Nonconstant Meromorphic Functions Research Articles

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

Meromorphic Functions Sharing Two Values

Let [see formula in PDF] and [see formula in PDF] be two nonconstant meromorphic functions, and [see formula in PDF] be a positive integer. If [see formula in PDF] and [see formula in PDF] share 1 CM (counting multiplicities), [see formula in PDF] and [see formula in PDF] share [see formula in PDF] IM (ignoring multiplicities), and [see formula in PDF], then either [see formula in PDF] or [see formula in PDF], where [see formula in PDF]. As an application, shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.

Submanifolds with ample normal bundle

AbstractWe construct germs of complex manifolds of dimension along projective submanifolds of dimension with ample normal bundle and without nonconstant meromorphic functions whenever . We also show that our methods do not allow the construction of similar examples when by establishing an algebraicity criterion for foliations on projective spaces that generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.

A power of a meromorphic function sharing a set with its higher order derivative

Abstract In this paper, we deduce the form of a nonconstant meromorphic function f when some power of f shares certain set counting multiplicities in the weak sense with the k-th derivative of the power. The results of this paper generalize the results due to Lahiri and Zeng [Afr. Mat. 27 (2016), 941-947].

Uniqueness of Derivatives and Shifts of Meromorphic Functions

Recently, some uniqueness theorems about meromorphic functions f(z) concerning their derivatives $$f'(z)$$ and shifts $$f(z+c)$$ with three CM sharing values have been obtained. In this paper, we continue to study this topic. We consider not only high order derivatives instead of just $$f'(z)$$ , but also IM sharing value instead of CM sharing value. In fact, we mainly prove that for a non-constant meromorphic function f(z) of hyper order strictly less than 1, if $$f^{(k)}(z)$$ and $$f(z+c)$$ share $$0,\infty $$ CM and 1 IM, then $$f^{(k)}(z)\equiv f(z+c)$$ , where c is a non-zero finite complex number. Our main theorem generalizes and greatly improves the related result due to Qi–Li–Yang. In addition, we give some discussion of this issue and obtain a uniqueness theorem concerning defective values in Sect. 3.

On the cardinality of a reduced unique range set

UDC 517.5Two meromorphic functions are said to share a set ignoring multiplicities (IM) if has the same pre-images under both functions. If any two nonconstant meromorphic functions, sharing a set IM, are identical, then the set is called a “reduced unique range set for meromorphic functions'' (in short, RURSM or URSM-IM). From the existing literature, it is known that there exists a RURSM with seventeen elements. In this article, we reduced the cardinality of an existing RURSM and established that there exists a RURSM with fifteen elements. Our result gives an affirmative answer to the question of L. Z. Yang (Int. Soc. Anal., Appl., and Comput., <strong>7</strong>, 551–564 (2000)).

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not K\"{a}hler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

On Uniqueness of Meromorphic Functions Partially Sharing Values with Their q-shifts

In this work, we give some uniqueness theorems for non-constant zero-order meromorphic functions when they and their q-shifts partially share values in the extended complex plane. This is a continuation of previous works of Charak et al. (J Math Anal Appl 435(2):1241–1248, 2016) and of Lin et al. (Bull Korean Math Soc 55(2):469–478, 2018). Furthermore, we show some uniqueness results in the case multiplicities of partially shared values are truncated to level $$m\ge 4$$ . As a consequence, we obtain a uniqueness result for an entire function of zero-order if it and its q-shift partially share three distinct values $$a_1, a_2, a_3$$ without truncated multiplicities, in which we do not need to count $$a_j$$ -points of multiplicities greater than 38 for all $$j=1,2,3$$ .

Uniqueness of meromorphic functions concerning their difference operators and derivatives

Let f and g be two nonconstant meromorphic functions. Shared value problems related to f and g are investigated in this paper. We give sufficient conditions in terms of weighted value sharing which imply that f is a linear transformation or inversion transformation of g. We also investigate the uniqueness problem of meromorphic functions with their difference operators and derivatives sharing some values.

Uniqueness theorems on meromorphic functions and their difference polynomials

ABSTRACT Let f be a nonconstant meromorphic function of finite order, and be a linear difference polynomial in f with small meromorphic coefficients. Some uniqueness theorems related to f sharing three values with are obtained. We also prove that f is identical with its difference operator , if they share three values CM. Some examples are given to show our results are best possible.

Meromorphic functions that share values with their shifts or their n-th order differences

Let f(z) be a non-constant meromorphic (entire) function of hyper-order strictly less than 1, n ≥ 3 (n ≥ 2) be an integer. It is shown that if fn(z) and fn(z + c) share a(≠ 0) ∈ ℂ and ∞ CM, then f(z) = t1f(z + c) or f(z) = t2f(z + 2c), where t1 and t2 satisfy $$t_i^n=1$$ , (i = 1, 2). Some examples are provided to show the sharpness of our results. In addition, we mainly obtain two uniqueness results of f(z) with their n-th order differences Δnf. For example, let f(z) be transcendental entire, and let a(≠ 0) ∈ ℂ. We show that, if f(z) and Δnf(≢ 0) share 0 CM and a IM, then f(z) = Δnf. And this research extends earlier results by Chen et al.

The zeros of f^{n}f^(k)-a and normal families of meromorphic functions

In this paper, we first prove that if f be a non-constant meromorphic function, all of whose zeros have multiplicity at least $k$, then f^{n}f^{(k)}-a has at least m+1 distinct zeros, where $k(\geq2),m(\geq1),n(\geq m+1)$ are three integers, and $a\in \mathbb{C}\cup\setminus\{0\}$.Also, in relation to this result, a normality criteria is given, which extends the related result.

MEROMORPHIC FUNCTIONS SHARING THREE VALUES AND A RESULT OF G. BROSCH

We prove a uniqueness theorem for two non-constant meromorphic functions sharing three values which improves a result of G. Brosch.

The algebraic dimension of compact complex threefolds with vanishing second Betti number

We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.

A Uniqueness Theorem for Meromorphic Functions Concerning Total Derivatives in Several Complex Variables

Motivated by many differences for the total derivative between entire functions and meromorphic functions, we mainly investigate the uniqueness problems for meromorphic functions in several complex variables concerning the total derivatives. Let f and g be two nonconstant meromorphic functions on $$\mathbb {C}^{m},$$ k be a positive integer such that $$f=0\Leftrightarrow g = 0, D^{k}f=\infty \Leftrightarrow D^{k}g=\infty , D^{k}f=1\Leftrightarrow D^{k}g=1.$$ We get that if $$2\delta (0, f)+(k+4)\varTheta (\infty , f)>k+5,$$ then $$\frac{D^{k}f-1}{D^{k}g-1}$$ is a nonzero constant. This is an extension of a uniqueness theorem for entire functions due to L. Jin. There are several examples to show that our result is sharp.

Uniqueness of meromorphic functions of differential polynomials sharing a small function with finite weight

Abstract Let f be a non-constant meromorphic function and {a=a(z)} ( {\not\equiv 0,\infty} ) a small function of f. Here, we obtain results similar to the results due to Indrajit Lahiri and Bipul Pal [Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 2017, 3, 825–838] for a more general differential polynomial by introducing the concept of weighted sharing.

Uniqueness of meromorphic functions sharing two sets with least possible cardinalities

Abstract Let f and g be two nonconstant meromorphic functions sharing two finite sets, namely S ⊂ ℂ and {∞}. We prove two uniqueness theorems under weaker conditions on ramification indices, reducing the cardinality of the shared set S and weakening the nature of sharing of the set {∞} which improve results of Fang-Lahiri [7], Lahiri [17], Banerjee -Majumder-Mukherjee [5] and others.

On the periodicity of meromorphic functions when sharing two sets IM

In this paper, we have investigated the sufficient conditions for periodicity of meromorphic functions and obtained two results directly improving the result of \emph{Bhoosnurmath-Kabbur} \cite{Bho & Kab-2013}, \emph{Qi-Dou-Yang} \cite{Qi & Dou & Yan-ADE-2012} and \emph{Zhang} \cite{Zha-JMMA-2010}. Let $\mathcal{S}_{1}=\left\{z:\displaystyle\int_{0}^{z-a}(t-a)^n(t-b)^4dt+1=0\right\}$ and $\mathcal{S}_{2}=\bigg\{a,b\bigg\}$, where $n\geq 4(n\geq 3)$ be an integer.\emph{ Let $f(z)$ be a non-constant meromorphic (entire) function satisfying $\ol E_{f(z)}(\mathcal{S}_j)=\ol E_{f(z+c)}(\mathcal{S}_j), (j=1,\;2)$ then $f(z)\equiv f(z+c)$.} We have exhibited some examples to show that, it is not necessary that the meromorphic function should be of finite order and also to show that the sets considered in this paper simply can't be replace by some arbitrary sets. At the last section, we have posed an open question for the future research.

Meromorphic functions that share four or three small functions with their difference operators

In this paper, we prove that non-constant meromorphic functions of finite order and their difference operators are identical, if they share four small functions “IM”, or share two small functions and ∞ CM. Our results show that a conjecture posed by Chen–Yi in 2013 is still valid for shared small functions, and improve some earlier results obtained by Li–Yi, Lü et al. We also study the uniqueness of a meromorphic function partially sharing three small functions with their difference operators.

Meromorphic Function Sharing Sets with Its Difference Operator or Shifts

Let f be a nonconstant meromorphic function, c ∊ ℂ, and let a(z)(≢ 0) ∊ S(f) be a meromorphic function. If f(z) and P(z, f(z)) share the sets {a(z), −a(z)}, {0} CM almost and share {∞} IM almost, where P(z, f(z)) is defined as (1.1), then f(z) ≡ ±P(z, f(z)) or f(z)P(z, f(z)) ≡ ±a2(z). This extends the results due to Chen and Chen (2013), Liu (2009) and Yi (1987).