María J. Martín
We review the relation between the classical formulas of the pre-Schwarzian and Schwarzian derivatives of locally univalent analytic functions and the derivatives of the generating functions of the methods due to Newton and Halley, respectively, for approximating zeros. We extend these relations to the cases when the functions considered are harmonic.
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In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation $ω_f$ of $f$. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.
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
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。
We introduce a new definition of pre-Schwarzian derivative for logharmonic mappings and basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzain is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic.
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.
Sharp multiplier estimates for the higher-order Schwarzian derivatives of the Koebe function
本文研究加权Bergman空间上乘法算子的乘子范数估计，其符号为单叶函数的高阶Schwarz导数，建立了Koebe函数的高阶Schwarz导数的尖锐乘子估计，推广了Shimorin的相关结果。
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.
In this paper we will show that the assumption on the negative Schwarzian derivative is redundant in the case of C^3 unimodal maps with a nonflat critical point. The following theorem will be proved: For any C^3 unimodal map of an interval with a nonflat critical point there exists an interval around the critical value such that the first entry map to this interval has negative Schwarzian derivative. Another theorem proved in the paper provides useful cross-ratio estimates. Thus, all theorems proved only for unimodal maps with negative Schwarzian derivative can be easily generalized.
The primary aim of this article is to extend certain inequalities concerning the pre-Schwarzian derivatives from the case of analytic univalent functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the h…
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.
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 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 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.
Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that the Schwarzian derivative does not have a direct transcendental singularity over infinity, and does not have infinity as a Borel exceptional value.
In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichmüller mappings, whose maximal dilatation estimate is asymptotically sharp.
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.
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.
For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.
On the pre-Schwarzian norm estimate of special close-to-convex harmonic mappings
研究单位圆盘内具有特殊解析部分的近凸调和映射类，得到了精确的预Schwarzian范数估计，并讨论了调和Bloch映射理论及解析与共解析部分的增长与偏差定理。
Analytic Besov functions, pre-Schwarzian derivatives, and integrable Teichmüller spaces
本文通过预Schwarzian导数研究可积Teichmüller空间$T_p$的嵌入问题，扩展了$p=1$情况下的结果，为复解析理论提供了统一框架。
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。
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.
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。
The notion of Schwarzian derivative for locally univalent holomorphic functions on complex plane was generalized for conformal diffeomorphisms by Osgood and Stowe in 1992 [27]. We shall identify a tensor that may serve as an analogue of the Schwarzian of Osgood and Stowe for CR mappings, and then use the tensor to define and study the CR Möbius transformations and metrics of pseudo-hermitian manifolds. We shall establish basic properties of the CR Schwarzian and a local characterization of CR spherical manifolds in terms of fully integrability of the CR Möbius equation. In another direction, w…
For a meromorphic function $f$ in the unit disk $U=\{z:\;|z|<1\}$ and arbitrary points $z_1,z_2$ in $U$ distinct from the poles of $f$, a sharp upper bound on the product $|f'(z_1)f'(z_2)|$ is established. Further, we prove a sharp distortion theorem involving the derivatives $f'(z_1)$, $f'(z_2)$ and the Schwarzian derivatives $S_f(z_1)$, $S_f(z_2)$ for $z_1,z_2\in U$. Both estimates hold true under some geometric restrictions on the image $f(U)$.
The Schwarzian Derivative for Convex Holomorphic Mappings in Several Complex Variables
本文研究了几复变量中凸全纯映射的Schwarz导数范数上界，推导出在多变量情况下的精确估计，扩展了经典一维结果，并给出了Roper-Suffridge扩展算子的最佳界。
A discrete Schwarzian derivative via circle packing
本文在基于圆堆积的离散解析函数理论中引入离散施瓦茨导数及其局部化概念内蕴施瓦茨导数，建立圆堆积花的内蕴施瓦茨导数参数化与堆积判据，并研究特殊花类。
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。
On a multiplier operator induced by the Schwarzian derivative of univalent functions
该研究考察由可拟共形延拓单叶函数的 Schwarz 导数诱导的乘子算子，并据此验证一大类拟圆盘满足 Brennan 猜想，同时给出渐近共形曲线和 Weil-Petersson 曲线的新刻画。
We argue relations between the Aharonov invariants and Tamanoi's Schwarzian derivatives of higher order and give a recursion formula for Tamanoi's Schwarzians. Then we propose a definition of invariant Schwarzian derivatives of a nonconstant holomorphic map between Riemann surfaces with conformal metrics. We show a recursion fomula also for our invariant Schwarzians.
We study necessary and sufficient conditions for a meromorphic quadratic differential with prescribed poles to be the Schwarzian derivative of a rational map. We give geometric interpretations of these conditions. We also study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case, the analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
Properties of Besov and $Q_p$ spaces in terms of the Schwarzian derivative of harmonic mappings
该研究借助Schwarzian导数与Carleson测度刻画单位圆盘内局部单叶保向调和映射的$\text{log }J_f$所属的Besov型与$Q_p$型空间，还引入基于雅可比算子的两类新空间并启动研究。
Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions
该研究针对单位圆盘内由特定曲线界定区域从属关系定义的单叶星形函数子类，求得了高阶Schippers-Schwarzian导数的精确上界及对应极值函数，并将其应用于Grunsky系数精确界的推导。
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.
For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm.
We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian derivative family. We also establish some criterion for the Schwarzian derivative family of a family $\F$ of analytic functions on a domain $D$ in the complex plane to be a normal family.
It is well-known that two locally univalent analytic functions $φ$ and $ψ$ have equal Schwarzian derivative if and only if there exists a non-constant Möbius transformation $T$ such that $φ=T\circ ψ$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.
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。
Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative
本文利用 Nevanlinna 理论研究了三个含 Schwarzian 导数的差分方程，获得了其次正规超越亚纯解存在的必要条件，并给出实例支持。
We consider the family of all meromorphic functions $f$ of the form $$ f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots $$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $α$, $0\le α<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\c…
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.
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。
Generalized Schwarzians and Normal Families
本文研究有界广义Schwarzian导数的函数族，证明其拟正规性，推导新公式并应用于局部单叶函数族和整函数，经典情形为k=2。
For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.
Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives
本文研究黎曼球面上的分段测地若尔当曲线，证明其共形焊接为分段莫比乌斯映射，且两区域均匀化映射的施瓦茨导数为有理函数，通过显式计算$C^1$测地对作为关键工具推进研究。
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.
On Koebe-type functions for harmonic quasiconformal mappings
本文研究了谐波准共形映射中Koebe型函数的性质，建立了相关的一致性条件、范数、系数不等式及生长和面积定理，改进了已有结果。
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。
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。
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 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…
We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions.
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 $β$.
The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class II
本研究第二部分在前期建立复Banach流形结构基础上，证明预Schwarzian导数映射像为实解析圆盘丛，为小Zygmund类子空间建立对应结果并构造商空间，证明其嵌入良定义且单射以赋予复结构。
We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Our auxiliary results include a characterization of …
Gears, Pregears and Related Domains
本文从 Schwarzian 导数视角研究单位圆盘到单齿齿轮状区域的共形映射，引入“预齿轮”概念并精确确定了一致性区域。
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。
Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.
Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules, in particular the Baker and tau-functions, which become operator-valued. Following from Part I we produce a pre-determinant structure for a class of tau-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly de…
Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions
本文研究圆盘到轴对称圆弧四边形的共形映射，数值求解附属参数与几何参数的关联，结合含Weierstrass P-函数的矩形映射问题，得到P函数零点及半周期像的新公式。
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