Non-real zeros of derivatives of meromorphic functions

Derivatives of Meromorphic functions with multiple zeros and elliptic functions

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple exept finitely many and T(r,h)=o{T(r,f)} as r tends to infinity, then f'=h has infinitely many solutions (including poles).

On the uniqueness of power of a meromorphic function sharing a set with its k-th derivative

In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more general settings namely set sharing

Further results on the uniqueness of meromorphic functions and their derivative counterpart sharing one or two sets

In this paper we prove a number of results concerning uniqueness of a meromorphic function as well as its derivative sharing one or two sets. In particular, we deal with the specific question raised in [18], [19], [20] and ultimately improve the result of Banerjee-Bhattacharjee [4].

Uniform boundedness of derivatives of meromorphic inner functions on the real line

Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivatives on the real line. This question was first studied by Louis de Brange in 1968 and was later revived by Anton Baranov in 2011.

A Note On Uniqueness Of Meromorphic Functions And Their Derivatives Sharing Two Sets

In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively. Moreover, we exhibit some examples to show the sharpness of some conditions used in our main result.

Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

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 $f(U)$.

亚纯函数的幂与其在 C^m 中的 k 阶方向导数共享值的性质

Power of a meromorphic function sharing value with its k-th order directional derivative in C^m

本文研究了多复变 C^m 中亚纯函数的幂与其 k 阶方向导数共享值的唯一性问题，将单变量唯一性定理推广至高维，并通过实例证明结果的最佳性。

解析函数与其一阶导数部分共享值的研究

Meromorphic functions that partially share values with their first derivative

研究解析函数与其导数部分共享值的唯一性，提出四个部分共享值的结论。

Non-real zeros of derivatives of real meromorphic functions

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.

Wiman-Valiron theory for a class of functions meromorphic in the unit disc

Analogues of the key results of Wiman-Valiron theory are proved for a class of functions meromorphic in the unit disc, based on an approach developed by Bergweiler, Rippon and Stallard for the plane setting. The results give local approximations for the function and its logarithmic derivative and, in the case of positive order of growth, for higher order logarithmic derivatives as well.

On Meromorphic Functions which are Brody Curves

We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P_1, i.e., which have bounded spherical derivative.

Necessary and Sufficient Condition for zeros of Derivative of Meromorphic and Entire Functions

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.

A note on spherical derivatives and normal families

We show that a family of meromorphic functions in the unit disk $\dk$ whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each $f$ meromorphic in $\dk$ we have $\inf_{z\in\dk} f^#(z)\le \frac{1}{2}$$ where $f^#$ denotes the spherical derivative of $f$.

Meromorphic functions with small Schwarzian derivative

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…

具有 Schwarzian 导数的差分方程的次正规超越亚纯解

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。

Zeros of Meromorphic function

We introduce and develop the root locus method in mathematics. And we study the distribution of zeros of meromorphic functions by root locus method.

On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $\log^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper \cite{CF} independent of Halburd et al \cite{HK-1}. We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic…

亚纯函数一阶导数与差分唯一性及整函数周期性的刻画

Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity

本文研究亚纯函数一阶导数与差分分担值的唯一性，解决Qi等人开放问题，并给出超越整函数周期性的充分条件。

Growth estimates for meromorphic solutions of higher order algebraic differential equations

We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class $X$ of meromorphic functions in the unit disc, defined by means of the spherical derivative, and $m \in \mathbb{N} \setminus \{1\}$, $f^m\in X$ implies $f\in X$. An affirmative answer to this is given for example in the case of $\mathord{\rm UBC}$, the $α$-normal functions with $α\ge1$ and certain (sufficiently large) Dirichlet type…

Linear differential polynomials in zero-free meromorphic functions

The paper determines all meromorphic functions with finitely many zeros in the plane having the property that a linear differential polynomial in the function, of order at least 3 and with rational functions as coefficients, also has finitely many zeros.

Zeros of differences of meromorphic functions

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.

Zeros of differential polynomials in real meromorphic functions

We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.

亚纯函数零点与极点的 zeta 函数、零点不存在性判据及其在 Cauchy 核之和中的应用

The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels

该研究通过联系亚纯函数零点与极点的 zeta 函数，提出零点不存在判据，并给出 Cauchy 核之和零点存在的充分条件。

关于导数的亚纯函数与其位移的唯一性

Uniqueness of Meromorphic Functions With Respect To Their Shifts Concerning Derivatives

本文研究了亚纯函数与其位移共享小函数时关于k阶导数的唯一性问题，采用新方法将现有整函数和一阶导数的结论推广至亚纯函数和k阶导数，并给出了k=0时的具体唯一性条件。

具有多重零点的亚纯函数族的正规性准则

Normality criteria for a family of meromorphic functions with multiple zeros

本文证明了具有多重零点的亚纯函数族的正规性准则，主要结果涉及微分多项式共享全纯函数，并推广了Fang、Zalcman和Chen等人的已有结果。

导数的非实零点

Non-real zeros of derivatives

该研究聚焦实亚纯函数导数的非实零点问题，证明了相关若干结果，且本文取代了此前arXiv上的同主题相关提交论文。

实亚纯函数中线性微分多项式的非实零点

Non-real zeros of linear differential polynomials in real meromorphic functions

该研究探讨实整函数相关微分多项式的零点分布问题，证明了仅具实零点的无限增长阶实整函数（或其倒数）对应的f''+ωf对任意正ω有无穷多非实零点。

Uniqueness theorems of meromorphic functions with their differential-difference operators in several complex variables

An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $πi$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We improves the author's result \cite{h} from entire function to meromorphic function, the first derivative to its differential-difference polynomial, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $ρ_{2}(f)<1$, l…

On the growth of logarithmic difference of meromorphic functions and a Wiman-Valiron estimate

The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.

关于亚纯函数的差商与导数商的一个新的邻近函数估计

A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function

该研究在亚纯函数增长性与值分布的特定假设下，得到了差分与导数商的邻近函数新估计，推导了Nevanlinna分歧项下界，并给出整函数值分布推论及多个复分析应用。

Value distribution and uniqueness for q-difference of meromorphic functions Sharing Two Sets

In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate the uniqueness of zero order meromorphic function with its q-difference operator sharing two sets with finite weight. Some examples have been exhibited which are relevant to the content of the paper.

Differential inequality of the second derivative that leads to normality

Let F be a family of functions meromorphic in a domain D. If {|f|/(1+|f|^3):f in F} is locally uniformly bounded away from zero, then F is normal.

形如 $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$ 的亚纯函数的零点

Zeros of meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$

本文研究了形如 $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$ 的亚纯函数的零点分布，证明了经典 Keldysh 定理的类比，并揭示了此类无零点函数与多项式系数二阶微分方程之间的关系。

Value distribution of q-differences of meromorphic functions in several complex variables

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.

Nevanlinna theory for meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold

The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for nonzero meromorphic functions on $\mathbb{C}^l.$ The second is to improve the definition of the non-integrated defect relation of H. Fujimoto \cite{F2} and to show two theorems on the new non-integrated defect relation of meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold. The third is to give a unicity theorem for meromor…

亚纯内函数与典范系统的唯一性定理

Uniqueness theorems for meromorphic inner functions and canonical systems

本文研究上半平面亚纯内函数的唯一性，利用谱、导数值与Clark测度等数据，推广Borg-Levinson两谱定理并推动典范Hamilton系统逆谱理论。

On certain type of difference polynomials of meromorphic functions

In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some inequalities for the relationship of the zero counting function of $f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function of $f(z)$. Based on these inequalities, we establish some difference analogues of a classical result of Hayman for meromorphic functions. Some special cases are also investigated. These results improve previous findin…

Rational decompositions of complex meromorphic functions

Let h be a complex meromorphic function decomposed in two different ways P(f) and Q(g), where f, g are meromorphic functions and P, Q are rational functions. We follow an approach due to C.-C. Yang, P. Li and K. H. Ha who handle similar decompositions, however, with P, Q polynomials, by applying the Second Nevanlinna Theorem. We establish relations between the number k of distinct zeros of P', the degrees of P and Q, and (unless f, g are analytic functions) the number of distinct zeros of the denominators of P and Q.

Normality Criteria for Families of Meromorphic Functions

In this paper we prove some normality criteria for a family of meromorphic functions, which involves the zeros of certain differential polynomials generated by the members of the family.

The winding number of PF+1 for polynomials P and meromorphic extendibility of F

Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD then f extends holomorphically through D. We prove a related result on meromorphic extendibility for smooth functions with finitely many zeros of finite order, which, in particular, implies that the conjecture holds for real analytic functions.

Difference Picard theorem for meromorphic functions of several variables

It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case where the hyper-order of f is less than 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the lemma on the logarithmic derivative and of the second …

一类亚纯函数的HRT猜想

The HRT conjecture for a class of meromorphic functions

本文证明非零亚纯函数的有限时频平移系统线性无关，从而证实HRT猜想对可延拓为除有限点外解析的平方可积函数成立。

亚纯函数的差分与导数之商的消失、无界及角向偏移

Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function

本文研究亚纯函数差分与导数之商的邻近函数，证明在周期消失或趋于无穷时其趋于零或为小函数，并给出角向偏移估计及亏量关系等应用。

Weighted Bergman kernel functions associated to meromorphic functions

We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.

Meromorphic functions with three radially distributed values

We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result.

On the convergence of multi-level Hermite-Padé approximants for a class of meromorphic functions

The present paper deals with the convergence properties of multi-level Hermite-Padé approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.

Higher-order interlacing for matrix-valued meromorphic Herglotz functions

Scalar-valued meromorphic Herglotz-Nevanlinna functions are characterized by the interlacing property of their poles and zeros together with some growth properties. We give a characterization of matrix-valued Herglotz-Nevanlinna functions by means of a higher-order interlacing property. As an application we deduce a matrix version of the classical Hermite-Biehler Theorem for entire functions.

Value distribution and linear operators

Nevanlinna's second main theorem is a far-reaching generalisation of Picard's Theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f' appear. In this paper we show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalisations of Picard's Theorem …

The Schwarzian derivative and the Wiman-Valiron property

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.

Zeta-functions for germs of meromorphic functions and Newton diagrams

For a germ of a meromorphic function f=P/Q, we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.

Malmquist 型 q-差分方程的零级亚纯解

Zero order meromorphic solutions of $q$-difference equations of Malmquist type

本文研究 Malmquist 型 q-差分方程 f(qz)^n=R(z,f)，证明当 |q|≠1 时其零级超越亚纯解仅存在于可约化为 q-差分线性或 Riccati 方程的方程中，并给出自治情形的显式解及差分 Painlevé 性质的应用。

Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values

In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition.

Normality and uniqueness property of meromorphic function in terms of some differential polynomials

In this paper, we will consider normality and uniqueness property of a family $\mathcal{F}$ of meromorphic functions when $[Q(f)]^{(k)}$ and $[Q(g)]^{(k)}$ share $α$ ignoring multiplicities, for any $f,g\in \mathcal{F}$, where $Q$ is a polynomial and $α$ is a small function. Our results do not need all of zeros of $Q$ have large order as other authors's results.