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In this paper, we prove an extension of Hadamard's classical theorem for determining the radius of mmeromorphy of an analytic function in terms of its Taylor coefficients. Our extension is expressed in terms of Fourier coefficients with respect to an orthonormal polynomial system on the unit circle. Our main result confirms a conjecture posed in [Dolomit Res Notes Approx 17 (2024):12-21]. [ABSTRACT FROM AUTHOR]

In this paper, we study unicity of meromorphic functions whose lower order is finite and noninteger and mainly prove: Let f and g be two nonconstant meromorphic functions, let n ≥ 6 be an integer, S = {z...-1 = 0}. If f and g share S, ∞ CM, and the lower order of f is finite and noninteger, then f ≡ g. This answers a question posed by Gross for meromorphic functions whose lower order is finite and noninteger. [ABSTRACT FROM AUTHOR]

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study coefficient bounds of a class of meromorphic functions involving higher order derivative. Several similar inequalities are also established for the inverse functions of the considered class of meromorphic functions. [ABSTRACT FROM AUTHOR]

We present the binary Bargmann symmetry constraint and algebro-geometric solutions for a semidiscrete integrable hierarchy with a bi-Hamiltonian structure. First, we derive a hierarchy associated with a discrete spectral problem by applying the zero-curvature equation and study its bi-Hamiltonian structure. Then, resorting to the binary Bargmann symmetry constraint for the potentials and eigenfunctions, we decompose the hierarchy into an integrable symplectic map and finite-dimensional integrable Hamiltonian systems. Moreover, with the help of the characteristic polynomial of the Lax matrix, we propose a trigonal curve encompassing two infinite points. On this trigonal curve, we introduce a stationary Baker–Akhiezer function and a meromorphic function, and analyze their asymptotic properties and divisors. Based on these preparations, we obtain algebro-geometric solutions for the hierarchy in terms of the Riemann theta function. [ABSTRACT FROM AUTHOR]

The existence of entire solutions of certain types of nonlinear homogeneous differential equations are investigated. We also find explicit forms of entire solutions of two types of nonlinear monomial differential equations, which are connected to trigonometric identities. Some further questions about nonlinear homogeneous differential equations are posed. [ABSTRACT FROM AUTHOR]

In the present work, the authors introduce a new linear differential operator associated with $ q $-calculus which uses the $ q $-derivative operator. For the new class involving the linear differential operator, interesting properties like coefficient inequality, growth, distortion, convolution and radii of starlikeness are established. Various connections of the present class with that of the earlier works are pointed out. [ABSTRACT FROM AUTHOR]

This paper deals with the extension of Hayman Conjecture to q-shift differential polynomials generated by meromorphic functions. We prove that the differential polynomials assumes every finite value infinitely often when sharing two Borel exceptional Values with appropriate conditions. Few examples are given to prove that the conditions are inevitable. [ABSTRACT FROM AUTHOR]

In this article, we study the uniqueness problem of meromorphic functions sharing two finite sets of values. The results proved in this paper significantly extend and also improve in some sense the results of Chen [3] and Chen [2], which were actually proved as an answer of a generalised version of a question asked in [9]. Some examples have also been provided in support of the accuracy of certain conditions used in the paper. [ABSTRACT FROM AUTHOR]

In this study, a broader class of rational functions r(u(z)) of degree mn, where u(z) is a polynomial of degree m is taken into consideration and obtain certain sharp compact generalization of well-known inequalities for rational functions. [ABSTRACT FROM AUTHOR]

A family of meromorphic functions must be normal if every function in the family shares a finite set of at least 3 elements with its derivatives, whereas a family of meromorphic functions that shares a set of infinite number of elements is usually not normal. In this paper, we prove a normal criterion of family of meromorphic functions for sharing infinite set under some additional constraints. Let F be a family of meromorphic functions on a region D, S be a bounded infinite set of complex numbers with at least 3 isolated points, if on the region D, for each f ∈ F, there is fS f S ∈⇔ ∈′, then F is normal on the region D. Based on further analysis, this paper speculates that if the aforementioned "bounded" condition of S is changed to "the distance between S and the origin is greater than 0, that is dist(,0) 0 S > ", then F will still be normal. [ABSTRACT FROM AUTHOR]

This paper delves into the uniqueness of finite-order meromorphic functions f(z) and g(z) over the extended complex plane, particularly when these functions share a small function a(z) under specific conditions. The study reveals new insights with significant applications, such as classifying different complexes within C based on their uniqueness. The primary goal is to explore the uniqueness of meromorphic functions that share a small function a(z) in the sense of IM (ignoring multiplicities) while constrained by finite order, alongside certain types of differential-difference polynomials. We focus on two non-constant meromorphic functions f(z) and g(z) of finite order, under the assumption that a small function a(z), relative to f(z), plays a crucial role in the analysis. The investigation centers on the uniqueness properties of a specific type of differential-difference polynomial of the form [fnP[f]H(z, f)], where P[f] is a differential polynomial of f(z) and H(z, f) is a difference polynomial of f(z), both defined in the equations (2) and (3), respectively. Importantly, these polynomials do not vanish identically and do not share common zeros or poles with either f(z) or g(z). The paper establishes conditions on several parameters, including k, n, d(P), Ψ, Q, t, and ξ, under which the shared value properties between f(z) and g(z) lead to two possible outcomes: either f(z) is a constant multiple of g(z), or f(z) and g(z) satisfy a specific algebraic difference equation. This result contributes to a deeper understanding of the relationship between shared values and the structural properties of meromorphic functions. As an application, the paper extends several previous results on meromorphic functions, including those by Dyavanal and M. M. Mathai, published in the Ukr. Math. J. (2019). Furthermore, by citing a particular example, we demonstrate that the established results hold true only under specific cases, highlighting the precision of the theorem. Finally, we offer a more compact version of the main theorem as an enhancement, providing a refined perspective on the uniqueness problem in the context of meromorphic functions. [ABSTRACT FROM AUTHOR]

In the paper, we study the uniqueness of meromorphic function with few poles sharing two pairs of small functions IM with its th derivatives and obtain two results which improve as well as generalize the recent result due to Huang et al. [1] in a large extend. [ABSTRACT FROM AUTHOR]

In this article, we study the growth of solutions of the homogeneous complex linear differential equation f(k) + Ak-1(z)f(k-1) + ··· + A1(z)f' + A0(z)f = 0, where the coefficients Aj(z) (j = 0, 1,..., k-1) are analytic or meromorphic functions in C\{z0}. Under the sufficient condition that there exists one dominant coefficient by its logarithmic lower order or by its logarithmic lower type, we extend some precedent results due to Liu, Long, and Zeng and others. [ABSTRACT FROM AUTHOR]

In this present investigation, for functions of the form f(z) = = 1/zp + ∑∞ k=1−p akz k unit disk U∗ = {z ∈ C : 0 < |z| < 1} we introduce a new subclass of meromorphically p-valent functions and investigate convolution properties, Coefficient estimates and contianment for this subclass. [ABSTRACT FROM AUTHOR]

In this paper, with the help of the notion of multiplicity, we study the uniqueness of differential polynomials that share two values and obtain some results that improve the results of Wang and Gao [8], Yang and Hau [11], and Fang and Qiu [2]. We also solve an open problem posed by Harina P. Waghamore and Ramya Maligi [9] [ABSTRACT FROM AUTHOR]

In this paper, we scrutinize a collection of meromorphic functions known as normal families, prove the theorem that normal families share a holomorphic function, and present several illustrative counterexamples. [ABSTRACT FROM AUTHOR]

The classical growth indicators of entire and meromorphic functions are order and type, which are generalized by several authors during the past decades. Chyzhykov et al. have first introduced the generalized growth scale, namely the φ-order (see [3]) taking φ as an increasing unbounded function. But, Heittokangas et al. [5] have introduced another new concept of φ-order of entire and meromorphic functions considering φ as subadditive function. Later, Bela¨ıdi et al. [1] have extended the above ideas and have introduced the definition of (α, β, γ)-order of entire and meromorphic functions, where α ∈ L¹-class, β ∈ L2-class, γ ∈ L3-class. In this paper, our motive is to develop the integral representations of (α, β, γ)- order and (α, β, γ)-lower order of a meromorphic function. We also investigate their equivalence relation under some certain conditions. [ABSTRACT FROM AUTHOR]

The paper proves a special case of a conjecture of Hellerstein, Shen, and Williamson concerning non-real zeros of derivatives of real meromorphic functions. [ABSTRACT FROM AUTHOR]

It is known that if f is holomorphic in the open unit disc D of the complex plane and if, for some c > 0, ∣f(z)∣ ⩽ 1/(1−∣z∣2)c, z ∈ D , then ∣f′(z)∣ ⩽ 2(c+1)/(1−∣z∣2)c+1. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in D. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class. [ABSTRACT FROM AUTHOR]

The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing. [ABSTRACT FROM AUTHOR]

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work's assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination. [ABSTRACT FROM AUTHOR]

In the paper, we discuss the uniqueness problem of meromorphic function when shares , and CM with , where and are two distinct finite values. The obtained result improves the recent result of Qi et al. [6] by dropping the condition "order of growth of is not an integer or infinite" by " ". Also in the paper we give an affirmative answer of the question raised by Wei et al. [9]. [ABSTRACT FROM AUTHOR]

In this article, we address the uniqueness problem of Selberg class L-functions in relation to a question posed in a previous work by Khoai (Ann Polon Math 126:265-278, 2021). The focus of our investigations is to determine the uniqueness of Selberg class L-functions when considered alongside an arbitrary meromorphic function f. The current study expands upon and generalizes the scope of a previous study presented in the paper Khoai (Ann Polon Math 126:265-278, 2021). Furthermore, to provide an answer to the same question that was initially raised in Khoai (Ann Polon Math 126:265-278, 2021), we conveniently tackle a previously unexplored aspect of the uniqueness problem in the context of Selberg class L-functions in Khoai (Ann Polon Math 126:265-278, 2021). [ABSTRACT FROM AUTHOR]

Our paper introduces a new linear operator using the convolution between a Mittag Leffler Function and basic hypergeometric function. Use of the linear operator creates a new class of meromorphic functions defined in the punctured open unit disk. Consequently, the paper examines different aspects Apps and assets like, extreme points, coefficient inequality, growth and distortion. In conclusion, the work discusses modified Hadamard product and closure theorems. [ABSTRACT FROM AUTHOR]

We investigate the uniqueness problem of entire functions that share two polynomials with their $k$th derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible.

In this article, we prove two results on the value distribution of meromorphic functions. Using the theorem of Yamanoi, the first result gives a precise estimation of the relationship between the characteristic function of a meromorphic function and its kkth derivative in a concise form. This result extends and improves some results of Shan, Singh, Gopalakrishna, Edrei, Weitsman, Yang, Wu and Wu, etc. The second result answers a conjecture posed by C. C. Yang. This conjecture turned to be false by a counter-example, but it will be true with an additional condition.

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory.

In this paper, we give some results about of the growth of transcendental meromorphic solutions of linear differential equations f(k) + Σj=0k-1 (Dj (z)+Aj(z) ePj(z)+Bj eQj(z)) f(j)= F, where Aj(z) (≠0), Bj (z) (≠0) and Dj (z) (j = 0,1,...,k-1) are meromorphic functions with finite order. Pj (z) = aj,nzn +...+ aj,o and Qj (z) = bj,nzn +...+bj,o (j=0,...,k-1) are nonconstant polynomials, aj,o.....aj,n, bj,o,....bj,a (j = 0,1,...,k-1) are complex numbers. Under some conditions, we prove that the hyper-order of every transcendental meromorphic solution of the above equation and of its corresponding homogeneous equation coincides with n. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set. [ABSTRACT FROM AUTHOR]

We first investigate the meromorphic solutions of a class of homogeneous second-order q-difference equations and the uniqueness problem for a meromorphic function with three shared values; then we discuss the uniqueness problem for the meromorphic solutions of a class of nonhomogeneous q-difference equations and a meromorphic function with four shared values. [ABSTRACT FROM AUTHOR]

Let f and g be two nonconstant meromorphic functions, and n be a positive integer. If fn(z) and gn(z) share 1 CM (counting multiplicities), f(z) and g(z) share ∞ IM (ignoring multiplicities), and ... + ... = S(r,f) + S(r,g), then either f(z) = tg (z) or f(z)g(z)=t, where tn = 1. As an application, shared values problems of a meromorphic function related to its shifts and difference operators are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study unicity of meromorphic functions concerning differential-difference polynomials and mainly prove: Let be nonnegative integers and max , let be the number of distinct values of , let be the number of distinct values of , let be a nonconstant meromorphic function of finite order satisfying , let be small functions of such that , let be distinct and let . If and share CM, then . Our results improve and extend some results due to [1, 18, 20]. [ABSTRACT FROM AUTHOR]

In this paper, we have dealt with the uniqueness problem of a general meromorphic function with function in terms of two shared sets. In our main theorem, we deal with general meromorphic functions instead of meromorphic functions having finitely many poles. As a corollary of our main theorem, we have shown that our result not only fills the gap of some theorems of [3] and [1] for but also reduces the cardinality of the main range set and hence our result significantly improves all the results in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we study the value distribution of $\mathcal{L}$-function in the extend Selberg class and a non-constant transcendental meromorphic $\mathsf{f}$ function with finitely many zeros of finite order, sharing a polynomial with its difference monomial. Meromorphic functions and $L$-functions are central to Nevanlinna theory, a branch of complex analysis focusing on the distribution of zeros and poles of analytic functions. The Meromorphic functions are unique due to their combination of meromorphicity and analyticity. They are defined on the entire complex plane except for isolated poles in the complex analysis. On the other hand, $L$-functions, arising notably in number theory and automorphic forms, exhibit unique properties related to their zeros and poles. One classic example of a prime number distribution is the Riemann $\zeta$-function. In this paper, we analyse the uniqueness results between a non-constant meromorphic function $\mathsf{f}$ having finitely many zeros and $L$-function, when their difference monomial $\mathsf{f}^{n}\prod\limits_{j=1}^{s}\mathsf{f}(z+c_{j})^{\mu_{j}}$ and $\mathcal{L}^{n}\prod\limits_{j=1}^{s}\mathcal{L}(z+c_{j})^{\mu_{j}}$ share a non-zero polynomial $p$. Our results extends and improves the earlier results of Harina P. Waghamore and Manjunatha B E\cite{bib6}.

We first investigate the meromorphic solutions of a class of homogeneous second-order q-difference equations and the uniqueness problem for a meromorphic function with three shared values; then we discuss the uniqueness problem for the meromorphic solutions of a class of nonhomogeneous q-difference equations and a meromorphic function with four shared values.

We study unique range sets of meromorphic functions over an angular domain in the light of weighted sharing. One of our main results generalizes and improves a result of Xu et al. (2014). Most importantly, we have pointed out a gap in the proofs of some main results of Rathod (2021) and subsequently rectifying the gap we have conveniently improved the results.

In this article, we have mainly focused on the uniqueness problem of an \(L\)-function \(\mathcal{L}\) with an \(L\)-function or a meromorphic function \(f\) under the condition of sharing the sets, generated from the zero set of some strong uniqueness polynomials. We have introduced two new definitions, which extend two existing important definitions of URSM and UPM in the literature and the same have been used to prove one of our main results. As an application of the result, we have exhibited a much improved and extended version of a recent result of Khoai-An-Phuong [23]. Our remaining results are about the uniqueness of \(L\)-function under weighted sharing of sets generated from the zeros of a suitable strong uniqueness polynomial, which improve and extend some results in [12].

In this paper, we scrutinize a collection of meromorphic functions known as normal families, prove the theorem that normal families share a holomorphic function, and present several illustrative counterexamples.