Difference analogues of the second main theorem for meromorphic functions in several complex variables

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard‐type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137 , 457–474 (2007); Comput. Meth. Funct. Theor. 12 , No. 1, 343–361 (2012)]. The method we used is very different from theirs.     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

On factorization of meromorphic functions

This paper improves on the results of Noda, Y., Li Baoqing and Song Quodong, and proves the following theorem: Letf(z) be a transcendental meromorphic function. Then the set {a∈C;(z−a)f(z) is not prime} is at most a countable set.     

多复变数的二阶微分从属

在文[1－2]的基础上讨论了多复变数中的二阶微分从属，取得了一些结果。这些结果是文[1-5]中的一些结果的推广。     

Small functions and uniqueness of meromorphic functions

We discuss the uniqueness of meromorphic functions sharing three weighted values and provide a complete answer to a question of T.C. Alzahary.     

Normal criteria for families of meromorphic functions

By using the Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family.     

Derivatives of meromorphic functions and exponential functions

We take up a new method to prove a Picard type theorem. Let f be a meromorphic function in the complex plane, whose zeros are multiple, and let R be a Möbius transformation. If \({\overline {\lim } _{r \to \infty }}\frac{{T\left( {r,f} \right)}}{{{r^2}}} = \infty \) then f′z) = R(e ^z ) has infinitely many solutions in the complex plane.     

On normal families of meromorphic functions

In this paper, we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions, which improve and generalize the related results of Gu, Bergweiler and Lin.     

Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

On one problem of uniqueness of meromorphic functions concerning small functions

In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.

     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensional case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 238–258, August, 1997. Translated by N. K. Kulman     

Local approximation of functions with several variables

This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave-convex functions are obtained as well.     

Univalence criterion for meromorphic functions and Loewner chains

The object of the present paper is to obtain a more general condition for univalence of meromorphic functions in the U^∗. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.     

On the unique range set of meromorphic functions

This paper studies the unique range set of meromorphic functions and shows that there exists a finite set such that for any two nonconstant meromorphic functions and the condition implies . As a special case this also answers an open question posed by Gross (1977) about entire functions and improves some results obtained recently by Yi.

     

On the results of Lahiri concerning uniqueness of meromorphic functions

In this paper we study the problem of meromorphic functions sharing three values with weight and obtain some theorems which improve the results given by Lahiri and others.     

Radially distributed values of meromorphic functions

Using the theory of normal families, we prove that a conjecture of Yanglo related to the radially distributed values of entire functions is positive under some additional conditions.     

Uniqueness of meromorphic functions whose derivatives share four small functions

In this paper, we will prove some uniqueness theorems of meromorphic functions whose derivatives share four distinct small functions. The results in this paper improve those given by R. Nevanlinna, L. Yang, G.D. Qiu, and other authors. An example is provided to show that the results in this paper are best possible.