Nikolai Nikolov, Peter Pflug
We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the "derivatives" of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds.
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We give an example showing that the Kobayashi-Royden pseudometric for a non-taut domain is, in general, not the derivative of the Lempert function.
We prove the the multipole Lempert function is monotone under inclusion of pole sets.
We study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.
Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors.
We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.
We find all matrices $A$ from the spectral unit ball $Ω_n$ such that the Lempert function $l_{Ω_n}(A,\cdot)$ is continuous.
An upper estimate for the Lempert function of any $C^{1+ε}$-smooth bounded domain in $\Bbb C^n$ is found in terms of the boundary distance.
This note should clarify how the behavior of certain invariant objects reflects the geometric convexity of balanced domains.
Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given.
In the paper we show that the Lempert theorem (i.e. the equality between the Lempert function and the Carathéodory distance) holds in the tetrablock, a bounded hyperconvex domain which is not biholomorphic to a convex domain.
Given a domain $Ω\subset \mathbb C$, the Lempert function is a functional on the space $Hol (\D,Ω)$ of analytic disks with values in $Ω$, depending on a set of poles in $Ω$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of…
The Lempert function for several poles $a_0, ..., a_N$ in a domain $Ω$ of $\mathbb C^n$ is defined at the point $z \in Ω$ as the infimum of $\sum^N_{j=0} \log|ζ_j|$ over all the choices of points $ζ_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $Ω$ sending 0 to $z$. This is always larger than the pluricomplex Green function for the same set of poles, and in general different. Here we look at the asymptotic behavior of the Lempert function for three poles in the bidisk (the origin and one on each axis) as they all tend to the origin. The limit of the…
The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.
Quasi Triangle Inequality for the Lempert function
研究证明在具有C^1光滑边界的伪凸域上，Lempert函数满足准三角不等式，并推导了与边界正则性相关的局部界。
For $Ω$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in Ω$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(Ω), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $Ω$ by $L_S (z) :=\inf \{\sum^N_{j=1}ν_j\log|ζ_j|: \exists ϕ\in \mathcal {O}(\mathbb D,Ω), ϕ(0)=z, ϕ(ζ_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is …
We give some further criteria for continuity or discontinuity of the Lempert funtion of the spectral ball $Ω_n$, with respect to one or both of its arguments, in terms of cyclicity the matrices involved.
We utilize the Legendre-Fenchel transform and weak geodesics for plurisubharmonic functions to construct a weight function that can be used in the Berndtsson-Lempert method, to give an Ohsawa-Takegoshi extension type of result. Theorem 4.1 and Theorem 0.1 in \cite{OT2017} (Theorem \ref{Theorem A} and \ref{Theorem B} below) follow as two special cases of this result, thus answering affirmatively a question posed by Ohsawa in remark 4.1 in \cite{OT2017}, on the Berndtsson-Lempert method.
We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert funct…
We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.
Pick interpolation and invariant functions
本文建立Pick体与不变函数的联系，证明Pick体关联的不变函数可决定Pick插值问题可解性并推广Carathéodory伪距离，还在开单位圆盘描述该函数并导出广义Lempert函数，证明特定条件下二者相等，且双/三圆盘在特殊情况有类似Lempert定理的结果。
It is shown that the Carathéodory distance and the Lempert function are almost the same on any strongly pseudoconvex domain in $\C^n;$ in addition, if the boundary is $C^{2+\eps}$-smooth, then $\sqrt{n+1}$ times one of them almost coincides with the Bergman distance.
Comparison and localization of invariant functions on strongly pseudoconvex domains
本文在强拟凸域上获得了 Lempert 函数、Carathéodory 距离及其无穷小形式的比较与定位结果，并证明了可见域和强完备域的相关结果。
In the paper we show the equality between the Lempert function and the Green function with two poles with equal weights in the bidisc thus giving the positive answer to a conjecture of Coman in the simplest unknown case. Actually, a slightly more general equality is proven which in some sense is natural when studied from the point of view of the Nevanlinna-Pick problem in the bidisc.
We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in C^n by Lempert and by Larusson and Sigurdsson.
In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.
We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also give some lower estimates. In order to obtain those results, and because it is of independent interest, we refine and unify some classical estimates about the Kobayashi distance and the Lempert function in such domains.
A simple proof is given of the known fact that an m-times continuously differentiable function on the real line can be approximated along with its derivatives by an entire function and its respective derivatives.
A result is proved concerning meromorphic functions of finite order in the plane such that all but finitely many zeros of the second derivative are zeros of the first derivative.
We characterize meromorphic function fields closed by partial derivatives in n variables.
Functional inequalities for starlike and convex functions associated with $g$-derivative operator
研究与g-导数算子相关的星形和凸函数，通过引入新算子定义新函数类，并利用解析函数从属原理进行系数估计，得出Fekete-Szegö不等式、Toeplitz行列式和Hankel行列式的界。
We prove a uniqueness theorem for an entire function, which shares certain values with its higher order derivatives.
We prove that the derivative of a non-linear entire function is unbounded on the preimage of an unbounded set.
A number of results are proved concerning the existence of non-real zeros of derivatives of strictly non-real meromorphic functions in the plane.
A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane
The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_α^p$ of the derivative of $B$ or the derivative of functions in its model space $H^2 \ominus BH^2$ is investigated.
First Derivative of Automorphic Function of Triangle Groups
本文针对三角形群的自守函数，利用高斯超几何函数给出了其一阶导数的显式表达式。
In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.
In this paper, we prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share three sets. The obtained results improve some recent existing results.
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 discuss some results around the following question: Let $f$ be a nonconstant complex entire function and $a$, $b$ two distinct complex numbers. If $f$ and its derivative $f'$ share their simple $a$-points and also share the value $b$, does this imply $f\equiv f'$?
We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.
In this paper, we continue to study the sharing value problems for higher order derivatives of meromorphic functions with its linear difference and $q$-difference operators. Some of our results generalize and improve the results of Meng--Liu (J. Appl. Math. and Informatics, 37(2019), 133--148) to a large extent.
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.
On manifold-like polyfolds as differential geometrical objects with applications in complex geometry
本文提出类流形多折叠作为微分几何对象，通过引入张量和证明各种几何结构的存在，推动复几何领域的发展。
The main result of the paper determines all real meromorphic functions of finite order in the plane for which the first derivative has finitely many zeros, while the function itself and one of its higher derivatives have finitely many non-real zeros.
The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function,Gamma function and digamma function in a new way.
For a rational function f we consider the norm of the derivative with respect to the spherical metric and denote by K(f) the supremum of this norm. We give estimates of this quantity K(f) both for an individual function and for sequences of iterates.
We investigate necessary and sufficient conditions under which entire functions in de Branges spaces can be recovered from function values and values of derivatives. Our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half-plane.
In this article we prove a general result which in particular suggests that, on a simply connected domain in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the operator of partial sums of the Taylor expansion with center z at 0, seen as functions of the center z. We also discuss a universality result of these operators.
We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the functional equation to obtain a second formulation in terms of fractional derivatives.
In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.
We derive a variational formula for the outward normal derivative of the Green function for the Schrödinger and Laplace--Beltrami operators, viewed as perturbations of the Laplacian. As an application we begin to characterize elliptic growth---the growth of a domain pushed outward by its own Green function.
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