In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not pre...In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not preserve convexity for some cases. Meanwhile, the growth theorem and di stortion theorem of the corresponding mappings are given.展开更多
In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Bana...In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.展开更多
In this note, the author introduces some new subcIasses of starlike mappings S^*Ωn1p2,…,pn(β,A,B)={f∈H(Ω):|itanβ+(1-itanβ)2/p(z)аp/аz(z)Jf^-1(z)f(z)-1-AB/1-B^2|〈B-A/1-B^2},on Reinhardt dom...In this note, the author introduces some new subcIasses of starlike mappings S^*Ωn1p2,…,pn(β,A,B)={f∈H(Ω):|itanβ+(1-itanβ)2/p(z)аp/аz(z)Jf^-1(z)f(z)-1-AB/1-B^2|〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n:|z1|^2+n∑j=2|zj|^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈(-π/2,π/2).Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F(z)=(f(z1)+f'(z1)Pm(z0),(f'(z1))^1/mz0)'where f is a normalized biholomorphic function on the unit disc D, z = (z1,z0) ∈Ωn1p2,…,pn,z0=(z2,…,zn)∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.展开更多
The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z...The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1|: (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α < 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn: ∑nj=1|zj|pj < 1} (pj > 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α < 1 and some suitable constants βj,γj,pj.展开更多
Suppose that X is a complex Banach space with the norm ||·|| and n is a positive integer with dim X ≥n≥ 2. In this paper,we consider the generalized Roper-Suffridge extension operator Φn,β2,γ2,...,βn+1,γn+...Suppose that X is a complex Banach space with the norm ||·|| and n is a positive integer with dim X ≥n≥ 2. In this paper,we consider the generalized Roper-Suffridge extension operator Φn,β2,γ2,...,βn+1,γn+1(f) on the domain Ωp1,p2,...,pn+1 defined展开更多
文摘In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not preserve convexity for some cases. Meanwhile, the growth theorem and di stortion theorem of the corresponding mappings are given.
基金This research is partly supported by the National Natural Science Foundation of China (10471048) the Doctoral Foundation of the Education Committee of China(20050574002)+1 种基金 the Natural Science Foundation of Fujian Province, China (Z0511013)the Education Commission Foundation of Fujian Province, China (JB04038)
文摘In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.
基金supported by the National Natural Science Foundation of China(11001246,11101139)Zhejiang Innovation Project(T200905)
文摘In this note, the author introduces some new subcIasses of starlike mappings S^*Ωn1p2,…,pn(β,A,B)={f∈H(Ω):|itanβ+(1-itanβ)2/p(z)аp/аz(z)Jf^-1(z)f(z)-1-AB/1-B^2|〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n:|z1|^2+n∑j=2|zj|^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈(-π/2,π/2).Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F(z)=(f(z1)+f'(z1)Pm(z0),(f'(z1))^1/mz0)'where f is a normalized biholomorphic function on the unit disc D, z = (z1,z0) ∈Ωn1p2,…,pn,z0=(z2,…,zn)∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No.10471048)the Research Fund for the Doctoral Program of Higher Education (Grant No.20050574002)+1 种基金the Natural Science Foundation of Fujian Province of China (Grant No.Z0511013)the Education Commission Foundation of Fujian Province of China (Grant No.JB04038)
文摘The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1|: (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α < 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn: ∑nj=1|zj|pj < 1} (pj > 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α < 1 and some suitable constants βj,γj,pj.
基金supported by the Doctoral Foundation of Ministry of Education of China (Grant No. 20050574002)the Natural Science Foundation of Fujian Province of China (Grant No. 2009J01007)
文摘Suppose that X is a complex Banach space with the norm ||·|| and n is a positive integer with dim X ≥n≥ 2. In this paper,we consider the generalized Roper-Suffridge extension operator Φn,β2,γ2,...,βn+1,γn+1(f) on the domain Ωp1,p2,...,pn+1 defined