Carlon-Schaffer算子在亚纯单叶函数上的应用

Some Applications of Carlon-Schaffer Operator in Meromorphic Univalent Functions

设∑表示形如f(z)=z-1+∑n=0 ∞anzn且在空心单位圆U0内解析的全体函数组成的类,Carlon-Schaffer算子为L(a,c)f(z)=z-1+∑n=0 ∞(a)n+1anzn/(c)n++1(n+1)!。利用算子L(a,c)定义了亚纯单叶函数的新子类:S*a,c(γ)={f∈∑:L(a,c)f(z)∈S*(γ)},Ca,c(γ)={f∈∑:L(a,c)f(z)∈C(γ)},Ka,c(β,γ)={f∈∑:L(a,c)f(z)∈K(β,γ)},Ka*,c(β,γ)={f∈∑:L(a,c)f(z)∈K*(β,γ)},并利用Miller引理建立了包含关系:在a+1-γ>0时,Sa*+1,c(γ)Sa*,c(γ),Ca+1,c(γ)Ca,c(γ),Ka+1,c(β,γ)Ka,c(β,γ),Ka*+1,c(β,γ)Ka*,c(β,γ);而c-γ>0时,Sa*,c-1(γ)Sa*,c(γ),Ca,c-1(γ)Ca,c(γ),Ka,c-1(β,γ)Ka,c(β,γ),Ka*,c-1(β,γ)Ka*,c(β,γ)。

Let ∑ be the class of fuhctions with form of f（z）=z^-1＋∑^∞ n=0 anz^n ,analytic in the open unit punch disk Uo. Carlon-Schaffer operator is defined as L（a,c）f（z）=z^-1＋∑^∞ n=0 （a）n＋1/（c）n＋1 anz^n/（n＋1）! the paper some new subclasses of meromorphic univalent functions were defined by Carlon-Schaffer Operator： S^＊ a,c（γ）={f∈∑：L（a,c）f（z）∈S＊（γ）},Ca,c（γ）={f∈∑：L（a,c）f（z）∈C（γ）},Ka,c（β,γ）={f∈∑：L（a,c）f（z）∈K（β,γ）},K^＊ a,c（β,γ）={f∈∑：L（a,c）f（z）∈K＊（β,γ）}. Inclusion relations were established with Miller Lemma ： if a ＋ 1 -γ〉 0 , then S^＊ a＋1,c（γ）S^＊ a,c（γ）,Ca＋1,c（γ）Ca,c（γ）,Ka＋1,c（β,γ）Ka,c（β,γ）,K^＊ a＋1,c（β,γ）K^＊ a,c（β,γ）;if c-γ〉0,then S^＊ a,c-1（γ）S^＊ a,c（γ）,Ca,c-1（γ）Ca,c（γ）,Ka,c-1（β,γ）Ka,c（β,γ）,K^＊ a,c-1（β,γ）K^＊ a,c（β,γ）.