Shankey Kumar, Swadesh Kumar Sahoo
In this article, we study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of such integrals.
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On the pre-Schwarzian norm of certain Logharmonic mappings
本研究建立对数调和映射前施瓦茨范数与解析函数前施瓦茨范数的关联，推导局部单叶对数调和映射范数有限及对数调和函数为Bloch函数的充要条件，还得到对数调和Bloch映射及其解析、余解析部分的范数与增长定理。
Pre-Schwarzian and Schwarzian norm Estimates for class of Ozaki Close-to-Convex functions
本文推导了Ozaki近凸函数的Schwarzian和pre-Schwarzian导数范数的精确界限，以及畸变和增长定理的精确界限，并探讨了解析部分固定的调和映射的pre-Schwarzian范数估计。
Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible $r\in (0,1)$ for which the pre-Schwarzian norm of the dilation $r^{-1}f(rz)$ is not greater than a prescribed number for normalized univalent functions $f(z)$ in the unit disk. Similar results concerning the Schwarzian derivative are also obtained
Pre-Schwarzian and Schwarzian norm Estimates for Robertson class
本文研究了Robertson类中满足特定条件的函数的预Schwarzian和Schwarzian范数估计，并给出了Schwarzian范数的精确上界。
Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic and harmonic mappings
本文研究了特定解析和调和映射类的Schwarzian导数和范数，建立了Schwarzian范数的精确估计，并通过预Schwarzian和Schwarzian范数得出单值性准则。
We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order $α,~0<α<1,$ and for uniformly convex functions.
For $-1\leq B<A\leq 1$, let $\mathcal{S}^*(A,B)$ denote the class of Janowski starlike functions which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp of pre-Schwarzian norm for the functions in the class $\mathcal{S}^*(A,B)$.
In this paper, we obtain sharp bounds for the norm of pre--Schwarzian derivatives of certain analytic functions. Initially this problem was handled by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud Univ Babeş--Bolyai Math {\bf61}(2): 155--162, 2016]. We pointed out that the proofs by Rahmatan et al. are incorrect and present correct proofs.
Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions
本文研究指数凸函数类的增长、畸变及Schwarzian范数估计，通过显式Schwarz函数推导参数依赖界，并分析极值问题。
Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part
本文针对单位圆盘内局部单叶调和函数，证明了一般预Schwarzian范数估计并纠正旧错误，引入新类F0并给出其预Schwarzian和Schwarzian范数的最佳估计，以及余解析函数的畸变和系数估计。
In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $α$ where $α\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.
Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class
本文研究广义罗伯特类函数的几何性质，推导其施瓦茨ian和预施瓦茨ian导数范数的精确界，并探讨函数的增长、扭曲及凹凸半径问题。
In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.
Sharp Pre-Schwarzian Norm Bounds for Ma-Minda Starlike Classes
本文建立统一框架计算 Ma-Minda 星形类的预 Schwarz 范数，给出直接通用方法，推导出经典和新引入星形子类的显式范数公式。
On the pre-Schwarzian norm estimate of special close-to-convex harmonic mappings
研究单位圆盘内具有特殊解析部分的近凸调和映射类，得到了精确的预Schwarzian范数估计，并讨论了调和Bloch映射理论及解析与共解析部分的增长与偏差定理。
On the Pre-Schwarzian Norm and Starlikeness of Certain Logharmonic Mappings
本文研究单位圆盘内的一类对数调和映射，得到其Pre-Schwarzian范数的精确界，讨论增长定理与星形性，推导遗传星形性的充分系数条件，并给出相关实例。
On the pre-Schwarzian and Schwarzian derivatives of log-harmonic mappings
本文为局部单叶对数调和映射定义前施瓦茨导数与施瓦茨导数，研究其几何函数论性质，并给出有限前施瓦茨范数的充要条件及其与解析函数范数的关系。
Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f'(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(β)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+β/2\}$, where $β>0$ and $\mathcal{F}(α)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>α\}$, where $-1/2\le α\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(β)$ and $\mathcal{F}(α)$.
Pre-Schwarzian norm estimate and characterization of certain harmonic mappings
研究单位圆盘上某类调和映射，给出预Schwarzian范数估计，证明其单叶性和近凸性，并讨论相关解析与共轭解析部分的增长与偏差定理，最后给出解析部分的系数估计。
Schwarzian Norm Estimate for Functions in Robertson Class
本文研究 Robertson 类 $\mathcal{S}_α$ 中规范化解析函数，给出并确定其 pre-Schwarzian 与 Schwarzian 范数的 sharp 估计。
Schwarzian norm estimates for some classes of analytic and harmonic mappings
本文研究了单位圆盘内满足特定实部条件的规范化解析函数类$\u27a9(β)$，建立了以$f''(0)$为参数的前Schwarzian范数和Schwarzian范数的精确界，得到了该类函数的增长、偏差及半径结果，并将其应用于调和映射的前Schwarzian范数估计。
Let $Co(α)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(α)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(α)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(α)$ whenever $f''(0)$ is fixed. We also give a characterization for co…
An estimation of the pre-Schwarzian norm for certain classes of analytic functions
本文研究了当φ(z)=1/(1-z)^s(0<s≤1)和φ(z)=(1+sz)^2(0<s≤1/√2)时，函数类S*(φ)和C(φ)的预Schwarz范数的精确估计。
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Möbius transformation. Some o…
The Schwarzian norm estimates for Janowski convex functions
本文研究Janowski凸函数类$\mathcal{C}(A,B)$的Schwarzian范数估计问题，借助Dieudonné引理得到精确估计结果，并通过新方法构造出达到该精确界的极值函数。
Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes
本文针对三类参数条件下的Ma-Minda型星形类与凸类函数，建立了预Schwarzian范数的精确估计。
Schwarzian Norm Estimate for Functions in Generalized Robertson Class
本文研究广义 Robertson 类 $\mathcal{S}_α(β)$，针对满足给定旋转实部不等式的规范化解析函数，推导其 pre-Schwarzian 与 Schwarzian 范数的 sharp 估计。
The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order $alpha$ in terms of the value of $f''(0)$, in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita and Yamashita for this particular case.
We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk and determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class $\mathcal{S}_H$ of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of $\mathcal{S}_H$ and the Schwarzian norm of the harmonic Koebe function.
Schwarzian Norm Estimates for Analytic Functions Associated with Convex Functions
本文针对与凸函数相关的解析函数类$\mathcal{F}(c)$，推导了Schwarzian和pre-Schwarzian导数的范数锐界，并给出了畸变与增长定理的锐界。
Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions
本文研究满足特定从属关系的 Ma-Minda 解析函数类，在两种特殊参数函数下给出函数及其 Alexander 变换的前 Schwarz 导数范数精确估计。
Sharp coefficient, logarithmic and Schwarzian estimates for $α$-convex functions on the real line
研究$\alpha$-凸函数类$M_\alpha$，推广了Chuaqui等人的定理，给出了尖锐的系数界、Fekete–Szegő不等式、阶和Schwarzian范数估计，并得到了$\alpha$-Koebe函数在$\alpha=1/n$时Schwarzian范数的显式公式。
The Schwarzian derivative in several complex variables univalence and geometry
本文综述多复变量中Schwarzian导数在局部双全纯映射上的性质，构造不变双线性算子度量曲率，证明小界蕴含单叶性，并推广Ahlfors--Weill定理。
We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.
A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to be in certain subclasses of functions are established in terms of the Schwarzian derivatives and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $α$-Bazileviĉ of order $β$.
We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result d…
In this article, we investigate some standard geometric properties of the integral operators $$ J_α[f](z)= \int_{0}^{z}\bigg(\frac{f(w)}{w}\bigg)^αdw, \,\,\, α\in \mathbb{C} \text{ and } |z|<1, $$ and $$ I_β[g](z)= \int_{0}^{z}\big(g'(w)\big)^βdw, \,\,\, β\in \mathbb{C} \text{ and } |z|<1, $$ where $f$ and $g$ are elements of certain classical families of normalized analytic functions defined on the unit disk. In particular, preserving properties of the Hornich sum of the operators $J_α$ and $I_β$ will be studied. Moreover, we also present sharp pre-Schwarzian norm estimate of such integrals.
Since the nonlinear integral transforms $J_α[f](z) = \int_{0}^{z}(f'(u))^α du$ and $I_α[f](z) =\int_0^z (f(u)/u)^α du$ with a complex number $α$ have been introduced, a great number of studies were dedicated to deriving sufficient conditions for univalence on the unit disk. On the other hand, little is known about the conditions that $J_α[f]$ or $I_α[f]$ produces a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper we discuss quasiconformal extendibility of the integral transforms $J_α[f]$ and $I_α[f]$ for holomorphic funct…
Dirichlet symbols and the nonlinear wave equation
本文研究与单位圆盘压缩算子关联的狄利克雷符号，分析其满足的非线性波动方程，引入Schwarzian渐近方差并得到相关振幅的上界，扩展了通用Teichmüller空间的适用范围。
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.
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