The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| ...The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.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 operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/...The generalized Roper-Suffridge extension operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/r)^(γ2)z_2,...,((rf(z_1)/r)/(z_1))^(β_n)(f′((z_1)/r))^(γ_n)z_n) for (z_1,z_2,...,z_n)∈Ω,where r=r(Ω) = sup{|z_1|:(z_1,z_2,...,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1, and we choose the branch of the power functions such that ((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1 and (f′(z_1))^(γ_j)|_(z_1=0)= 1,j=2,...,n.In this paper,we prove that the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) is from the subset of S_α~*(U) to Sα~*(Ω)(0≤α<1) onΩand the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) preserves the starlikeness of order a or the spirallikeness of typeβon D_p for some suitable constantsβ_j,γ_j,p_j,where D_p= {(z_1,z_2,...,z_n)∈C^n:∑_(j=1)~n|z_j|^(p_j)<1}(p_j>0,j=1,2...,n),U is the unit disc in the complex plane C,and S_α~* (Ω) is the class of all normalized starlike mappings of orderαonΩ.We also obtain thatΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)∈S_α~*(D_p) if and only if f∈S_α~*(U) for 0≤α<1 and some suitable constantsβ_j,γ_j,p_j.展开更多
It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism.and its explicit expression is given.
The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only...The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only on zj for 1≤j≤n,where k is the natural number that satisfies k<p≤k+1.When p∞,this gives the result on the unit polydisc obtained by Sulfridge in 1970.展开更多
For a class of Reinhardt domains, we prove that the holomorphic sectional curvatures are upper-bounded by a negative constant. Then we obtain a comparison theorem for the Kobayashi and Bergman metrics on the domains.
Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…...Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.展开更多
In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locall...In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locally biholomorphic function on the unit disc D, r = sup{｜z1｜ ： z =（z1, ···, zn） ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.展开更多
In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounde...In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.展开更多
In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, ...In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.展开更多
In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike map...In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike mappings is obtained.展开更多
In this paper, by the definition of spirallike mapping of type β and order α ,we discuss that the generalized Roper-Suffridge extension operator preserves spirallikeness of type β and order α in complex Banach spa...In this paper, by the definition of spirallike mapping of type β and order α ,we discuss that the generalized Roper-Suffridge extension operator preserves spirallikeness of type β and order α in complex Banach spaces.展开更多
Let pj ∈ N and pj ≥-1, j = 2,...,n be a fixed positive integer. In this paper a generalized Roper-Suffridge extension operator F（z） ={f（Z1）＋f＇（z1）} on Reinhardt domain is defined. Some different conditions f...Let pj ∈ N and pj ≥-1, j = 2,...,n be a fixed positive integer. In this paper a generalized Roper-Suffridge extension operator F（z） ={f（Z1）＋f＇（z1）} on Reinhardt domain is defined. Some different conditions for Pj areestablished under which the operator preserves an almost spirallike mapping of type fl and order a and spirallike mapping of type β and order α, respectively. In particular, our results reduce to many well-known results.展开更多
The Bergman kernel function K(z,), z, w∈Ω for a domain ΩC^n is the kernel of the Bergman projection operator, the operator projecting L^2(Ω) onto its holomorphic subspace. In this note, we consider the Reinhardt
In this paper, we consider a class of bounded Reinhardt domains Dα（m, n1,…,nm）. The Bergman kernel function K（z,z^）, the Bergman metric matrix T（z,z^）, the Cauchy-Szegoe kernel function S（z,ζ^） are obtained...In this paper, we consider a class of bounded Reinhardt domains Dα（m, n1,…,nm）. The Bergman kernel function K（z,z^）, the Bergman metric matrix T（z,z^）, the Cauchy-Szegoe kernel function S（z,ζ^） are obtained. Then we prove that the formal Poisson kernel function is not a Poisson kernel function. At last, we prove that Dα is a quasiconvex domain and Dα is a stronger quasiconvex domain if and only if Dα is a hypersphere.展开更多
The boundary behavior of the Bergman kernel function of a kind of Reinhardt domain is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points ( Z, Z) Let Q be the Reinhardt dom...The boundary behavior of the Bergman kernel function of a kind of Reinhardt domain is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points ( Z, Z) Let Q be the Reinhardt domainwhere is the Standard Euclidean norm in and let K( Z, W) be the Bergman kernel function of Ω. Then there exist two positive constants m and M, and a function F such thatholds for every Z∈Ω . Hereand is the defining function of Ω The constants m and M depend only on Ω = This result extends some previous known results.展开更多
We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and ∇ operators to the set . All relevant ...We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and ∇ operators to the set . All relevant properties relating to the effectiveness in Reinhardt and hyperelliptic domains of these new sets are properly deduced. The case of classical orthogonal polynomials is investigated in details and the results are given in a table. Notations are also provided at the end of a table. 展开更多
基金This work was supported by 973 Project, the National Natural Science Foundation of China (Grant No. 19871081) the Natural Science Foundation of Guangdong Province and Anhui Province.
文摘The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.
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 operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/r)^(γ2)z_2,...,((rf(z_1)/r)/(z_1))^(β_n)(f′((z_1)/r))^(γ_n)z_n) for (z_1,z_2,...,z_n)∈Ω,where r=r(Ω) = sup{|z_1|:(z_1,z_2,...,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1, and we choose the branch of the power functions such that ((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1 and (f′(z_1))^(γ_j)|_(z_1=0)= 1,j=2,...,n.In this paper,we prove that the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) is from the subset of S_α~*(U) to Sα~*(Ω)(0≤α<1) onΩand the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) preserves the starlikeness of order a or the spirallikeness of typeβon D_p for some suitable constantsβ_j,γ_j,p_j,where D_p= {(z_1,z_2,...,z_n)∈C^n:∑_(j=1)~n|z_j|^(p_j)<1}(p_j>0,j=1,2...,n),U is the unit disc in the complex plane C,and S_α~* (Ω) is the class of all normalized starlike mappings of orderαonΩ.We also obtain thatΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)∈S_α~*(D_p) if and only if f∈S_α~*(U) for 0≤α<1 and some suitable constantsβ_j,γ_j,p_j.
基金Project supported by the National Natural Science Foundation of China(No.19631010)
文摘It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism.and its explicit expression is given.
基金Project supported in part by the National Natural Science Foundation of China.
文摘The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only on zj for 1≤j≤n,where k is the natural number that satisfies k<p≤k+1.When p∞,this gives the result on the unit polydisc obtained by Sulfridge in 1970.
基金Project supported partly by the National Natural Science Foundation of China
文摘For a class of Reinhardt domains, we prove that the holomorphic sectional curvatures are upper-bounded by a negative constant. Then we obtain a comparison theorem for the Kobayashi and Bergman metrics on the domains.
基金Supported by the National Natural Science Foundation of China(11001074,11061015,11101124)
文摘Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.
文摘In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locally biholomorphic function on the unit disc D, r = sup{｜z1｜ ： z =（z1, ···, zn） ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471111, 11571105 and 11671362)the Natural Science Foundation of Zhejiang Province (Grant No. LY16A010004)
文摘In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.
基金the Natural Science Foundation of China (Grant No.10671194 and 10731080/A01010501)
文摘In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.
文摘In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike mappings is obtained.
文摘In this paper, by the definition of spirallike mapping of type β and order α ,we discuss that the generalized Roper-Suffridge extension operator preserves spirallikeness of type β and order α in complex Banach spaces.
文摘Let pj ∈ N and pj ≥-1, j = 2,...,n be a fixed positive integer. In this paper a generalized Roper-Suffridge extension operator F（z） ={f（Z1）＋f＇（z1）} on Reinhardt domain is defined. Some different conditions for Pj areestablished under which the operator preserves an almost spirallike mapping of type fl and order a and spirallike mapping of type β and order α, respectively. In particular, our results reduce to many well-known results.
文摘The Bergman kernel function K(z,), z, w∈Ω for a domain ΩC^n is the kernel of the Bergman projection operator, the operator projecting L^2(Ω) onto its holomorphic subspace. In this note, we consider the Reinhardt
基金Supported by the NSF of Henan University(04YBRW043)
文摘In this paper, we consider a class of bounded Reinhardt domains Dα（m, n1,…,nm）. The Bergman kernel function K（z,z^）, the Bergman metric matrix T（z,z^）, the Cauchy-Szegoe kernel function S（z,ζ^） are obtained. Then we prove that the formal Poisson kernel function is not a Poisson kernel function. At last, we prove that Dα is a quasiconvex domain and Dα is a stronger quasiconvex domain if and only if Dα is a hypersphere.
文摘The boundary behavior of the Bergman kernel function of a kind of Reinhardt domain is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points ( Z, Z) Let Q be the Reinhardt domainwhere is the Standard Euclidean norm in and let K( Z, W) be the Bergman kernel function of Ω. Then there exist two positive constants m and M, and a function F such thatholds for every Z∈Ω . Hereand is the defining function of Ω The constants m and M depend only on Ω = This result extends some previous known results.
基金The first author's work was supported in part by the National Natural Science Foundation of China (Grant No.10571135)the Doctoral Program Foundation of the Ministry of Education of China (Grant No. 20050240711)
文摘It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in C^2 is an automorphism.
文摘We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and ∇ operators to the set . All relevant properties relating to the effectiveness in Reinhardt and hyperelliptic domains of these new sets are properly deduced. The case of classical orthogonal polynomials is investigated in details and the results are given in a table. Notations are also provided at the end of a table.
