Nevanlinna's lemma on the logarithmic derivative of a meromorphic function

In proving the second fundamental theorem for functions meromorphic in the half-plane, Nevanlinna has asserted (in 1925) that they satisfy a lemma similar to the well-known lemma on the logarithmic derivative, but his proof was based on additional assumptions. These assumptions were later relaxed by Dufresnoy (in 1939) and Ostrovskii (in 1961). Here we shall show that in the general case, functions which are meromorphic in the half-plane do not satisfy a lemma similar to the lemma on the logarithmic derivative.     

线性差分多项式的值分布

利用亚纯函数Nevanlinna理论的差分对应物,研究了亚纯函数的线性差分多项式的值分布,建立了具有最大亏量和的亚纯函数与其差分多项式的特征函数的关系,所得结果推广了现有的一些相关结果.     

On an estimate of the size of the exceptional set in the lemma on the logarithmic derivative

In the paper, an exact estimate of the size of the exceptional set in the lemma on the logarithmic derivative is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 603–607, April, 2000.     

向量值Nevanlinna第二基本定理的余项S(r)的估计

首先通过建立并证明推广的对数导数基本引理,讨论了１２π∫２π０ｌｏｇ＋‖ｆ′（ｒｅｉθ）‖‖ｆ（ｒｅｉθ）‖ｄθ的增长问题,进而对向量值Ｎｅｖａｎｌｉｎｎａ第二基本定理的余项进行了估计     

推广的Schwarz-Pick引理

本文给出了Schwarz-Pick引理中单位圆到单位圆内的解析映射f的n阶导数|f(n)(z)|的进一步估计,并且给出了n=2时的精确估计．     

Direct approach to quantum extensions of Fisher information

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.      

On fixed-points and singular values of transcendental meromorphic functions

In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic function. To attain our purpose we first establish a fundamental inequality for the modulus of derivative of a holomorphic covering mapping whose image is an annulus by virtue of the hyperbolic metric. The inequality is of independent significance. We make a simple survey on some domain constants for ...     

Logarithmic derivatives of diffusion measures in a Hilbert space

For the logarithmic derivative of transition probability of a diffusion process in a Hilbert space, we construct a sequence of vector fields on Riemannian n-dimensional manifolds that converge to this derivative.     

The compound logarithmic function

We define the compound logarithmic function on simple connected region. As an application of such function, we point out a slight mistake of Hayman in his proof for logarithmic derivative theorem (Lemma 2.3, Page 36), in his book Meromorphic Functions, published in 1964. We modify his proof.     

The Maximum Principle for Parabolic Inequalities on Stratified Sets

In this paper, we consider the heat conduction operator with elliptic part of divergent type on a stratified set (i.e., on the set of manifolds of various dimension). We prove an analog of the lemma on the normal derivative and the weak and strong maximum principles for parabolic inequalities on this set.     

The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function f(z), f(0) = 1 (0 < r="> m( r,\fracf￠f ) < ln+ { \fracT(r,f)r\fracrr- r } + 5.8501.m\left( {r,\frac{{f'}}{f}} \right)< \ln + \left\{ {\frac{{T(\rho ,f)}}{r}\frac{\rho }{{\rho - r}}} \right\} + 5.8501.     

The normal derivative lemma for the Laplacian on a polyhedral set

An analog of the Oleinik-Hopf normal derivative lemma for the Laplace operator on a polyhedral set is considered.     

Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space $$B^{\frac{1}{2}}_{2,1}$$. Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.     

On the multiple values and uniqueness of algebroid functions on annuli

In this paper, we discuss the influence of multiple values and Valiron deficiencies on the uniqueness problem of algebroid functions on annuli, we get the several uniqueness theorems of algebroid functions on annuli, and also we extend the Nevanlinna value distribution theory for algebroid functions on annuli.     

SEVERAL UNIQUENESS THEOREMS OF ALGEBROID FUNCTIONS ON ANNULI

In this paper, we discuss the uniqueness problem of algebroid functions on annuli, we get several uniqueness theorems of algebroid functions on annuli, which extend the Nevanlinna value distribution theory for algebroid functions on annuli.     

Nevanlinna theory of meromorphic functions on annuli

In this survey paper, we discuss the recent development of Nevanlinna theory of meromorphic functions on annuli, which extends results in Nevanlinna theory in the complex plane or in a disk. In particular, we show that the approach taken on annuli is a unified treatment of functions meromorphic in the complex plane, a disk and an annulus. It allows one to obtain many results in the complex plane and in a disk as corollaries of our results in annuli.     

圆环中的亚纯函数及其导数在重值条件下的一个唯一性定理

In this paper, we first obtain the famous Xiong Inequality of meromorphic functions on annuli. Next we get a uniqueness theorem of meromorphic function on annuli concerning to their multiple values and derivatives by using the inequality.     

On surface radiation conditions for high-frequency wave scattering

A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist–Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.     

Estimates of the heat kernel on a manifold of nonpositive curvature

On a Riemannian manifold of nonpositive curvature, we obtain dimension-independent estimates for the fundamental solution of a parabolic equation and for the logarithmic derivative of this solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1129–1136, August, 1998.