On Sufficient Conditions for p-valently Starlikeness

Let An(p)(p,n ∈ N = {1,2,3,...}) denote the class of functions of the form f(z) = zp +ap+nzp+n +...which are analytic in the unit disk E = {z:|z| < 1}.A function f(z) in A(p) is said to be in the class Sp(a,b) if it satisfies zf(z) pf(z) 1 + az 1 + bz for some a and b(-1 ≤ b < a ≤ 1).In this paper,using the method of differential subordinations,we give new criteria for f(z) to be in the classes Sp(a,b)(-1 ≤ b < a ≤ 1),that is,if f(z) ∈ A(p) satisfies f(z) = 0 in 0 < |z| < 1,f(z) = 0 when μ = 1,and zf(z) f(z) μ 1 + zff((zz)) + λzff(z(z))h(z),where h(z) = p1+μ(1 + λ) 11 ++ abzz 1+μ + pμ(1 + az)1-μ(1+bz)1+μ,λ≥-1,μ∈ {-1,0,1},then f(z) ∈ Sp(a,b).Our results improve or extend some results due to Owa,Nunokawa,Padmanabhan,Silverman,Obradovic,Yang and others[3-10].

STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION

Let q_(λ)(z)=1+λsinh(ζ),0<λ<1/sinh(1)be a non-vanishing analytic function in the open unit disk.We introduce a subclass S^(*)(q_(λ))of starlike functions which contains the functions f such that zf'/f is subordinated by q_(λ).We establish inclusion and radii results for the class S^(*)(q_(λ))for several known classes of starlike functions.Furthermore,we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class S^(*)(q_(λ)).We also find a sharp bound for the third Hankel determinant for the caseλ=1/2.

CERTAIN SUFFICIENT CONDITIONS FOR STARLIKENESS AND CONVEXITY OF MEROMORPHICALLY MULTIVALENT FUNCTIONS

In this paper we derive certain sufficient conditions for starlikeness and convexity of order α of meromorphically multivalent functions in the punctured unit disk.

RADIUS OF STARLIKENESS FOR THE CLASS S(α,n)

In this paper we shall investigate the bound of starlikeness r for the classS(α, n) in [1]. We obtain the following sharp result:where α_0 is the unique solution of the equationα[1+(1-2α)r^(2n)]/1-r^(2n)={1-(1-2α)r^n/1+r^n}~2with respect to α in the open interval (0, 1). This result is also the extension of somewell-known ones in [2], [3, Theorem 1] and [4, Theorem 2]

ON CRITERIA FOR STARLIKENESS AND CONVEXITY OF ANALYTIC FUNCTIONS

In this paper the authors study (1-γ+zf(?)(z)/f'(z))/(zf'(z)/f(z)) as a criteria for starlikeness and convexity. Sharp upper bound of |α2| and of the Fekete-Szego functional |α3-μα22| is given for a class of analytic functions defined by using this expression.

STARLIKENESS AND SUBORDINATION PROPERTIES OF A LINEAR OPERATOR

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator Lαk . For functions in this class Hαk （p,λ;h） we estimate the coefficients. Furthermore, some subordination properties related to the operator Lαk are also derived.

On Sufficient Conditions for p-valently Close-to-convexity

Let A p(n)(p, n∈N=｛1,2,…｝) denote the class of functions of the form f(z)=z p+a p+n z p+n +… which are analytic in the unit disc E=｛z:｜z｜<1｝. By using the method of differential subordinati ons we give some sufficient conditions for a function f(z)∈A p(n) to be a certain subclass R p(n,k) of p-valently close-to-convexity funct ions.

Radius of Starlikeness for Class S(a,n)and Its Extension

This paper obtain that the radius of starlikeness for class S(α,n)in [1] is,tespectivety, where α_ is unique solution of equation (αα)^(1/2)=σwith a in (0.1),and α-[1+(1-2α)r^(2n)]/(1-r^(2n)),σ =[1-(1-2α)r~]/(1+r~).Futhermore,we consider an extension of class S(α,n):Let S(α、β、n) denote the class of functions f(z)=z+α_z^(n+1)+…(n≥1)that are analytie in |z|<1 such that f(z)/g (z)∈p(α,n)[1],where g(z)∈S~*(β)[2].This paper prove that the radius of starlikeness of class S(α, β,n) is given by the smallest positive root(less than 1)of the following equations (1-2α)(1-2β)r^(2)-2[1-α-β-n(1-α)]r^+1=0.0≤α≤α_0, (1-α)[1-(1-2β)r~]-n[r^(1+r^)=0.,α_0≤α<1. where α=[1+(1-2α)r^(2)]/(1-r^(2)(0≤r<1),α_0(?(0,1) is some fixed number.This result is also the cxtension of well-known results[T.Th3] and [8,Th3]

The Growth Theorem for Almost Starlike Mappings of Order α on Bounded Starlike Circular Domains

In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it. This type of domain on which we discuss is rather general,in the sense that the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it.

有关共轭点和对称共轭点的星像函数逆函数的Hermitian Toeplitz行列式

S_(c)^(*)和S_(sc)^(*)分别称为有关共轭点的星像函数类、有关对称共轭点的星像函数类,得到了其逆函数的Hermitian Toeplitz行列式的估计.

The Growth Theorem for Strongly Starlike Mappings of Order α on Bounded Starlike Circular Domains

The authors obtain the growth and covering theorems for strongly starlike mappings of order α on bounded starlike circular domains.This kind of domain discussed is rather general,since the domain must be starlike if exists a normalized biholomorphic starlike mapping on it.

On a Class of p-valent Analytic Functions

In this paper we consider the extreme points of closed convex hull of the class T σ(p,α) and then it is used to determine the coefficient bounds. Some other interesting properties of the class T σ(p,α) are also investigated.

ON THE COEFFICIENTS OF CERTAIN CLASSES OF p-VALENT FUNCTIONS

In this paper ,we point out that Theorem 3 of Aouf [1] and Theorem 1 of Aouf [2] are in general false for n≥p +2 and correct these

On the Second-order Item Coefficients for ε-starlike Mappings in C^n

Let B n be the unit ball in C n, we study ε-starlike mappings on B n. The upper bounds of second order item coefficients of homogeneous expansion for ε-starlike mappings are obtained.

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN C^n

In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

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.

DISTORTION THEOREMS FOR SUBCLASSES OF STARLIKE MAPPINGS ALONG A UNIT DIRECTION IN C^n

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.

UNIFORMLY STARLIKE MAPPINGS AND UNIFORMLY CONVEX MAPPINGS ON THE UNIT BALL B^n

In this article, we extend the definition of uniformly starlike functions and uni- formly convex functions on the unit disk to the unit ball in C^n, give the discriminant criterions for them, and get some inequalities for them.

Convexity for New Integral Operator on k-uniformly p-valent a-convex Functions of Complex Order

In this paper, we obtain the convexity of new general integral operator on some classes of k-uniformly p-valent a-convex functions of complex order. These results extend some known theorems.

Third Hankel Determinant for the Inverse of Starlike and Convex Functions

Denote S to be the class of functions which are analytic,normalized and univalent in the open unit disk U={z:|z|<1}.The important subclasses of S are the class of starlike and convex functions,which we denote by S and C.In this paper,we obtain the third Hankel determinant for the inverse of functions f(z)=z+∞Σn=2 anz^n belonging to S^*and C.