Uniqueness of meromorphic functions sharing small function with their difference operators and derivatives

To consolidate or adapt to many studies on meromorphic functions, we define a new subclass of meromorphic functions of complex order involving a differential operator. The defined function class combines the concept of spiral-like functions with other studies pertaining to subclasses of multivalent meromorphic functions. Inclusion relations, integral representation, geometrical interpretation, coefficient estimates and solution to the Fekete-Szegö problem of the defined classes are the highlights of this present study. Further to keep up with the present direction of research, we extend the study using quantum calculus. Applications of our main results are given as corollaries.

We investigate the uniqueness of transcendental analytic fun- ctions that share three values DM in one angular domain instead of the whole complex plane. 1. Introduction and main results In this paper, a transcendental meromorphic (analytic) function is mero- morphic (analytic) in the whole complex plane C and not rational. We as- sume that the reader is familiar with the Nevanlinna's theory of meromorphic functions and the standard notations such as m(r, f ), T (r, f ). For references, see (2). We say that two meromorphic functions f and g share the value a (a ∈ C = C ∪ {∞}) in X ⊆ C provided that in X, we have f (z) = a if and only if g(z) = a. We will state whether a shared value is by DM (differential mul- tiplicities), or by IM (ignoring multiplicities). R. Nevanlinna (see (4)) proved that if two meromorphic functions f and g have five distinct IM shared values in X = C, then f (z) ≡ g(z). After his very work, the uniqueness of meromor- phic functions with shared values in the whole complex plane attracted many investigations (for references, see (7)). E. Mues consider DM shared values and proved the following theorem.

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM.Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

A note on partial sharing of values of meromorphic functions with their shifts

We fix z 0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜 z 0 := the field of germs of meromorphic functions at z 0. We fix f 1, …, fr ∈ 𝓜 z 0 and we consider the 𝔽-algebras S := 𝔽[f 1, …, fr ] and S ¯ := F [ f 1 ± 1 , … , f r ± 1 ] . $\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings S h o l := F [ f a := f 1 a 1 ⋯ f r a r : ( a 1 , … , a r ) ∈ N r and f a is holomorphic at z 0 ] , S ¯ h o l := F [ f a := f 1 a 1 ⋯ f r a r : ( a 1 , … , a r ) ∈ Z r and f a is holomorphic at z 0 ] , $$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$ and we tackle in detail the case 𝔽 = 𝓜<1, the field of meromorphic functions of order < 1, and fj ’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.

Meromorphic functions on annuli sharing finite sets with truncated multiplicity

In this work the question of the relation of the growth of a meromorphic function and that of its derivative is considered. In the plane J.M. Whittaker proved that and have the same order. Here it is shown that for meromorphic functions in the order of growth of and are the same at zero and infinity. This can also be extended to meromorphic functions , that is, meromorphic functions in the plane punctured at a finite number of points. A natural question is whether this always happens at a neighbourhood of a transcendental singularity. The answer to this question is unknown to the author.

Inspired by a lot of studies on the uniqueness results of a L-function with a meromorphic function, in this article, we examine the uniqueness of two differential polynomials, one generated by a meromorphic function with finitely many poles and another by a L-function, when they share two values with some weight. The results of our examination extend, generalize as well as improve the results of Hao and Chen [2, 3].

We study the normality of families of meromorphic functions defined in terms of certain omitted functions. In particular, we prove the following results. Firstly, if $\mathcal{F}$ is a family of meromorphic functions in a domain D ⊂ ℂ, and a(z), b(z) and c(z) are distinct meromorphic functions in D and if, for all f ∈ $\mathcal{F}$ and all z ∈ D, f(z) ≠ a(z), f(z) ≠ b(z) and f(z) ≠ c(z), then $\mathcal{F}$ is normal in D. Secondly, letting R(w) be a rational function of degree greater than or equal to 3 and $\mathcal{F}$ be a family of functions meromorphic in a domain D ⊂ ℂ, if there exists a non-constant meromorphic function α(z) in D such that, for all f ∈ $\mathcal{F}$ and z ∈ D, R(f(z)) ≠ α(z), then $\mathcal{F}$ is normal in D.

Let E be the infinite dimensional locally convex space equipped with the finite open topology τ0, M be a complex manifold modelled with the locally convex space E, (Ω, φ) be a Riemann domain over the manifold M, (Ω̄, φ̄) be the locally pseudoconvex hull of the domain (Ω, φ) over M. Let X be a complex Banach manifold with the weak disk property and f: Ω → X be a meromorphic map. The meromorphic mapping f is extended to a meromorphic mapping f : Ω̄ → X if and only if the set of points of indeterminacy of the mapping f is extended to an analytic thin set of Ω̄.KeywordsProduct SpaceComplex ManifoldConvex SpaceHausdorff SpaceStein ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The discrete coupled nonlinear Schrodinger (DCNLS) hierarchy associated with a discrete $$3\times 3$$ matrix spectral problem is derived, which are composed of the positive and negative flows. Utilizing the characteristic polynomial of Lax matrix for the DCNLS hierarchy, we introduce a trigonal curve with three infinite points and three zero points, from which we establish the associated Baker–Akhiezer function and meromorphic functions. The DCNLS equations are decomposed into a system of Dubrovin-type ordinary differential equations. Using the theory of the trigonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire DCNLS hierarchy.

In the present paper, the authors introduce and investigates two new subclasses Mγ(κ) and Nη(κ) of meromorphic functions which are defined by terms of the quasi-subordination. The coefficients estimate inclusive of the classical Fekete-Szegö functional |a1−λa02| for functions belonging to this class are then derived. Investigate a majorization problem for the class’s meromorphic functions associated with these classes is also pointed out.

On the instability of the Riemann hypothesis for curves over finite fields

Meromorphic and Entire Approximation in BMO-Norm

IRVING GERSTIntroduction.Let /(z) be a nonconstant meromorphic function.Further, let there exist two numbers my^O, 1 and hy^O, and two rational functions R and S, such thatand(2)Then the function f(z) has both a rational multiplication and a rational addition theorem.We propose to determine all such functions.Our problem is a generalization of one treated in a paper by Ritti1), in which all periodic meromorphic functions having a multiplication theorem were determined.This is equivalent to taking S(z) =z in (2).For this case, Ritt proved that \m\ ^ 1 ; and if \m\ > 1, then f(z) is a linear function of one of the functions eaz, cos(az+ß), p(z+ß); when gs = 0, g»2(z+j3); when g2 = 0, S>'(z+ß) and j?3(z+j3).Here a is arbitrary while ß is restricted to certain values.If \m\ =1, then it was shown that m is either -1 or a third, fourth or sixth root of unity.The forms for the function f(z) were also explicitly given in this case.Use will be made of the results quoted and also of the methods of Ritt's paper.We distinguish the following three cases :(A) m >1;(B) m ¿1, with m not a rational root of unity; (C) m a primitive wth root of unity.For Case (A), which is the one usually considered in multiplication theorems, we can restate the problem in a way which shows its connection with the theory of a class of functions first systematically studied by Poincaré(2).He provided an existence theorem for meromorphic functions satisfying (1) assuming that 7?(z) has a fixed point(3) a for which 7?'(a) has a modulus greater than unity.Thus our problem is equivalent to that of finding all Poincaré functions having a rational addition theorem.