Loewner chains associated with the generalized Roper-Suffridge extension operator

In this paper, we introduce two classes of generalized Roper-Suffridge extension operators and prove that they can be embedded in Loewner chains. In particular, our proof shows that these two classes of operators preserve starlikeness and spirallikeness of type α on two important classes of Reinhardt domains in Cn, respectively. Finally, some other related results are given.     

On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings

In this paper, we consider a normalized biholomorphic mapping f(x) defined on the unit ball in a complex Banach space, where the origin 0 is a zero of order k+1 of f(x)−x. The precise growth and covering theorem for f(x) is obtained when f(x) is a starlike mapping or a starlike mapping of order α. Especially, the precise growth and covering theorem for f(x) is also established when f(x) is a quasi-convex mapping. Moreover, the precise distortion theorem for f(x) is given when f(x) is a convex mapping. Our result includes many known results.     

The generalized Roper-Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p₁?2 (or 0<p₁?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that

     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Some characterizations of the Besov space and the α-Bloch space

In this paper we obtain some characterizations of Besov spaces and α-Bloch spaces on the unit ball of Cn, which extend some results in the settings of the unit disk.     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces.Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C~n. Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.     

Convex and alpha-prestarlike subordination chains

In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over (0,1] on the unit disc in the complex plane. We obtain sufficient conditions for a mapping f(z,t) to be an α-p.s.c. over (0,1], which generalize a well-known result of Ruscheweyh [St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montreal, 1982]. We also obtain sufficient conditions for injectivity for nonanalytic mappings on the unit disc, and give certain examples of α-c.s.c. over (0,1] on the unit disc.     

Characterizations of α-Bloch spaces on the unit ball

We obtain a characterization of the α-Bloch space for any α>0 in the unit ball of Cn in terms of |f(z)−f(w)|/|z−w|. Moreover, a new characterization for the Bloch space is given.     

Norm estimates for the Alexander transforms of convex functions of order alpha

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0?α<1.     

一类α次殆星形映照的偏差定理

近年来,双全纯凸(准凸)映照的偏差定理的研究已经获得一些可喜成果,但目前星形映照的偏差定理研究成果还较少.利用α次殆星形映射的增长估计,我们得到了C~n中开单位球B~n上一类α次殆星形映射的偏差估计.对复Banach空间单位球B和Reinhard域Ω_p1,…,p_n,可得到同样的结论,正如猜想的一样.     

On some classes of bounded univalent mappings in several complex variables

Let B ⁿ be the Euclidean unit ball in C ⁿ . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B ⁿ . We prove that K-quasiregular strongly starlike mappings of order α on B ⁿ have continuous and univalent extensions to ${\overline{B}^n}$ . We show that bounded convex mappings on B ⁿ are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B ⁿ . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B ⁿ .     

On the Fekete and Szegö problem for starlike mappings of order <Emphasis Type="Italic">α</Emphasis>

Let S _α ^* be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class S _α ^* , and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C ⁿ .     

The practical research on flood risk analysis based on IIOSM and fuzzy α-cut technique

Flood disasters are one of the most common and destructive natural hazards all over the world. In this paper, improved interior-outer-set model (IIOSM) based on information diffusion theory is introduced in detail to assess flood risk in an effort to obtain accurate analytical results that represent the actual situation. Then fuzzy α-cut technique is applied to calculate the fuzzy expected values under the possibility–probability distribution (PPD) calculated by IIOSM. Taking the value of α throughout the interval (0, 1], we correspondingly get access to the conservative risk value (R_C) and venture risk value (R_V). Selection of α, R_C and R_V is dependent on present technical conditions and risk preference of different people. To illustrate the procedure of IIOSM and fuzzy α-cut technique, we employ them respectively to analyze the flood risk in Sanshui District, located in the center of Guangdong province in China. The results, such as risk value estimations, as well as fuzzy expected values, i.e. R_C and R_V under the given α-cut level, can reflect the flood risk quite accurately. The outcomes of this research based on IIOSM and fuzzy α-cut technique offer new insights to carry out an efficient way for various flood protection strategies.     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

The stability of the entropy of degree alpha

In this paper, we first prove that the generalized fundamental equation of information depending on a positive real parameter α, is stable in the sense of Hyers and Ulam provided that α≠1, then we apply this result to prove the stability of a system of functional equations that characterizes the entropy of degree alpha or Havrda-Charvát entropy which has recently often been called the Tsallis entropy.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

Convolutions of meromorphic multivalent functions with respect to n-ply points and symmetric conjugate points

It is well-known that the classes of starlike, convex and close-to-convex univalent functions are closed under convolution with convex functions. In this paper, closure properties under convolution of general classes of meromorphic p-valent functions that are either starlike, convex or close-to-convex with respect to n-ply symmetric, conjugate and symmetric conjugate points are investigated.     

A class of semigroups regularized in space and time

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.     

Global integral criteria for composition operators

Let D_α denote the Dirichlet-type space of functions analytic on the unit disk U and Q_α the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator C_φ acting on D_α, respectively, Q_α by C_φf=f°φ, where f∈D_α, respectively, f∈Q_α. The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.