Value distribution and uniqueness results of L-functions concerning certain linear differential polynomials

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study a transcendental entire function f (z) that shares a non-zero polynomial a (z) with f′(z), together with its linear differential polynomials of the form: L[f] = a2(z)f″(z) + a3 (z)f′′′(z) + … + am (z)f(m) (z) (am (z) $\not\equiv$ 0), where the coefficients ak (z) (k = 2, 3, ..., m) are rational functions.

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.

We study the uniqueness of entire functions, when they share a linear polynomial, in particular, fixed points, with their linear differential polynomials.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by \begin{equation*} L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0. \end{equation*}In this paper, we solve an open problem posed by Li [J. Math. Anal. Appl. 310 (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials $ L_k[f] $. Furthermore, we study the Fermat-type functional equations of the form $ f^n+g^n=1 $ to find the meromorphic solutions (f, g) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.

We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.

Linear complete differential resultants and the implicitization of linear DPPEs

An Entire Function that Shares a Small Function with its Derivative and Linear Differential Polynomial

We use the idea of normal family to investigate the uniqueness problems for entire functions: $f$ shares one finite non-zero value with its first derivative $f^{(1)}$ together with its certain type of linear differential polynomial $L(f)$ of constant coefficients. The results here improve and generalize results obtained by Yang, Li and Yang, and Li

This paper is devoted to studying the uniqueness problem of entire functions sharing a small function with their linear differential polynomials.

Linear sparse differential resultant formulas

We give a slightly different proof of a theorem by N, Steinmetz: Suppose that f is a transcendental meromorphic function Then for ∊ > 0 and distinct functions a1...aq such that T(r. aj) = 0(T(r,f) we have For the proof we use a lemma on the zeros of linear differential polynomials of the form W(a1…aq,f).

On the inversion of a linear differential polynomial

Preface Geometric value distribution theory Barsegian, G.: A new program of investigations in analysis: Gamma-lines approaches Sukiasyan, G.: On level sets of quasiconforrnal mappings Classical value distribution theory Alonso, A., Fernandez, A. and Perez, J.: On the unintegrated Nevanlinna fundamental inequality for meromorphic functions of slow growth Barsegian, G. and Yang, C.-C.: On some new concept of exceptional values Ciechanowicz, E. and Marchenko, I.: Maximum modulus points, deviations and spreads of meromorphic functions Craven, T. and Csordas, G.: Composition theorems, multiplier sequences and complex zero decreasing sequences Korhonen, R.: Nevanlinna theory in an annulus Marchenko, I. and Nikolenko, I.: On strong asymptotic tracts of functions holomorphic in a disk Complex differential and functional equations Barsegian, G., Sarkisian, A. and Yang, C.-C.: A new trend in complex differential equations: quasimeromorphic solutions Ha, H.K. and Yang, C.-C.: On the functional equation P(f) = Q(g) He, Y.: Value distribution of the higher order analogues of the first Painleve equation Yang, C.-C. and Li, P.: Some further results on the functional equation P(f) = Q(g) Several variables theory Aihara, Y.: Recent topics in uniqueness problem for meromorphic mappings Berenstein, C. and Li, B.Q.: On interpolation problems in Cn Hu, P.-C. and Yang, C.-C. : Jet bundles and its applications in value distribution of holomorphic mappings Tu, Z.-H.: Normal families of meromorphic mappings of several complex variables into the complex projective space

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

In this paper, we will give suitable conditions on differential polynomials [Formula: see text] such that they take every finite nonzero value infinitely often, where [Formula: see text] is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf ]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials [Formula: see text] to have some terms of any degree of [Formula: see text] and also the hypothesis [Formula: see text] in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11(2) (1995) 355–373] is replaced by [Formula: see text] in our result.