由微分從屬和卷積定義的解析函數(shù)類的包含性質(zhì)

都俊杰，秦　川，鄒發(fā)偉，李小飛

(1.長江大學(xué)工程技術(shù)學(xué)院，中國 荊州　434020; 2.長江大學(xué)信息與數(shù)學(xué)學(xué)院，中國 荊州　434000;3.澳門大學(xué)科技學(xué)院，中國 澳門　519040)

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由微分從屬和卷積定義的解析函數(shù)類的包含性質(zhì)

都俊杰1*，秦川1，鄒發(fā)偉1，李小飛2,3

(1.長江大學(xué)工程技術(shù)學(xué)院，中國 荊州434020; 2.長江大學(xué)信息與數(shù)學(xué)學(xué)院，中國 荊州434000;3.澳門大學(xué)科技學(xué)院，中國 澳門519040)

摘要本文由微分從屬和卷積定義了在單位圓盤U={z∈C:|z|<1}內(nèi)的三類單葉解析函數(shù)類Pa1,…,aq;b1,…,bs(μ,h,λ),Ta1,…,aq;b1,…,bs(μ,h,α),Ra1,…,aq;b1,…,bs(μ,h,α),并利用從屬性質(zhì)和凸函數(shù)的理論，研究得到了它們的包含關(guān)系.

關(guān)鍵詞從屬；卷積；包含性質(zhì)；星象函數(shù)；凸函數(shù)

記N表示由單位圓盤U內(nèi)的單葉解析凸的函數(shù)h(z)組成的正實(shí)部函數(shù)類，即滿足Re{h(z)}>0.Ozkan和Altintas[1]定義了下面的函數(shù)類:

Ra(h,α):={f:(ka*f)′(z)+αz(ka*f)″(z)

Ra,c(h,α):={f:(L(a,c)f)′(z)+αz(L(a,c)f)″(z)

Ra1,…,aq;b1,…,bs(h,α):={f:(H1(a1,…,aq;b1,…,bs)f)′(z)+αz(H1(a1,…,aq;b1,…,bs)f)″(z)

z(hμ(a1,…,aq;b1,…,bs;z)*f(z))′=hμ(a1,…,aq;b1,…,bs;z)*zf′(z),

(1)

hμ(a1,…,aq;b1,…,bs;z)*z2f″(z)=z2(Hμq,s(a1,…,aq;b1,…,bs)f)″(z).

(2)

h(z),f∈A,z∈U},0≤λ≤1;h∈N;

(3)

h(z),f∈A,z∈U},α≥0;h∈N;

(4)

h(z),f∈A,z∈U},α≥0;h∈N.

(5)

本文利用從屬性質(zhì)與凸函數(shù)的理論，研究得到上述函數(shù)類的包含性質(zhì).

1基本引理

引理2[12]設(shè)0<α≤β,若β≥2或α+β≥3,則

引理3[13]若f(z)∈K,g(z)∈S*,則對于U內(nèi)任意解析函數(shù)h(z),都有

2主要結(jié)論

定理1設(shè)g(z)=λzf′(z)+(1-λ)f(z),則f(z)∈Pa1,…,aq;b1,…,bs(μ,h,λ)當(dāng)且僅當(dāng)g(z)∈Pa1,…,aq;b1,…,bs(μ,h,0).

證利用(1)、(2)式，得

因此，g(z)∈Pa1,…,aq;b1,…,bs(μ,h,0).反過來，若g(z)∈Pa1,…,aq;b1,…,bs(μ,h,0),按同樣的方法容易得到f(z)∈Pa1,…,aq;b1,…,bs(μ,h,λ).

證設(shè)f(z)∈Pa1,…,aq;b1,…,bs(μ,h,λ),zφ′(z)=λzf′(z)+(1-λ)f(z)=g(z),由于g(z)∈Pa1,…,aq;b1,…,bs(μ,h,0),故zφ′(z)∈Pa1,…,aq;b1,…,bs(μ,h,0).經(jīng)計(jì)算，得

因此，φ(z)=Pa1,…,aq;b1,…,bs(μ,h,1).

定理3若f(z)∈Ta1,…,aq;b1,…,bs(μ,h,α),則f(z)∈Ta1,…,aq;b1,…,bs(μ,h,0).

定理4若α>β≥0,f(z)∈Ta1,…,aq;b1,…,bs(μ,h,α),則f(z)∈Ta1,…,aq;b1,…,bs(μ,h,β).

證當(dāng)β=0時，由定理3容易得到結(jié)論.當(dāng)β>0時，設(shè)f(z)∈Ta1,…,aq;b1,…,bs(μ,h,α),由函數(shù)類的定義和從屬性質(zhì)知

故f(z)∈Ta1,…,aq;b1,…,bs(μ,h,β).

定理5若f(z)∈Ra1,…,aq;b1,…,bs(μ,h,α),則f(z)∈Ra1,…,aq;b1,…,bs(μ,h,0).

定理6若α>β≥0,f(z)∈Ra1,…,aq;b1,…,bs(μ,h,α),則f(z)∈Ra1,…,aq;b1,…,bs(μ,h,β).

證當(dāng)β=0時，由定理5容易得到結(jié)論.當(dāng)β>0時，設(shè)f(z)∈Ra1,…,aq;b1,…,bs(μ,h,α),由函數(shù)類的定義和從屬性質(zhì)知

故f(z)∈Ra1,…,aq;b1,…,bs(μ,h,β).

定理7f(z)∈Ra1,…,aq;b1,…,bs(μ,h,α)?zf′(z)∈Ta1,…,aq;b1,…,bs(μ,h,α),

(6)

Ra1,…,aq;b1,…,bs(μ,h,α)?Ta1,…,aq;b1,…,bs(μ,h,α).

(7)

證由函數(shù)類的定義和式(5)、(6)并經(jīng)計(jì)算得

上式意味著式(6)成立.設(shè)f(z)∈Ra1,…,aq;b1,…,bs(μ,h,α),并令

證設(shè)f(z)∈Pa1,…,aq;b1,…,bs(μ,h,λ),則存在Schwarz函數(shù)w(z)使得

h(w(z)).

即f(z)∈Pa1,…,aq;b1,…,bs(μ,h,λ).

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(編輯HWJ)

Inclusion Properties for Subclasses of Analytic Functions Defined by Differential Subordination and Convolution

DUJun-jie1*,QINChuan1,ZOUFa-wei1,LIXiao-fei2,3

(1.College of Engineering and Technology, Yangtze University, Jingzhou 434020, China;2.School of Information and Mathematics, Yangtze University, Jingzhou 434020, China;3.College of Science and Technology, University of Macau, Macau, 519040, China)

AbstractIn this article, we define three subclasses of analytic functions Pa1,…,aq;b1,…,bs(μ,h,λ),Ta1,…,aq;b1,…,bs(μ,h,α),Ra1,…,aq;b1,…,bs(μ,h,α) by using of differential subordination and convolution in the open disc U={z∈C:|z|<1}. Inclusion properties of these subclasses are obtained by employing properties of subordination and theories of convex functions.

Key wordssubordination; convolution; inclusion properties; starlike function; convex function

中圖分類號O174.51

文獻(xiàn)標(biāo)識碼A

文章編號1000-2537(2016)02-0077-05

*通訊作者,E-mail：29149875@qq.com

基金項(xiàng)目：湖北省自然科學(xué)基金資助項(xiàng)目(2013CFAO053);湖北省教育廳科研基金資助項(xiàng)目(B2013281)；長江大學(xué)科研基金資助項(xiàng)目(2013cjy01);長江大學(xué)工程技術(shù)學(xué)院科技創(chuàng)新基金資助項(xiàng)目(2015J0802)

收稿日期：2015-11-12

DOI:10.7612/j.issn.1000-2537.2016.02.013