占龍俊,黃心中
(華僑大學(xué)數(shù)學(xué)科學(xué)學(xué)院,福建泉州362021)
調(diào)和映照與像域?yàn)榫€性連結(jié)的剪切函數(shù)的關(guān)系
占龍俊,黃心中
(華僑大學(xué)數(shù)學(xué)科學(xué)學(xué)院,福建泉州362021)
調(diào)和映照;線性連結(jié);調(diào)和擬共形映照;α近于凸.
Huang[3]對(duì)Chuaqui等[2]所得的結(jié)論進(jìn)一步推廣,得到不少有趣的成果.Huang[3]還特別指出,在一定條件下,f(D)的線性連結(jié)性與f(z)的擬共形性,及h(D)線性連結(jié)性與h(z)的單葉性是一對(duì)不變量.文獻(xiàn)[6-8]分別對(duì)單葉調(diào)和映照的性質(zhì)及穩(wěn)定性問(wèn)題進(jìn)行研究.
Chen等[10]證明了定理C[2].
證明 假設(shè)F(z)=h(z)-g(z),若h(z)在D上不單葉,則存在z1,z2∈D,且z1≠z2,使得h(z1)=h(z2).由h(z)=F(z)+g(z),則F(z2)-F(z1)=g(z1)-g(z2).根據(jù)F(z)的單葉性,令w=F(z)有w2-w1=g(F-1(w1))-g(F-1(w2)).由于F(D)是M-線性連結(jié)區(qū)域.則存在連結(jié)w1-w2∈F(D)的可求長(zhǎng)曲線γ∈F(D)滿足l(γ)≤M|w1-w2|,即
這與w2-w1=g(F-1(w1))-g(F-1(w2))矛盾,從而說(shuō)明h(z)在D上單葉.
令ξ=h(F-1(w))=w+g(F-1(w)),任意ξ1,ξ2∈h(D),存在w1,w2∈F(D),使ξ1=h(F-1(w1)),ξ2=h(F-1(w2)).由于F(D)是M-線性的連結(jié)區(qū)域,則存在γ為連結(jié)w1,w2的曲線,使l(γ)≤M|w1-w2|.取Γ=h(F-1(γ)),有
又
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相應(yīng)于定理C的結(jié)果,進(jìn)一步研究其參數(shù)化的情況,得到定理2.
證明 因F(z)為D上α近于凸映照,由文獻(xiàn)[6]得F(z)在D上的單葉,且F(D)是1/cos(απ/2)-線性連結(jié)區(qū)域,則由定理C可得h(z)在D上的單葉.
令ξ=h(F-1(w))=w+g(F-1(w)),任意ξ1,ξ2∈h(D),存在w1,w2∈F(D),使ξ1=h(F-1(w1)),ξ2=h(F-1(w2)).由于F(D)是1/cos(απ/2)-線性連結(jié)區(qū)域,則存在γ為連結(jié)w1,w2的曲線,使l(γ)≤1/cos(απ/2|w1-w2|).取Γ=h(F-1(γ)),有
又
又
注1 注意到當(dāng)|λ|<1時(shí),這時(shí)t可以取到小于2的常數(shù).
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對(duì)于fλ(z)=h(z)+λg(z)時(shí),采用相同的證明方法,可得到定理3中的結(jié)論.定理3證畢.
在定理1,2,3中,若F(z)=h(z)+g(z)有相應(yīng)的假定,結(jié)論仍是成立的.
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Relation Between Harmonic Mapping and Its Shear Function With Linearly Connected Image Domain
ZHAN Long-jun,HUANG Xin-zhong
(School of Mathematical Sciences,Huaqiao University,Quanzhou 362021,China)
harmonic mapping;linearly connected domain;harmonic quasiconformal mapping;α-close-to convex
陳志賢 英文審校:黃心中)
O174.51;O174.55文獻(xiàn)標(biāo)志碼: A
1000-5013(2015)05-0603-06 doi:10.11830/ISSN.1000-5013.2015.05.0603
2015-01-05
黃心中(1957-),男,教授,博士,主要從事函數(shù)論的研究.E-mail:huangxz@hqu.edu.cn.
國(guó)家自然科學(xué)基金資助項(xiàng)目(11471128);福建省自然科學(xué)基金資助項(xiàng)目(2014J01013)