Search

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]

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 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 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 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 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]

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]

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]

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.

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.

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]

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]

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.

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].

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 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]

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 complex analysis, order and type are classical growth indicators of entire and meromorphic functions. During the past decades, several authors have made the close investigations on the properties of entire and meromorphic functions in different directions using the concepts of order, iterated p-order [8, 11], (p, q)-th order [6, 7], (p, q)-φ order [10] and achieved many valuable results. But in [3], Chyzhykov et al. showed that both definitions of iterated porder and (p, q)-th order have the disadvantage that they do not cover arbitrary growth (see [3, Example 1.4]). They used more general scale, called the φ-order (see [3]). On the other hand, Heittokangas et al. [4] have introduced another new concept of φ-order of entire and meromorphic functions considering φ as subadditive function. Considering all these aspects, Belaıdi et al. [1, 2] have extended the above ideas and have introduced the definitions of (α, β, γ)- order and (α, β, γ)-type of entire and meromorphic functions. In this paper, we establish the integral representations of (α, β, γ)-type and (α, β, γ)-weak type of a meromorphic function. We also investigate their equivalence relation under some certain conditions. [ABSTRACT FROM AUTHOR]

We investigate the uniqueness problem of entire functions that share two polynomials with their kth 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. [ABSTRACT FROM AUTHOR]

We obtain new integral formulas for holomorphic, meromorphic, subharmonic and -subharmonic functions on a concentric annulus. These formulas relate integrals from such functions to their distributions of zeros and poles, as well as Riesz measures and charges, respectively. Our new formulas are applied to obtaining inequalities for such functions and to new completeness conditions for exponential systems in Banach spaces of functions on a convex compact set in the complex plane. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the uniqueness problem of derivatives of a meromorphic function and its difference polynomial in view of a partially sharing. As a consequence of the main result, we improve the recent result of W. J. Chen and Z. G. Huang with the weaker hypotheses and also supplement several results in particular cases. [ABSTRACT FROM AUTHOR]

We mainly study the possible value distribution and uniqueness results of the general differential-difference polynomials of transcendental entire functions with finite order when they share a small function under weighted sharing and weakly weighted sharing hypotheses. These are significant generalization of earlier results. As a very special case, we obtain the results of Majumder et al. (A note on the uniqueness of certain types of differential-difference polynomials, Ukrainian Mathematical Journal, 73(5), 791-810(2021)). We fortify some examples to claim that the used conditions are best possible in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we intend to find out some inequalities of generalized relative order (α, β), generalized relative type (α, β) and generalized relative weak type (α, β) of a meromorphic function f with respect to an entire function g when generalized relative order (γ, β), generalized relative type (γ, β) and generalized relative weak type (γ, β) of f and generalized relative order (γ, α), generalized relative type (γ, α) and generalized relative weak type (γ, α) of g with respect another entire function h are given, where α, β, γ are slowly increasing functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniqueness of meromorphic functions of finite order concerning their difference operators and derivatives and prove that if and share , , CM, where and are two distinct polynomials, then they assume one of following cases: ; reduces to a polynomial and , where is a nonzero constant and are all constants. This generalizes the corresponding results due to Qi et al. and Deng et al. [ABSTRACT FROM AUTHOR]

A new subclass of meromorphic univalent functions by using the qhypergeometric and Hurwitz-Lerch Zeta functions is defined. Also, by applying the generalized Liu-Srivastava operator on meromorphic functions, some geometric properties of the new defined subclass such as coefficient estimates, extreme points, convexity and connected set structure are investigated. [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}.

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.

Geometric function theory (GFT) is one of the richest research disciplines in complex analysis. This discipline also deals with the extended differential inequality theory, known as the differential subordination theory. Based on these theories, this study focuses on analyzing intriguing aspects of the geometric subclass of meromorphic functions in terms of a linear complex operator and a special class of Hurwitz-Lerch-Zeta functions. Hence, several of its geometric attributes are deduced. Furthermore, the paper highlights the different fascinating advantages and applications of various new geometric subclasses in relation to the subordination and inclusion theorems.