Meromorphic Research Articles (Page 1)

This study introduces a new integral operator Ip,μδ that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses: Σp+δ,μ,α, consisting of functions with nonnegative coefficients, and Σp+δ,μ,α,c, which further fixes the second positive coefficient. For these classes, we establish a necessary and sufficient coefficient condition, which serves as the foundation for deriving a set of sharp results. These include accurate coefficient bounds, distortion theorems for functions and derivatives, and radii of starlikeness and convexity of a specific order. Furthermore, we demonstrate the closure property of the class Σp+δ,μ,α,c, identify its extreme points, and then construct a neighborhood theorem. All the findings presented in this paper are sharp. To demonstrate the practical utility of our symmetric operator paradigm, we apply it to a canonical fractional electrodynamics problem. We demonstrate how sharp distortion theorems establish rigorous, time-invariant upper bounds for a solitary electrostatic potential and its accompanying electric field, resulting in a mathematically guaranteed safety buffer against dielectric breakdown. This study develops a symmetric and consistent approach to investigating the geometric characteristics of meromorphic multivalent functions and their applications in physical models.

Abstract We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre’s conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert’s 17th problem on analytic surfaces.

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this study is to provide an investigation on the argument properties of multivalent meromorphic functions in a punctured open unit disc and to obtain some results for its subclass.

UDC 512.5 We introduce new classes of functions generalizing the well-known classes of functions of complex variable, such as the entire functions, meromorphic functions, rational functions, and polynomial functions, which take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend some recently obtained characterization theorems presented in [B. Q. Li, Amer. Math. Monthly, 122, № 2, 169–172 (2015)] and [B. Q. Li, Amer. Math. Monthly, 132, № 3, 269–271 (2025)] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems generalize a recently established version of the fundamental theorem of algebra presented in [V. M. Abramov, Amer. Math. Monthly, 132, № 4, 356–360 (2025)].

Consider a sequence of meromorphic functions $(f_n)_n$. This paper presents a technique that enables the transfer of convergence properties from $(f_n^{(m+1)}/f_n^{(m)})_n$ to subsequences of $(f_n^{(m)}/f_n^{(m-1)})_n$. As an application, we will show that the families of functions with bounded Schwarzian derivative are quasi-normal.

We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative \(S_k(f)\). We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives and logarithmic derivatives, and prove several (quasi-)normality results. Moreover, we derive a new formula for \(S_k(f)\), which yields a result for families \(F\subseteq H\mathbb{D}\) of locally univalent functions that satisfy\(S_k(f)(z)\neq b(z)\) for some \(b\in M(\mathbb{D})\) and all \(f\in F\), \(z\in\mathbb{C}\)and for entire functions \(g\) with \(S_k(g)(z)\neq 0\) and \(S_k(g)(z)\neq\infty\) for all \(z\in\mathbb{C}\). The classical Schwarzian derivative \(S_f\) is contained as the case \(k=2\).

In this paper, we present an effective two-stage numerical algorithm for the simultaneous finding of all roots of meromorphic functions in a region within the complex plane. At the first stage, we construct a polynomial with the same roots as the ones of the considered function; at the next step, we apply some method for the simultaneous approximation of its roots. To show the efficiency and applicability of our algorithm together with its advantages over the classical Newton, Halley and Chebyshev’s iterative methods, we conduct three numerical examples, where we apply it to two test functions and to an important engineering problem.

Abstract The paper deals with the uniqueness problem of derivatives of meromorphic functions when they share certain values with their shift and difference operators. In the case of shift operators, our result improves and extends all the existing results in this direction. We establish two uniqueness theorems for difference operators that also extend and improve certain theorems in this direction. Furthermore, we exhibit some examples pointing out the sharpness of some of our conditions.

The purpose of this paper is to derive many third-order differential superordination and subordination results

In the paper, we apply the concept of weighted sharing to study the uniqueness problems of differential polynomial of meromorphic function of zero order with its $q$-shift. The results of the paper improve and extend some recent results due to H. P. Waghamore and M. M. Manakame [Int. J. Open Problems Compt. Math., 18 (2025), 22-34]. A typical theorem obtained in the paper is as follows: Let $P$ be a polynomial, $f(z)$ be a non-constant meromorphic function of zero-order. Suppose that $q$ is a non-zero complex constant, $\eta \in \mathbb{C}$ and $n$ is an integer satisfying $n\geq m+3\tau+3\Omega +6$, where $m=\deg P$, $\tau =\sum_{j=1}^{s}\mu _{j}$ and $\Omega =\sum_{j=1}^{s}j\mu _{j}.$ If $f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}$ and $f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}$ share $(1,2)$ and $(\infty,\infty)$, then $\displaystyle f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}\equiv f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}.$ Three other similar theorems are also obtained in the paper.

UDC 514.7 We establish the second main theorems for meromorphic mappings of complete Kähler manifolds into complex projective spaces with hyperplanes in the general position. This is a continuation of the works by Atsuji [J. Math. Soc. Japan, 60, № 2, 471–493 (2008)] and Dong [J. Inst. Math. Jussieu, 1–29 (2022)]. We extend the results of these works to targets of higher dimension. As an application, we show that every meromorphic mapping of a complete Kähler manifold of nonpositive Ricci curvature and maximal volume growth into complex projective spaces with small growth, as compared to a half of the traceless Ricci curvature of the manifold, must be constant.

Abstract We consider the series $$\sum _{n=1}^{\infty } z^{n} (a_{n} + x)^{-s}$$ ∑ n = 1 ∞ z n ( a n + x ) - s where $$\{a_{n}\}$$ { a n } satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex s-plane. Thus we may associate a Lerch-type zeta function $$\varphi (z,s,x)$$ φ ( z , s , x ) to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees 2 and 3. Our method generalizes a formula of Ramanujan for the classical Hurwitz–Riemann zeta functions. We determine the poles and residues of $$\varphi $$ φ , which turn out to be polynomials in x. In addition we study the dependence of $$\varphi (z,s,x)$$ φ ( z , s , x ) on x and z, and its properties as a function of three complex variables.

In this investigation, we introduce a new meromorphic operator defined by meromorphic univalent functions. A new class of meromorphic functions is introduced by this operator, which can generate several distinct subclasses depending on the values of its parameters. Within the framework of this class of functions, we obtain several significant algebraic and geometric properties, including coefficient estimates, distortion theorems, the radius of starlikeness, convex combination closure, extreme point characterization, and neighborhood structure. Our findings are sharp, offering accurate and significant insights into the mathematical structure and behavior of these functions. In addition, we present several applications of these results in fluid mechanics, like identifying stagnation points in vortex flows, predicting velocity decline in source/sink systems, and determining stability thresholds that protect crucial streamlines from perturbations, which demonstrates that the introduced operator and class characterize critical properties of 2D potential flows.

UDC 517.5 We present some results on the asymptotic paths and asymptotic values for meromorphic functions in the $n$-fold extended punctured plane $\widehat{\mathbb{C}}\backslash\{p_{1},\ldots,p_{n}\} .$ These results extend some classical results obtained for analytic and meromorphic functions in the complex plane $\mathbb{C}.$ In particular, in this more general setting, we give a version of F. Iversen's result on the existence of asymptotic values for entire functions in the plane. We also obtain a bound for the number of isolated directly critical singularities of a meromorphic function of finite order $k$ and a finite number of poles.

This paper extends the investigation into the uniqueness problems concerning meromorphic functions and their differences or difference polynomials and explores the conditions under which two transcendental meromorphic functions 𝑓 (𝑧) and 𝑔(𝑧), with hyper-order less than one, they share certain values or small functions. For instance, 𝑓 (𝑧)𝑛 and share common values together with 𝑔(𝑧)𝑛 and share common values. Moreover, we address scenarios where 𝑓 (𝑧)𝑛 and 𝑔(𝑘)(𝑧 + 𝑐) share values 𝐼𝑀 together with 𝑔(𝑧)𝑛 and 𝑓 (𝑘)(𝑧 + 𝑐) share values 𝐼𝑀, leading to conclusions about the relationship between the functions.

Truncated Sharing of Subsets and Uniqueness of Meromorphic Functions in a Non-Archimedean Field

There are some uniqueness problems with meromorphic functions with difference operators that we looked into in this paper. We looked at them in the light of partial sharing. Specifically, we have obtained two uniqueness results by considering sharing and partial sharing of small functions. In the first theorem and shares CM, whereas in the second theorem and partially share CM. 2010 AMS Classification: 30D35, 30D45 Keywords and phrases: Uniqueness, Meromorphic function, Difference operator, Small function, Partial sharing.

Abstract Let F ( s ) F\left(s) be a function from the extended Selberg class. We consider decompositions F ( s ) = f ( h ( s ) ) F\left(s)=f\left(h\left(s)) , where f f and h h are meromorphic functions. Among other things, we show that F F is prime if and only if the greatest common divisor of the orders of all zeros and the pole of F F is 1.

Abstract In this paper, a continuum mechanics model of graphene is proposed, and its analytical solution is derived. Graphene is modeled as a doubly-periodic thin elastic plate with a hexagonal cell having a circular hole at the hexagon center. Graphene is characterized by a general chiral vector and is subject to remote tension. For the solution, the Filshtinskii solution obtained for the symmetric case is generalized for any chirality. The method uses the doubly-periodic Kolosov–Muskhelishvili complex potentials, the theory of the elliptic Weierstrass function and quasi-doubly-periodic meromorphic functions and reduces the model to an infinite system of linear algebraic equations with complex coefficients. Analytical expressions and numerical values for the stresses and displacements are obtained and discussed. The displacements expressions possess the Young modulus and Poisson ratio of the graphene bonds. They are derived as functions of the effective graphene moduli available in the literature.

Let f be an entire function of finite exponential type less than or equal to \sigma which is bounded by 1 on the real axis and satisfies f(0) = 1 . Under these assumptions, Hörmander showed that f cannot decay faster than \cos(\sigma x) on the interval (-\pi/\sigma,\pi/\sigma) . We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite–Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently, in model spaces generated by meromorphic inner functions), generalizing some recent results of Brevig, Chirre, Ortega-Cerdà, and Seip.