Application of fractional calculus operators for a new class of univalent functions with negative coefficients defined by Hohlov operator

In this paper, we introduce a new class W(a, b, c, γ, β) which consists of analytic and univalent functions with negative coefficients in the unit disc defined by Hohlov operator, we obtain distortion theorem using fractional calculus techniques for this class. Also coefficient inequalities and some results for this class are obtained.     

A CLASS OF HARMONIC STARLIKE FUNCTIONS WITH RESPECT TO SYMMETRIC POINTS ASSOCIATED WITH WRIGHT GENERALIZED HYPERGEOMETRIC FUNCTION

Making use of Wright operator we introduce a new class of complex-valued harmonic functions with respect to symmetric points which are orientation preserving, univalent and starlike. We obtain coefficient conditions, extreme points, distortion bounds, and convex combination.     

Conformal invariants and higher-order schwarz lemmas

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.     

关于面积平均p叶函数（Ⅰ）

Ｌｅｂｅｄｅｖ－Ｍｉｌｉｎ方法是研究单叶函数的强有力的方法，本文建立了几个不等式，使得用这个方法来研究面积平均ｐ叶函数成为可能．本文还给出一个应用一－ＨａｍｉＩｔｏｎ猜测的证明。利用上述不等式和方法我们还能得到一些有趣的结果，这将另文讨论。     

On conformal distortion and Sullivan's sector theorem

Using general bounds on the conformal distortion of univalent maps, we prove a strong version of Sullivan's sector theorem, which gives certain sufficient conditions for an arbitrarily long composition of univalent Herglotz functions to map the upper half-plane into a proper sub-sector.

     

The Schwarz Inequality on the Boundary for Functions Regular in the Disk

The classical Schwarz inequality on the boundary of the disk for the functions regular in the disk is refined in various directions. Inequalities involving zeros of the function, an inequality for points mapped to symmetric points on the circle, and an inverse estimate for univalent functions are presented. Other inequalities are discussed, and the possibility of applying them to estimates of polynomials and rational functions is indicated. Bibliography: 17 titles.     

Symmetric strong vector quasi-equilibrium problems

In this paper, we introduce the symmetric strong vector quasi-equilibrium problem. We then demonstrate that the symmetric strong vector quasi-equilibrium problem is solvable under the suitable assumptions. As an application, we get an existence theorem of the strong saddle points of vector-valued functions. In addition, we give a characterization of vector-valued properly quasi-convex functions. This work was supported by the Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China     

On Grunsky inequalities and the exact domain of variability of the Grunsky functional

We give a generalized version of the classical Grunsky inequalities. As an application, we study the exact domain of variability of the Grunsky functional for univalent functions with quasiconformal extensions.     

P-Bloch函数的偏差定理和单叶半径

In this paper, we establish distortion theorems for both normalized p-Bloch functions with branch points and normalized locally univalent p-Bloch functions defined on the unit disk, respectively. These distortion theorems give lower bounds on |f′(z)| and ■f′(z). As applications of these distortion theorems, the lower bounds of the radius of the largest schlicht disk on these Bloch functions are given, respectively. Notice that when p = 1, our results reduce to that of Liu and Minda.     

一类新的单叶调和函数(英文)

A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) by generalized Salagean operator of harmonic univalent functions.We give sufficient coefficient conditions for normalized harmonic functions in the class LH_λ(α,β).These conditions are also shown to be necessary when the coefficients are negative.This leads to distortion bounds and extreme points.     

用Salagean算子定义的缺系数的单叶调和函数类

本文研究了用Salagean算子定义的缺系数单叶调和函数类.利用从属关系和算子理论得到类中函数的系数估计、极值点、偏差定理、卷积性质、凸性组合与凸半径,推广了已有的一些结果.     

Boundary Distortion Estimates for Holomorphic Maps

We establish some estimates of the angular derivatives from below for holomorphic self-maps of the unit disk ${\mathbb {D}}$ at one and two fixed points of the unit circle provided there is no fixed point inside ${\mathbb {D}}$ . The results complement Cowen–Pommerenke and Anderson–Vasil’ev type estimates in the case of univalent functions. We use the method of extremal length and a semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to obtain a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.     

由卷积和从属关系刻画的单叶调和函数某些子类

Let SH be the class of functions f = h + ˉg that are harmonic univalent and sensepreserving in the open unit disk U = {z ∈ C : |z| 1} for which f(0) = f′(0)-1 = 0. In the present paper, we introduce some new subclasses of SH consisting of univalent and sensepreserving functions defined by convolution and subordination. Sufficient coefficient conditions,distortion bounds, extreme points and convolution properties for functions of these classes are obtained. Also, we discuss the radii of starlikeness and convexity.     

A new class of Salagean-type harmonic univalent functions

We define and investigate a new class of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convex combination and radii of convex for the above class of harmonic univalent functions.     

Certain geometric properties of normalized Bessel functions

In this paper, we give a set of sufficient conditions for the normalized form of the generalized Bessel function to be univalent in the open unit disk, and further we obtain certain inequalities containing normalized Bessel functions.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Some inequalities for symmetric functions and an application to orthogonal polynomials

We present some sharp inequalities for symmetric functions and give an application to orthogonal polynomials.     

Fixed points of a mapping and Hyers–Ulam stability

We show that many general results on Hyers–Ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem. The theorem is formulated for self-maps of some subsets of the space of functions from a nonempty set into the set of reals. We also give some applications of that theorem, e.g., in investigations of solutions of some difference equations and functional inequalities.     

Second order evolution equations with time-dependent subdifferentials

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.