Univalence of Integral Operators on Neighborhoods of Analytic Functions

摘要

Assume that $$Delta $$ Δ is the unit disk in the complex plane and $$mathcal {A}$$ A is the class of analytic functions f in $$Delta $$ Δ with normalization conditions $$f(0)=f^{prime }(0)-1=0$$ f ( 0 ) = f ′ ( 0 ) - 1 = 0 . For $$lambda _{i},mu _{i}in mathbb {C}$$ λ i , μ i ∈ C and $$f_{i}in mathcal {A}$$ f i ∈ A $$(1le ile n)$$ ( 1 ≤ i ≤ n ) , consider the integral operator: $$begin{aligned} F(z):=F_{lambda ,mu }[(f_{1},ldots ,f_{n})](z)=int _{0}^{z}prod _{i=1}^{n}(f_{i}^{prime }(t))^{lambda _{i}} left( dfrac{f_{i}(t)}{t}right) ^{mu _{i}}mathrm{d}t qquad (zin Delta ), end{aligned}$$ F ( z ) : = F λ , μ [ ( f 1 , … , f n ) ] ( z ) = ∫ 0 z ∏ i = 1 n ( f i ′ ( t ) ) λ i f i ( t ) t μ i d t ( z ∈ Δ ) , where $$lambda =(lambda _{1},ldots ,lambda _{n})$$ λ = ( λ 1 , … , λ n ) and $$mu =(mu _{1},ldots ,mu _{n})$$ μ = ( μ 1 , … , μ n ) . For $$delta >0$$ δ > 0 , define $$begin{aligned} V_{delta }(f):=left{ gin mathcal {A}^{n}:||f^{prime }-g^{prime }||_{infty }le delta right} , end{aligned}$$ V δ ( f ) : = g ∈ A n : | | f ′ - g ′ | | ∞ ≤ δ , where $$||f^{prime }-g^{prime }||_{infty }:=max _{zin Delta ,~1le ile n}|f_{i}^{prime }(z)-g_{i}^{prime }(z)|$$ | | f ′ - g ′ | | ∞ : = max z ∈ Δ , 1 ≤ i ≤ n | f i ′ ( z ) - g i ′ ( z ) | , a neighborhood of f , $$f=(f_{1},ldots ,f_{n})$$ f = ( f 1 , … , f n ) , $$g=(g_{1},ldots ,g_{n})$$ g = ( g 1 , … , g n ) , $$fin mathcal {A}^{n}$$ f ∈ A n , $$mathcal {A}^{n}=mathcal {A}times cdots times mathcal {A}$$ A n = A × ⋯ × A and $$times $$ × is the Cartesian product. In this paper, we determine the radii of $$V_{delta }(f)$$ V δ ( f ) , such that the integral operator F ( z ) carries the neighborhood into the class $$mathcal {S}$$ S (class of univalent functions), where $$f_{i}~(1le ile n)$$ f i ( 1 ≤ i ≤ n ) belongs to the universal linear invariant families or satisfies certain conditions.