐(𝜆,𝜇;𝜙)(𝑐0;𝜙𝑁).

Proof. Let 𝑓𝑆𝑘𝑐(𝜆,𝜇+1;𝜙), and from (1.6), we obtain that 𝜇𝐼𝑘𝑐(𝜆,𝜇+1)𝑓(𝑧)𝐼𝑘𝑐=𝑧𝐼(𝜆,𝜇)𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)+(𝜇1).(2.5) Making use of the differentiation on both sides in (2.5) and setting 𝑝(𝑧)=(𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)])/(𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)), we get the following: 𝑧𝐼𝑘𝑐(𝜆,𝜇+1)𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇+1)𝑓(𝑧)=𝑝(𝑧)+𝑧𝑝(𝑧)𝑝(𝑧)+(𝜇1)𝜙(𝑧).(2.6) Since 𝜇1 and Re{𝜙(𝑧)+(𝜇1)}>0, using Lemma 2.1, we conclude that 𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙).

Corollary 2.5. Let 𝜆0,𝜇1, and 1𝐵<𝐴1. Then 𝑆𝑘+1,𝑐[𝜇;𝐴,𝐵]𝑆𝑘,𝑐[𝜇;𝐴,𝐵] and 𝑆𝑘,𝑐[𝜇+1;𝐴,𝐵]𝑆𝑘,𝑐[𝜇;𝐴,𝐵].

Theorem 2.6. Let 𝜆0 and 𝜇1. Then 𝐶𝑐𝑘+1(𝜆,𝜇;𝜙)𝐶𝑘𝑐(𝜆,𝜇;𝜙) and 𝐶𝑘𝑐(𝜆,𝜇+1;𝜙)𝐶𝑘𝑐(𝜆,𝜇;𝜙).

Proof. Using (1.10) and Theorem 2.3, we observe that 𝑓(𝑧)𝐶𝑐𝑘+1(𝜆,𝜇;𝜙)𝑧𝑓(𝑧)𝑆𝑐𝑘+1(𝜆,𝜇;𝜙)𝑧𝑓(𝑧)𝑆𝑘𝑐(𝜆,𝜇;𝜙)𝐼𝑘𝑐(𝜆,𝜇)𝑧𝑓(𝑧)𝑆𝐼(𝜙)𝑧𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝑆(𝜙)𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐶(𝜙)𝑓𝐶𝑘𝑐(𝜆,𝜇;𝜙).(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 𝜆0,𝑐0 and Re{(1𝜆+𝑐)/𝜆}>0.
Then 𝐾𝑐𝑘+1(𝜆,𝜇;𝜙,𝜓)𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓) and 𝐾𝑘𝑐(𝜆,𝜇+1;𝜙,𝜓)𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓)(𝜙,𝜓𝑁).

Proof. Let 𝑓𝐾𝑐𝑘+1(𝜆,𝜇;𝜙,𝜓). In view of the definition of the class 𝐾𝑐𝑘+1(𝜆,𝜇;𝜙,𝜓), there is a function 𝑔𝑆𝑐𝑘+1(𝜆,𝜇;𝜙) such that 𝑧𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)𝐼𝑐𝑘+1(𝜆,𝜇)𝑔(𝑧)𝜓(𝑧).(2.8) Applying Theorem 2.3, then 𝑔𝑆𝑘𝑐(𝜆,𝜇;𝜙) and let 𝑞(𝑧)=(𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧)])/(𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧))𝜙(𝑧).
Let the analytic function 𝑝 with 𝑝(0)=1 as 𝑧𝐼𝑝(𝑧)=𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐.(𝜆,𝜇)𝑔(𝑧)(2.9) Thus, rearranging and differentiating (2.9), we have 𝐼𝑘𝑐(𝜆,𝜇)𝑧𝑓(𝑧)𝐼𝑘𝑐=𝐼(𝜆,𝜇)𝑔(𝑧)𝑝(𝑧)𝑘𝑐(𝜆,𝜇)𝑔(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧)+𝑝(𝑧).(2.10) Making use (1.5), (2.9), (2.10), and 𝑞(𝑧), we obtain that 𝑧𝐼𝑐𝑘+1(𝜆,𝜇)𝑓(𝑧)𝐼𝑐𝑘+1=𝐼(𝜆,𝜇)𝑔(𝑧)𝑐𝑘+1(𝜆,𝜇)𝑧𝑓(𝑧)𝐼𝑐𝑘+1=(𝜆,𝜇)𝑔(𝑧)(1𝜆+𝑐)𝐼𝑘𝑐(𝜆,𝜇)𝑧𝑓𝐼(𝑧)+𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑧𝑓(𝑧)(1𝜆+𝑐)𝐼𝑘𝑐𝐼(𝜆,𝜇)𝑔(𝑧)+𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑔(𝑧)=(1𝜆+𝑐)𝐼𝑘𝑐(𝜆,𝜇)𝑧𝑓/𝐼(𝑧)𝑘𝑐+𝐼(𝜆,𝜇)𝑔(𝑧)𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑧𝑓(𝑧)/𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧)𝐼(1𝜆+𝑐)+𝜆𝑧𝑘𝑐(𝜆,𝜇)𝑔(𝑧)/𝐼𝑘𝑐=𝑝(𝜆,𝜇)𝑔(𝑧)(1𝜆+𝑐)𝑝(𝑧)+𝜆(𝑧)𝑞(𝑧)+𝑝(𝑧)(1𝜆+𝑐)+𝜆𝑞(𝑧)=𝑝(𝑧)+𝑧𝑝(𝑧)𝑞(𝑧)+(1𝜆+𝑐)/𝜆𝜓(𝑧).(2.11) Since 𝑞(𝑧)𝜙(𝑧) and Re{(1𝜆+𝑐)/𝜆}>0, then Re{𝑞(𝑧)+(1𝜆+𝑐)/𝜆}>0. Using Lemma 2.2, we conclude that 𝑝(𝑧)𝜓(𝑧) and thus 𝑓𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓). 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 [2124]𝐹𝑐[]=𝑓(𝑧)𝑐+1𝑧𝑐𝑧0𝑡𝑐1𝑓(𝑡)𝑑𝑡(𝑐>1,Re𝑐0)=𝑧+𝑛=2𝑐+1𝑎𝑛+𝑐𝑛𝑧𝑛(3.1) and satisfies the following:𝑐𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝐼𝑓(𝑧)+𝑧𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)=(𝑐+1)𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧).(3.2)

Theorem 3.1. If 𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙), then 𝐹𝑐𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙).

Proof. Let 𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙), then (𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)])/(𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧))𝜙(𝑧). Taking the differentiation on both sides of (3.2) and multiplying by 𝑧, we obtain 𝑧𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐=𝑧𝐼(𝜆,𝜇)𝑓(𝑧)𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[𝑓]+𝑧𝑧𝐼(𝑧)𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)/𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)𝑧𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)/𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[].𝑓(𝑧)+𝑐(3.3) Setting 𝑝(𝑧)=(𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[𝑓(𝑧)]])/(𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[𝑓(𝑧)]), we have 𝑧𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)=𝑝(𝑧)+𝑧𝑝(𝑧).𝑝(𝑧)+𝑐(3.4) Lemma 2.1 implies (𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[𝑓(𝑧)]])/(𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[𝑓(𝑧)])𝜙(𝑧). Hence 𝐹𝑐𝑓𝑆𝑘𝑐(𝜆,𝜇;𝜙).

Theorem 3.2. Let 𝑓𝐶𝑘𝑐(𝜆,𝜇;𝜙), then 𝐹𝑐𝑓𝐶𝑘𝑐(𝜆,𝜇;𝜙).

Proof. By using (1.10) and Theorem 3.1, it follows that 𝑓𝐶𝑘𝑐(𝜆,𝜇;𝜙)𝑧𝑓(𝑧)𝑆𝑘𝑐(𝜆,𝜇;𝜙)𝐹𝑐𝑧𝑓(𝑧)𝑆𝑘𝑐𝐹(𝜆,𝜇;𝜙)𝑧𝑐[]𝑓(𝑧)𝑆𝑘𝑐(𝜆,𝜇;𝜙)𝐹𝑐[]𝑓(𝑧)𝐶𝑘𝑐(𝜆,𝜇;𝜙).(3.5)

Theorem 3.3. Let 𝜙,𝜓𝑁, and 𝑓𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓), then 𝐹𝑐𝑓𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓).

Proof. Let 𝑓𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓), then there exists function 𝑔𝑆𝑘𝑐(𝜆,𝜇;𝜙) such that (𝑧[𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)])/(𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧))𝜓(𝑧). Since 𝑔𝑆𝑘𝑐(𝜆,𝜇;𝜙) therefore from Theorem 3.1, 𝐹𝑐[𝑓(𝑧)]𝑆𝑘𝑐(𝜆,𝜇;𝜙). Then let 𝑧𝐼𝑞(𝑧)=𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑔(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑔(𝑧)𝜙(𝑧).(3.6) Set 𝑧𝐼𝑝(𝑧)=𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[].𝑔(𝑧)(3.7) By rearranging and differentiating (3.7), we obtain that 𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐𝑧𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]=𝐼𝑔(𝑧)𝑝(𝑧)𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑔(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[]+𝐼𝑔(𝑧)𝑘𝑐(𝜆,𝜇)𝐹𝑐[]𝑝𝑔(𝑧)(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝐹𝑐[].𝑔(𝑧)(3.8) Making use (3.2), (3.7), and (3.6), it can be derived that 𝑧𝐼𝑘𝑐(𝜆,𝜇)𝑓(𝑧)𝐼𝑘𝑐(𝜆,𝜇)𝑔(𝑧)=𝑝(𝑧)+𝑧𝑝(𝑧).𝑐+𝑞(𝑧)(3.9) Hence, applying Lemma 2.2, we conclude that 𝑝(𝑧)𝜓(𝑧), and it follows that 𝐹𝑐[𝑓(𝑧)]𝐾𝑘𝑐(𝜆,𝜇;𝜙,𝜓).

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.

References

  1. A. Cǎtaş, “On certain classes of p-valent functions defined by new multiplier transformations,” TC Istanbul Kultur University Publications, TC Istanbul kultur University, vol. 91, pp. 241–250, 2008, Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA '07), Istanbul, Turkey, August 2007.
  2. A. Cǎtaş, G. I. Oros, and G. Oros, “Differential subordinations associated with multiplier transformations,” Abstract and Applied Analysis, vol. 2008, Article ID 845724, 11 pages, 2008.
  3. A. Cǎtaş, “Neighborhoods of a certain class of analytic functions with negative coefficients,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 111–121, 2009. View at Scopus
  4. N. E. Cho and K. I. Noor, “Sandwich-type theorems for a class of multiplier transformations associated with the Noor integral operators,” Abstract and Applied Analysis, vol. 2012, Article ID 904272, 13 pages, 2012.
  5. R. M. El-Ashwah, M. K. Aouf, and S. M. El-Deeb, “On a class of multivalent functions defined by an extended multiplier transformations,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 623–628, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. A. A. Lupaş, “A note on a subclass of analytic functions defined by Ruscheweyh derivative and multiplier transformations,” International Journal of Open Problems in Complex Analysis, vol. 2, no. 2, pp. 60–66, 2010.
  7. K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, pp. 71–80, 1999.
  8. S. R. Mondal and A. Swaminathan, “Geometric properties of generalized polylogarithm,” Integral Transforms and Special Functions, vol. 21, no. 9, pp. 691–701, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. N. E. Cho and J. A. Kim, “Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation,” Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 323–330, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. J. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral operators,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432–445, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. O. S. Kwon and N. E. Cho, “Inclusion properties for certain subclasses of analytic functions associated with the Dziok-Srivastava operator,” Journal of Inequalities and Applications, vol. 2007, Article ID 51079, 10 pages, 2007.
  12. N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,” Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 39–49, 2003. View at Scopus
  13. F. M. Al-Oboudi, “On univalent functions defined by derivative operator,” International Journal of Mathematics and Mathematical Sciences, vol. 27, pp. 1429–1436, 2004.
  14. G. S. Salagean, “Subclasses of univalent functions,” in Proceedings of the Complex Analysis 5th Romanian-Finnish Seminar, Part 1, vol. 1013, pp. 362–372, Springer, 1983.
  15. B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in Analytic Function Theory, pp. 371–374, World Scientific, River Edge, NJ, USA, 1992.
  16. W. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang, Eds., pp. 157–169, International Press, Cambridge, Mass, USA, 1992.
  17. W. Janowski, “Some extremal problems for certain families of analytic functions I,” Annales Polonici Mathematici, vol. 28, pp. 297–326, 1973.
  18. R. Omar and S. A. Halim, “Classes of functions defined by Dziok-Srivastavaoperator,” Far East Journal of Mathematical Sciences. In press.
  19. P. Enigenberg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a Briot-Bouquet differential subordination,” General Inequalities, vol. 3, pp. 339–348, 1983.
  20. S. S. Miller and P. T. Mocanu, “Differential subordination and univalent functions,” The Michigan Mathematical Journal, vol. 28, pp. 157–171, 1981.
  21. S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.
  22. I. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138–147, 1993. View at Publisher · View at Google Scholar · View at Scopus
  23. R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965.
  24. A. E. Livington, “On the radius of univalence of certain analytic functions,” Proceedings of the American Mathematical Society, vol. 17, pp. 352–357, 1966.