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Abstract and Applied Analysis
Volumeย 2008, Article IDย 246876, 8 pages
http://dx.doi.org/10.1155/2008/246876
Research Article

Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator

Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea

Received 8 August 2007; Revised 29 October 2007; Accepted 23 November 2007

Academic Editor: John Michaelย Rassias

Copyright ยฉ 2008 Nak Eun Cho. 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

The purpose of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions associated with a family of linear operators, which are defined by means of the Hadamard product (or convolution). Some integral preserving properties are also considered.

1. Introduction

Let ๐’œ denote the class of functions of the form ๐‘“(๐‘ง)=๐‘ง+โˆž๎“๐‘˜=2๐‘Ž๐‘˜๐‘ง๐‘˜(1.1) which are analytic in the open unit disk ๐•Œ={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1}. If ๐‘“ and ๐‘” are analytic in ๐•Œ, we say that ๐‘“ is subordinate to ๐‘”, written ๐‘“โ‰บ๐‘” or ๐‘“(๐‘ง)โ‰บ๐‘”(๐‘ง) if there exists an analytic function ๐‘ค in ๐•Œ with ๐‘ค(0)=0 and |๐‘ค(๐‘ง)|<1 for ๐‘งโˆˆ๐•Œ such that ๐‘“(๐‘ง)=๐‘”(๐‘ค(๐‘ง)). We denote by ๐’ฎโˆ—, ๐’ฆ, and ๐’ž the subclasses of ๐’œ consisting of all analytic functions which are, respectively, starlike, convex, and close-to-convex in ๐•Œ.

Let ๐’ฉ be the class of all functions ๐œ™ which are analytic and univalent in ๐•Œ and for which ๐œ™(๐•Œ) is convex with ๐œ™(0)=1 and Re{๐œ™(๐‘ง)}>0 for ๐‘งโˆˆ๐•Œ.

Making use of the principle of subordination between analytic functions, many authors investigated the subclasses ๐’ฎโˆ—(๐œ™), ๐’ฆ(๐œ™), and ๐’ž(๐œ™,๐œ“) of the class ๐’œ for ๐œ™,๐œ“โˆˆ๐’ฉ (cf. [1, 2]), which are defined by ๐’ฎโˆ—๎‚†(๐œ™)โˆถ=๐‘“โˆˆ๐’œโˆถ๐‘ง๐‘“๎…ž(๐‘ง)๎‚‡,๎‚†๐‘“(๐‘ง)โ‰บ๐œ™(๐‘ง)in๐•Œ๐’ฆ(๐œ™)โˆถ=๐‘“โˆˆ๐’œโˆถ1+๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž๎‚‡,๎‚†(๐‘ง)โ‰บ๐œ™(๐‘ง)in๐•Œ๐’ž(๐œ™,๐œ“)โˆถ=๐‘“โˆˆ๐’œโˆถโˆƒ๐‘”โˆˆ๐’ฎโˆ—(๐œ™)s.t.๐‘ง๐‘“๎…ž(๐‘ง)๎‚‡.๐‘”(๐‘ง)โ‰บ๐œ“(๐‘ง)in๐•Œ(1.2) For ๐œ™(๐‘ง)=๐œ“(๐‘ง)=(1+๐‘ง)/(1โˆ’๐‘ง) in the definitions defined above, we have the well-known classes ๐’ฎโˆ—, ๐’ฆ, and ๐’ž, respectively. Furthermore, for the function classes ๐’ฎโˆ—[๐ด,๐ต] and ๐’ฆ[๐ด,๐ต] investigated by Janowski [3] (also see [4]), it is easily seen that ๐’ฎโˆ—๎‚€1+๐ด๐‘ง๎‚1+๐ต๐‘ง=๐’ฎโˆ—๐’ฆ๎‚€[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1),1+๐ด๐‘ง๎‚1+๐ต๐‘ง=๐’ฆ[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1).(1.3)

We now define the function โ„Ž(๐‘Ž,๐‘)(๐‘ง) by โ„Ž(๐‘Ž,๐‘)(๐‘ง)โˆถ=โˆž๎“๐‘˜=0(๐‘Ž)๐‘˜(๐‘)๐‘˜๐‘ง๐‘˜+1,๎‚€๐‘งโˆˆ๐•Œ;๐‘Žโˆˆโ„;๐‘โˆˆโ„โงตโ„คโˆ’0;โ„คโˆ’0๎€ฝ๎€พ๎‚,โˆถ=0,โˆ’1,โˆ’2,โ€ฆ(1.4) where (๐œˆ)๐‘˜ is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by (๐œˆ)๐‘˜โˆถ=ฮ“(๐œˆ+๐‘˜)=๎‚ป๎€ฝ0๎€พ,๎€ฝ๎€พฮ“(๐œˆ)1if๐‘˜=0,๐œˆโˆˆโ„‚โงต๐œˆ(๐œˆ+1)โ‹ฏ(๐œˆ+๐‘˜โˆ’1)if๐‘˜โˆˆโ„•โˆถ=1,2,โ€ฆ,๐œˆโˆˆโ„‚.(1.5)

We also denote by ๐ฟ(๐‘Ž,๐‘): ๐’œโ†’๐’œ the operator defined by ๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)=โ„Ž(๐‘Ž,๐‘)(๐‘ง)โˆ—๐‘“(๐‘ง)(๐‘งโˆˆ๐•Œ;๐‘“โˆˆ๐’œ),(1.6) where the symbol (โˆ—) stands for the Hadamard product (or convolution). Then it is easily observed from definitions (1.4) and (1.6) that ๐ฟ(2,1)๐‘“(๐‘ง)=๐‘ง๐‘“๎…ž(๐‘ง) and ๐‘ง๎‚€๎‚๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)๎…ž=๐‘Ž๐ฟ(๐‘Ž+1,๐‘)๐‘“(๐‘ง)โˆ’(๐‘Žโˆ’1)๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง).(1.7) Furthermore, we note that ๐ฟ(๐‘›+1,1)๐‘“(๐‘ง)=๐ท๐‘›๐‘“(๐‘ง)(๐‘›>โˆ’1), where the symbol ๐ท๐‘› denotes the familiar Ruscheweyh derivative [5] (also, see [6]) for ๐‘›โˆˆโ„•0โˆถ=โ„•โˆช{0}. The operator ๐ฟ(๐‘Ž,๐‘) was introduced and studied by Carlson and Shaffer [7] which has been used widely on the space of analytic and univalent functions in ๐•Œ (see also [8]).

By using the operator ๐ฟ(๐‘Ž,๐‘), we introduce the following classes of analytic functions for ๐œ™,๐œ“โˆˆ๐’ฉ, ๐‘Žโˆˆโ„ and ๐‘โˆˆโ„โงตโ„คโˆ’0: ๐’ฎ๐‘Ž,๐‘๎‚†(๐œ™)โˆถ=๐‘“โˆˆ๐’œโˆถ๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)โˆˆ๐’ฎโˆ—๎‚‡,๐’ฆ(๐œ™)๐‘Ž,๐‘๎‚†๎‚‡,๐’ž(๐œ™)โˆถ=๐‘“โˆˆ๐’œโˆถ๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)โˆˆ๐’ฆ(๐œ™)๐‘Ž,๐‘๎‚†๎‚‡.(๐œ™,๐œ“)โˆถ=๐‘“โˆˆ๐’œโˆถ๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)โˆˆ๐’ž(๐œ™,๐œ“)(1.8) We also note that ๐‘“(๐‘ง)โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™)โŸบ๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™).(1.9) In particular, we set ๐’ฎ๐‘Ž,๐‘๎‚€1+๐ด๐‘ง๎‚1+๐ต๐‘ง=๐’ฎ๐‘Ž,๐‘๐’ฆ[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1),๐‘Ž,๐‘๎‚€1+๐ด๐‘ง๎‚1+๐ต๐‘ง=๐’ฆ๐‘Ž,๐‘[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1).(1.10)

In this paper, we investigate several inclusion properties of the classes ๐’ฎ๐‘Ž,๐‘(๐œ™), ๐’ฆ๐‘Ž,๐‘(๐œ™), and ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“). The integral preserving properties in connection with the operator ๐ฟ(๐‘Ž,๐‘) are also considered. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out.

2. Inclusion Properties Involving the Operator ๐ฟ(๐‘Ž,๐‘)

The following lemmas will be required in our investigation.

Lemma 2.1 (See [9, Pages 60-61]). Let ๐‘Ž2โ‰ฅ๐‘Ž1>0. If ๐‘Ž2โ‰ฅ2 or ๐‘Ž1+๐‘Ž2โ‰ฅ3, then the function โ„Ž(๐‘Ž1,๐‘Ž2)(๐‘ง) defined by (1.4) belongs to the class ๐’ฆ.

Lemma 2.2 (See [10]). Let ๐‘“โˆˆ๐’ฆ and ๐‘”โˆˆ๐’ฎโˆ—. Then for every analytic function ๐‘„ in ๐•Œ, ๎€ท๐‘“โˆ—๐‘„๐‘”)(๐‘“โˆ—๐‘”)(๐•Œ)โŠ‚co๐‘„(๐•Œ),(2.1) where co๐‘„(๐•Œ) denote the closed convex hull of ๐‘„(๐•Œ).

Theorem 2.3. Let ๐‘Ž2โ‰ฅ๐‘Ž1>0, ๐‘โˆˆโ„โงตโ„คโˆ’0, and ๐œ™โˆˆ๐’ฉ. If ๐‘Ž2โ‰ฅ2 or ๐‘Ž1+๐‘Ž2โ‰ฅ3, then ๐’ฎ๐‘Ž2,๐‘(๐œ™)โŠ‚๐’ฎ๐‘Ž1,๐‘(๐œ™).(2.2)

Proof. Let ๐‘“โˆˆ๐’ฎ๐‘Ž2,๐‘(๐œ™). Then there exists an analytic function ๐‘ค in ๐•Œ with |๐‘ค(๐‘ง)|<1(๐‘งโˆˆ๐•Œ) and ๐‘ค(0)=0 such that ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž2๎‚๎‚,๐‘๐‘“(๐‘ง)๎…ž๐ฟ๎‚€๐‘Ž2๎‚๎‚€๎‚,๐‘๐‘“(๐‘ง)=๐œ™๐‘ค(๐‘ง)(๐‘งโˆˆ๐•Œ).(2.3) By using (1.6) and (2.3), we have ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž1๎‚๎‚,๐‘๐‘“(๐‘ง)๎…ž๐ฟ๎‚€๐‘Ž1๎‚=๐‘ง๎‚€โ„Ž๎‚€๐‘Ž,๐‘๐‘“(๐‘ง)1๎‚๎‚,๐‘(๐‘ง)โˆ—๐‘“(๐‘ง)๎…žโ„Ž๎‚€๐‘Ž1๎‚=๐‘ง๎‚€โ„Ž๎‚€๐‘Ž,๐‘(๐‘ง)โˆ—๐‘“(๐‘ง)2๎‚๎‚€๐‘Ž,๐‘(๐‘ง)โˆ—โ„Ž1,๐‘Ž2๎‚๎‚(๐‘ง)โˆ—๐‘“(๐‘ง)๎…žโ„Ž๎‚€๐‘Ž2๎‚๎‚€๐‘Ž,๐‘(๐‘ง)โˆ—โ„Ž1,๐‘Ž2๎‚=โ„Ž๎‚€๐‘Ž(๐‘ง)โˆ—๐‘“(๐‘ง)1,๐‘Ž2๎‚๎‚€๐ฟ๎‚€๐‘Ž(๐‘ง)โˆ—๐‘ง2๎‚๎‚,๐‘๐‘“(๐‘ง)๎…žโ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€๐‘Ž(๐‘ง)โˆ—๐ฟ2๎‚=โ„Ž๎‚€๐‘Ž,๐‘๐‘“(๐‘ง)1,๐‘Ž2๎‚๎‚€๎‚๐ฟ๎‚€๐‘Ž(๐‘ง)โˆ—๐œ™๐‘ค(๐‘ง)2๎‚,๐‘๐‘“(๐‘ง)โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€๐‘Ž(๐‘ง)โˆ—๐ฟ2๎‚.,๐‘๐‘“(๐‘ง)(2.4) It follows from (2.3) and Lemma 2.1 that ๐ฟ(๐‘Ž2,๐‘)๐‘“(๐‘ง)โˆˆ๐’ฎโˆ— and โ„Ž(๐‘Ž1,๐‘Ž2)(๐‘ง)โˆˆ๐’ฆ, respectively. Then by applying Lemma 2.2 to (2.4), we obtain ๎‚†โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€๐‘Ž(๐‘ง)โˆ—๐œ™(๐‘ค)๐ฟ2๎‚๐‘“๎‚‡,๐‘๎‚†โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€๐‘Ž(๐‘ง)โˆ—๐ฟ2๎‚๐‘“๎‚‡,๐‘(๐•Œ)โŠ‚๎‚€๎‚co๐œ™(๐•Œ)โŠ‚๐œ™(๐•Œ),(2.5) since ๐œ™ is convex univalent. Therefore, from the definition of subordination and (2.5), we have ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž1๎‚๎‚,๐‘๐‘“(๐‘ง)๎…ž๐ฟ๎‚€๐‘Ž1๎‚,๐‘๐‘“(๐‘ง)โ‰บ๐œ™(๐‘ง)(๐‘งโˆˆ๐•Œ),(2.6) or, equivalently, ๐‘“โˆˆ๐’ฎ๐‘Ž1,๐‘(๐œ™), which completes the proof of Theorem 2.3.

Theorem 2.4. Let ๐‘Žโˆˆโ„, ๐‘2โ‰ฅ๐‘1>0 and ๐œ™โˆˆ๐’ฉ. If ๐‘2โ‰ฅ2 or ๐‘1+๐‘2โ‰ฅ3, then ๐’ฎ๐‘Ž,๐‘1(๐œ™)โŠ‚๐’ฎ๐‘Ž,๐‘2(๐œ™).(2.7)

Proof. (๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘1(๐œ™)). Using a similar argument as in the proof of Theorem 2.3, we obtain ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž,๐‘2๎‚๎‚๐‘“(๐‘ง)๎…ž๐ฟ๎‚€๐‘Ž,๐‘2๎‚=โ„Ž๎‚€๐‘Ž๐‘“(๐‘ง)1,๐‘Ž2๎‚๎‚€๎‚๐ฟ๎‚€(๐‘ง)โˆ—๐œ™๐‘ค(๐‘ง)๐‘Ž,๐‘1๎‚๐‘“(๐‘ง)โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€(๐‘ง)โˆ—๐ฟ๐‘Ž,๐‘1๎‚,๐‘“(๐‘ง)(2.8) where ๐‘ค is an analytic function in ๐•Œ with |๐‘ค(๐‘ง)|<1(๐‘งโˆˆ๐•Œ) and ๐‘ค(0)=0. Applying Lemma 2.1 and the fact that ๐ฟ(๐‘Ž,๐‘1)๐‘“(๐‘ง)โˆˆ๐’ฎโˆ—, we see that ๎‚†โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€๐‘Žโˆ—โ„Ž(๐‘ค)๐ฟ2๎‚๐‘“๎‚‡,๐‘๎‚†โ„Ž๎‚€๐‘Ž1,๐‘Ž2๎‚๎‚€โˆ—๐ฟ๐‘Ž,๐‘1๎‚๐‘“๎‚‡(๐•Œ)โŠ‚๎‚€๎‚co๐œ™(๐•Œ)โŠ‚๐œ™(๐•Œ),(2.9) since ๐œ™ is convex univalent. Thus the proof of Theorem 2.3 is completed.

Corollary 2.5. Let ๐‘Ž2โ‰ฅ๐‘Ž1>0, ๐‘2โ‰ฅ๐‘1>0, and ๐œ™โˆˆ๐’ฉ. If ๐‘Ž2โ‰ฅmin{2,3โˆ’๐‘Ž1} and ๐‘2โ‰ฅmin{2,3โˆ’๐‘1}, then ๐’ฎ๐‘Ž2,๐‘1(๐œ™)โŠ‚๐’ฎ๐‘Ž2,๐‘2(๐œ™)โŠ‚๐’ฎ๐‘Ž1,๐‘2(๐œ™).(2.10)

Theorem 2.6. Let ๐‘Ž2โ‰ฅ๐‘Ž1>0, ๐‘2โ‰ฅ๐‘1>0 and ๐œ™โˆˆ๐’ฉ. If ๐‘Ž2โ‰ฅmin{2,3โˆ’๐‘Ž1} and ๐‘2โ‰ฅmin{2,3โˆ’๐‘1}, then ๐’ฆ๐‘Ž2,๐‘1(๐œ™)โŠ‚๐’ฆ๐‘Ž2,๐‘2(๐œ™)โŠ‚๐’ฆ๐‘Ž1,๐‘2(๐œ™).(2.11)

Proof. Applying (1.9) and Corollary 2.5, we observe that ๐‘“(๐‘ง)โˆˆ๐’ฆ๐‘Ž2,๐‘1๎‚€๐‘Ž(๐œ™)โŸบ๐ฟ2,๐‘1๎‚๎‚€๐ฟ๎‚€๐‘Ž๐‘“(๐‘ง)โˆˆ๐’ฆ(๐œ™)โŸบ๐‘ง2,๐‘1๎‚๎‚๐‘“(๐‘ง)๎…žโˆˆ๐’ฎโˆ—๎‚€๐‘Ž(๐œ™)โŸบ๐ฟ2,๐‘1๎‚๎‚€๐‘ง๐‘“๎…ž๎‚(๐‘ง)โˆˆ๐’ฎโˆ—(๐œ™)โŸบ๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž2,๐‘1(๐œ™)โ‡’๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž2,๐‘2๎‚€๐‘Ž(๐œ™)โŸบ๐ฟ2,๐‘2๎‚๎‚€๐‘ง๐‘“๎…ž๎‚(๐‘ง)โˆˆ๐’ฎโˆ—๎‚€๐ฟ๎‚€๐‘Ž(๐œ™)โŸบ๐‘ง2,๐‘2๎‚๎‚๐‘“(๐‘ง)๎…žโˆˆ๐’ฎโˆ—๎‚€๐‘Ž(๐œ™)โŸบ๐ฟ2,๐‘2๎‚๐‘“(๐‘ง)โˆˆ๐’ฆ(๐œ™)โŸบ๐‘“(๐‘ง)โˆˆ๐’ฆ๐‘Ž2,๐‘2(๐œ™),๐‘“(๐‘ง)โˆˆ๐’ฆ๐‘Ž2,๐‘2๎‚€๐‘Ž(๐œ™)โŸบ๐ฟ2,๐‘2๎‚๎‚€๐‘Ž๐‘“(๐‘ง)โˆˆ๐’ฆ(๐œ™)โŸบ๐ฟ2,๐‘2๎‚๎‚€๐‘ง๐‘“๎…ž๎‚(๐‘ง)โˆˆ๐’ฎโˆ—(๐œ™)โ‡’๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž1,๐‘2๎‚€๐ฟ๎‚€๐‘Ž(๐œ™)โŸบ๐‘ง1,๐‘2๎‚๎‚๐‘“(๐‘ง)๎…žโˆˆ๐’ฎโˆ—(๐œ™)โŸบ๐‘“(๐‘ง)โˆˆ๐’ฆ๐‘Ž1,๐‘2(๐œ™),(2.12) which evidently proves Theorem 2.6.

Taking ๐œ™(๐‘ง)=(1+๐ด๐‘ง)/(1+๐ต๐‘ง)(โˆ’1โ‰ค๐ต<๐ดโ‰ค1;๐‘งโˆˆ๐•Œ) in Corollary 2.5 and Theorem 2.6, we have the following corollary.

Corollary 2.7. Let ๐‘Ž2โ‰ฅ๐‘Ž1>0 and ๐‘2โ‰ฅ๐‘1>0. If ๐‘Ž2โ‰ฅmin{2,3โˆ’๐‘Ž1} and ๐‘2โ‰ฅmin{2,3โˆ’๐‘1}, then ๐’ฎ๐‘Ž2,๐‘1[๐ด,๐ต]โŠ‚๐’ฎ๐‘Ž2,๐‘2[๐ด,๐ต]โŠ‚๐’ฎ๐‘Ž1,๐‘2๐’ฆ[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1),๐‘Ž2,๐‘1[๐ด,๐ต]โŠ‚๐’ฆ๐‘Ž2,๐‘2[๐ด,๐ต]โŠ‚๐’ฆ๐‘Ž1,๐‘2[๐ด,๐ต](โˆ’1โ‰ค๐ต<๐ดโ‰ค1).(2.13)

To prove the theorems below, we need the following lemma.

Lemma 2.8. Let ๐œ™โˆˆ๐’ฉ. If ๐‘“โˆˆ๐’ฆ and ๐‘žโˆˆ๐’ฎโˆ—(๐œ™), then ๐‘“โˆ—๐‘žโˆˆ๐’ฎโˆ—(๐œ™).

Proof. Let ๐‘žโˆˆ๐’ฎโˆ—(๐œ™). Then ๐‘ง๐‘ž๎…ž๎‚€๎‚(๐‘ง)=๐‘ž(๐‘ง)๐œ™๐œ”(๐‘ง)(๐‘งโˆˆ๐•Œ),(2.14) where ๐œ” is an analytic function in ๐•Œ with |๐‘ค(๐‘ง)|<1(๐‘งโˆˆ๐•Œ) and ๐‘ค(0)=0. Thus we have ๐‘ง๎‚€๎‚๐‘“(๐‘ง)โˆ—๐‘ž(๐‘ง)๎…ž=๐‘“(๐‘ง)โˆ—๐‘ž(๐‘ง)๐‘“(๐‘ง)โˆ—๐‘ง๐‘ž๎…ž(๐‘ง)=๎‚€๎‚๐‘“(๐‘ง)โˆ—๐‘ž(๐‘ง)๐‘“(๐‘ง)โˆ—๐œ™๐œ”(๐‘ง)๐‘ž(๐‘ง)๐‘“(๐‘ง)โˆ—๐‘ž(๐‘ง)(๐‘งโˆˆ๐•Œ).(2.15) By using similar arguments to those used in the proof of Theorem 2.3, we conclude that (2.15) is subordinated to ๐œ™ in ๐•Œ and so ๐‘“โˆ—๐‘žโˆˆ๐’ฎโˆ—(๐œ™).

Theorem 2.9. Let ๐‘Ž2โ‰ฅ๐‘Ž1>0, ๐‘2โ‰ฅ๐‘1>0 and ๐œ™,๐œ“โˆˆ๐’ฉ. If ๐‘Ž2โ‰ฅmin{2,3โˆ’๐‘Ž1} and ๐‘2โ‰ฅmin{2,3โˆ’๐‘1}, then ๐’ž๐‘Ž2,๐‘1(๐œ™,๐œ“)โŠ‚๐’ž๐‘Ž2,๐‘2(๐œ™,๐œ“)โŠ‚๐’ž๐‘Ž1,๐‘2(๐œ™,๐œ“).(2.16)

Proof. First of all, we show that ๐’ž๐‘Ž2,๐‘1(๐œ™,๐œ“)โŠ‚๐’ž๐‘Ž2,๐‘2(๐œ™,๐œ“).(2.17) Let ๐‘“โˆˆ๐’ž๐‘Ž2,๐‘1(๐œ™,๐œ“). Then there exists a function ๐‘ž2โˆˆ๐’ฎโˆ—(๐œ™) such that ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž2,๐‘1๎‚๐‘“๎€ท๎‚๐‘ง)๎…ž๐‘ž2(๐‘ง)โ‰บ๐œ“(๐‘ง)(๐‘งโˆˆ๐•Œ).(2.18) From (2.18), we obtain ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž2,๐‘1๎‚๎‚๐‘“(๐‘ง)๎…ž๎‚€๎‚=๐œ“๐‘ค(๐‘ง)(๐‘งโˆˆ๐•Œ),(2.19) where ๐‘ค is an analytic function in ๐•Œ with |๐‘ค(๐‘ง)|<1(๐‘งโˆˆ๐•Œ) and ๐‘ค(0)=0. By virtue of Lemmas 2.1 and 2.8, we see that โ„Ž(๐‘Ž1,๐‘Ž2)(๐‘ง)โˆ—๐‘ž2(๐‘ง)โ‰ก๐‘ž1(๐‘ง) belongs to ๐’ฎโˆ—(๐œ™). Then we have ๐‘ง๎‚€๐ฟ๎‚€๐‘Ž2,๐‘2๎‚๎‚๐‘“(๐‘ง)๎…ž๐‘ž1=โ„Ž๎‚€๐‘(๐‘ง)1,๐‘2๎‚๎‚€๐ฟ๎‚€๐‘Ž(๐‘ง)โˆ—๐‘ง2,๐‘1๎‚๎‚๐‘“(๐‘ง)๎…žโ„Ž๎‚€๐‘1,๐‘2๎‚(๐‘ง)โˆ—๐‘ž2=โ„Ž๎‚€๐‘(๐‘ง)1,๐‘2๎‚๎‚€๎‚๐‘ž(๐‘ง)โˆ—๐œ“๐‘ค(๐‘ง)2(๐‘ง)โ„Ž๎‚€๐‘1,๐‘2๎‚(๐‘ง)โˆ—๐‘ž2(๐‘ง)โ‰บ๐œ“(๐‘ง)(๐‘งโˆˆ๐•Œ),(2.20) which implies that ๐‘“โˆˆ๐’ž๐‘Ž1,๐‘(๐œ™,๐œ“).
Moreover, the proof of the second part is similar to that of the first part and so we omit the details involved.

3. Inclusion Properties Involving Various Operators

The next theorem shows that the classes ๐’ฎ๐‘Ž,๐‘(๐œ™), ๐’ฆ๐‘Ž,๐‘(๐œ™), and ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“) are invariant under convolution with convex functions.

Theorem 3.1. Let ๐‘Ž>0, ๐‘โˆˆโ„โงตโ„คโˆ’0, ๐œ™,๐œ“โˆˆ๐’ฉ and let ๐‘”โˆˆ๐’ฆ. Then (i)๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™)โ‡’๐‘”โˆ—๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™),(ii)๐‘“โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™)โ‡’๐‘”โˆ—๐‘“โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™),(iii)๐‘“โˆˆ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“)โ‡’๐‘”โˆ—๐‘“โˆˆ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“).

Proof. (i) Let ๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™). Then we have ๐‘ง๎‚€๎‚๐ฟ(๐‘Ž,๐‘)(๐‘”โˆ—๐‘“)(๐‘ง)๎…ž=๎‚€๎‚๐ฟ(๐‘Ž,๐‘)(๐‘”โˆ—๐‘“)(๐‘ง)๐‘”(๐‘ง)โˆ—๐‘ง๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)๎…ž.๐‘”(๐‘ง)โˆ—๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)(3.1) By using the same techniques as in the proof of Theorem 2.3, we obtain (i).
(ii) Let ๐‘“โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™). Then, by (1.9), ๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™) and hence from (i), ๐‘”(๐‘ง)โˆ—๐‘ง๐‘“๎…ž(๐‘ง)โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™). Since ๐‘”(๐‘ง)โˆ—๐‘ง๐‘“๎…ž(๐‘ง)=๐‘ง(๐‘”โˆ—๐‘“)๎…ž(๐‘ง),(3.2) we have (ii) applying (1.9) once again.
(iii) Let ๐‘“โˆˆ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“). Then there exists a function ๐‘žโˆˆ๐’ฎโˆ—(๐œ™) such that ๐‘ง๎‚€๎‚๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)๎…ž๎‚€๎‚=๐œ“๐‘ค(๐‘ง)๐‘ž(๐‘ง)(๐‘งโˆˆ๐•Œ),(3.3) where ๐‘ค is an analytic function in ๐•Œ with |๐‘ค(๐‘ง)|<1(๐‘งโˆˆ๐•Œ) and ๐‘ค(0)=0. From Lemma 2.8, we have that ๐‘”โˆ—๐‘žโˆˆ๐’ฎโˆ—(๐œ™). Since ๐‘ง๎‚€๎‚๐ฟ(๐‘Ž,๐‘)(๐‘”โˆ—๐‘“)(๐‘ง)๎…ž=๎‚€๎‚(๐‘”โˆ—๐‘ž)(๐‘ง)๐‘”(๐‘ง)โˆ—๐‘ง๐ฟ(๐‘Ž,๐‘)๐‘“(๐‘ง)๎…ž=๎‚€๎‚๐‘”(๐‘ง)โˆ—๐‘ž(๐‘ง)๐‘”(๐‘ง)โˆ—๐œ“๐‘ค(๐‘ง)๐‘ž(๐‘ง)๐‘”(๐‘ง)โˆ—๐‘ž(๐‘ง)โ‰บ๐œ“(๐‘ง)(๐‘งโˆˆ๐•Œ),(3.4) we obtain (iii).

Now we consider the following operators [5, 11] defined by ฮจ1(๐‘ง)=โˆžโˆ‘๐‘˜=11+๐‘๐‘ง๐‘˜+๐‘๐‘˜๎‚€๎€ฝ๐‘๎€พ๎‚,ฮจReโ‰ฅ0;๐‘งโˆˆ๐•Œ21(๐‘ง)=๎‚ƒ1โˆ’๐‘ฅlog1โˆ’๐‘ฅ๐‘ง||๐‘ฅ||๎‚.1โˆ’๐‘ง๎‚„๎‚€log1=0;โ‰ค1,๐‘ฅโ‰ 1;๐‘งโˆˆ๐•Œ(3.5) It is well known ([12], see also [5]) that the operators ฮจ1 and ฮจ2 are convex univalent in ๐•Œ. Therefore, we have the following result, which can be obtained from Theorem 3.1 immediately.

Corollary 3.2. Let ๐‘Ž>0, ๐‘โˆˆโ„โงตโ„คโˆ’0, ๐œ™,๐œ“โˆˆ๐’ฉ and let ฮจ๐‘–(๐‘–=1,2) be defined by (3.5). Then (i)๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™)โ‡’ฮจ๐‘–โˆ—๐‘“โˆˆ๐’ฎ๐‘Ž,๐‘(๐œ™),(ii)๐‘“โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™)โ‡’ฮจ๐‘–โˆ—๐‘“โˆˆ๐’ฆ๐‘Ž,๐‘(๐œ™),(iii)๐‘“โˆˆ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“)โ‡’ฮจ๐‘–โˆ—๐‘“โˆˆ๐’ž๐‘Ž,๐‘(๐œ™,๐œ“).

Acknowledgments

The author would like to express his gratitude to the referees for their valuable suggestions. This work was supported by Pukyong National University Research Fund in 2007 (PK-2007-013).

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