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For systems of linear differential equations whose dimension can be lowered, we estimate the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of the coefficients of the system.

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish., Comment: Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions

This paper is devoted to studying the growth of solutions of $ f''+A(z)f'+B(z)f = 0 $, where $ A(z) $ and $ B(z) $ are meromorphic functions. With some additional conditions, we show that every non-trivial solution $ f $ of the above equation has infinite order. In addition, we also obtain the lower bound of measure of the angular domain, in which the radial order of $ f $ is infinite.

The arithmetic, p-arithmetic, and incoherent intersections between pairs of closed geodesics on a modular or Shimura curve are defined, and some of their expected algebraicity and factorisation properties are examined. These properties follow in a special case from the conjectures of [DV] on the RM values of rigid meromorphic cocycles, and are inspired from the recent work of James Rickards [Ri] and Xavier Guitart, Marc Masdeu and Xavier Xarles [GMX] in the quaternionic setting.

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation. In the second part, we prove a generalization that accommodates certain non-spherical spaces with spherical quotients. As an application, the resulting theory applies to the prehomogeneous vector space underlying Bhargava's cubes, which is also considered by F. Sato and Suzuki–Wakatsuki in their study of toric periods.

In this paper, we introduce a class of meromorphic starlike function by subordination relationship and generalized $\lambda$-Hadamard product. We obtain the necessary and sufficient conditions and closure properties of the class. In addition, some new results of the class are given.

We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality 3 that are unavoidable with respect to the set of meromorphic functions on $\mathbb C$ . We further obtain families consisting of less than three functions that are unavoidable with respect to certain subsets of meromorphic functions. In the other direction, we show that for every meromorphic function f, there exists an entire function that avoids f on $\mathbb C$ .

This paper is related to fuzzy differential subordinations for meromorphic functions. Fuzzy differential subordination results are obtained using a new operator which is the combination Hadamard product and integral operator for meromorphic function.

The outcome of the research presented in this paper is the definition and investigation of two new subclasses of meromorphic functions. The new subclasses are introduced using a differential operator defined considering second-order differential polynomials of meromorphic functions in U\{0}=z∈C:0

A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.

In [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions F in a domain D⊂C, and for a positive constant e, if for each f∈F there exist meromorphic functions af,bf,cf such that f omits af,bf,cf in D and min{ρ(af(z),bf(z)),ρ(bf(z),cf(z)),ρ(cf(z),af(z))}≥e for all z∈D, then F is normal in D. Here, ρ is the spherical metric in Cˆ. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc △:={z∈C:|z

We give a lower bound for the Hausdorff dimension of the support of the Green current associated with some meromorphic maps.

Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $\varphi(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-\varphi(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-\varphi(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq1)$ be positive integers. If for each $f\in \mathscr{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal on domain $D$., Comment: 8 pages

In this paper, in the light of weighted sharing of sets, we investigate the possible uniqueness of meromorphic function of restricted hyper order with its linear c-shift operator. Our first two theorems improve a number of earlier results. Our last theorem together with a corollary improves and extends a result due to Li, Lu and Xu [14]. Most importantly, our another corollary deducted from the last theorem not only provides an answer to an open question posed by Liu [16] but also noticeably improves two results of Chen and Chen [4]. A number of examples have been exhibited by us pertinent with the content of the paper. At the penultimate section which is also the application part of our result, we extend a recent result of Liu, Ma and Zhai [17]. Finally, presenting two examples, we conjecture that one of our result may hold for a larger class of functions and we place it as an open question for future investigations.

As a starting point for higher-dimensional patching in the Berkovich setting, we show that this technique is applicable around certain fibers of a relative Berkovich analytic curve. As a consequence, we prove a local–global principle over the field of overconvergent meromorphic functions on said fibers. By showing that these germs of meromorphic functions are algebraic, we also obtain local–global principles over function fields of algebraic curves defined over a class of (not necessarily complete) ultrametric fields, thus generalizing the results of Mehmeti(Compos Math 155:2399–2438, 2019).

Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by Andre in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor, which measure irregularity along divisors on blowups of the original space; this generalizes another result of Andre on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties., 27 pages; v4: refereed version; some technical edits in 2.2

Taking into account of all the previous results, in this paper, we first resolve the “2 CM+1 IM” small functions sharing issue of a meromorphic function of restricted hyper order and its linear shift delay differential operator. We answer two open questions addressed by Qi-Yang (Analysis Math 46(4):843–865, 2020) in a convenient manner. Our second result, related to “2 IM” small function sharing problem of the same system, improves a result of (Analysis Math, 46(4):843–865, 2020). Moreover, concerning shift operator, we have pointed out a shortcoming in the sharing conditions in an example exhibited by Charak-Korhonen-Kumar (J Math Anal Appl 435(2):1241–1248, 2016). In an attempt to resolve the issue, we have solved the “2 IM+1 CM” small functions sharing problem which in turn extend a result of Li-Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

UDC 517.9 We study the uniqueness problems of certain types of differential-difference polynomials sharing a small function.In this paper, we not only solve the open problem occurred in [A. Banerjee, S. Majumder, On the uniqueness of certain types of differential-difference polynomials, Anal. Math., 43, № 3, 415-444 (2017)], but also present our main results in a more generalized way.

In this paper, we introduce certain subclasses of meromorphic harmonic univalent functions, which are defined by using generalized (p, q)-post quantum calculus operators as well as subordination relationship. Sufficient coefficient conditions, extreme points, distortion bounds and convolution properties for functions belonging to the subclasses are obtained.

"Two meromorphic functions f and g having the same set of poles are known as equipolar. In this paper we study some uniqueness results of equi-polar meromorphic functions sharing a finite set and improve some recent results of Bhoosnurmath-Dyavanal [4] and Banerjee-Mallick [3] by removing some unnec- essary conditions on rami cation indices as well as relaxing the condition on the nature of sharing of the value infinity by f and g from counting multiplicity to ignoring multiplicity."

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.

The purpose of the paper is to rectify a series of errors occurred in (Banerjee and Kundu in Lithuanian Math J 41: 379–392, 2020, Sahoo and Sarkar in Ann Alexandru Ioan Cuza Univ Math 66:81–92, 2020, Yuan et al. in Lithuanian Math J 58:249–262, 2018) for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set (Frank and Reinders in Complex Var Theory Appl 37:185–193, 1998) to settle a long time unsolved problem.

UDC 517.5 The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically starlike functions defined here by means of a certain class of -derivative (or -difference) operators. The familiar concept of neighborhood for meromorphic functions are also considered. Moreover, by using a Ruscheweyh-type -derivative operator, we define and study another new class of functions emerging from the class of normalized meromorphic functions.

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors

In this paper, using definition of subordination and Mittag-Leffler function we introduce classes of p-valent meromorphic functions and obtain some subordination results for these classes.

The φ-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of me...

In this article, we study the relationship between solutions and their arbitrary-order derivatives of the higher order non-homogeneous linear differential equation \begin{document}$ \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) \end{equation*} $\end{document} in the unit disc $ \bigtriangleup $ with analytic or meromorphic coefficients of finite $ [p, q] $-order. We obtain some oscillation theorems for $ f^{(j)}(z)-\varphi(z) $, where $ f $ is a solution and $ \varphi(z) $ is a small function.

We consider Dirac operators with dislocation potentials on the line. The dislocation potential is a periodic potential for x 0 and the same potential but shifted by t ∈ R for x > 0 . Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: eigenvalues or resonances. These poles are called states and there are no other poles. We prove: 1) states are continuous functions of t, and we obtain their local asymptotics; 2) for each t states in the gap are distinct; 3) states can be monotone or non-monotone functions of t; 4) we construct examples of operators with different types of states in gaps.

The major target of this paper is to study a confirmed class of meromorphic univalent functions . We procure several results, such as those related to coefficient estimates, distortion and growth theorem, radii of starlikeness, and convexity for this class, n additionto hadamard product, convex combination, closure theorem, integral operators, and neighborhoods.

The uniqueness problems of meromorphic functions of differential polynomials sharing value have been studied and many results on this topic have been obtained. In this paper, we investigate the uniqueness problems sharing one small function of meromorphic functions concerning differential polynomials. In particular, the results of the paper improve the results due to Zhang and Wu (J Inequal Appl 2019: 1–14, 2019).

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples to show that these results are optimal. As a byproduct, we also show that analytic dependence implies algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax–Schanuel theorem will be discussed.

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $$D\subset {\mathbb {C}}^m$$ into $${\mathbb {P}}^n$$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto’s work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of $${\mathbb {P}}^n$$ -valued holomorphic mappings.

In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class.

By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form \begin{document}$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $\end{document} \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $\end{document} and \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $\end{document} are investigated, where $ n\geq 2 $ is an integer, $ P_{d}(z, f) $ is a difference polynomial in $ f $ of degree $ d\leq n-1 $ with small coefficients, $ u(z) $ is a non-zero polynomial, $ v(z) $ is a non-constant polynomial, $ \lambda, p_{j}, \alpha_{j}\; (j = 1, 2) $ are non-zero constants. Some examples are also presented to show our results are best in certain sense.

In this paper we formulate classical theorems Plesner and Meyer on the boundary behavior of meromorphic functions and their refinement and strengthening - Gavrilov's and Kanatnikov's theorems. An application of these theorems to classes of meromorphic functions with integrable spherical derivative and annular holomorphic functions is presented. Collingwood's theorem on boundary singularities of the Tsuji function as well as Kanatnikov's theorems are formulated. Kanatnikov's theorems strengthen and generalize Collingwood's theorem to broader classes of meromorphic functions with summable spherical derivatives. Special attention is paid to the boundary properties of annular holomorphic functions. The behavior of annular holomorphic functions on the boundary of the unit circle is considered. It is shown that Gavrilov's P-sequences play an important role in the study of the boundary properties of holomorphic and meromorphic functions.

Define the differential operators ϕn for n ∈ ℕ inductively by ϕ1 [f](z)= f (z) and $${\phi _{n + 1}}[f](z) = f(z){\phi _n}[f](z) + {d \over {dz}}{\phi _n}[f](z)$$ . For a positive integer k ≥ 2 and a positive number δ, let $${\cal F}$$ be the family of functions f meromorphic on domain D ⊂ ℂ such that ϕk[f](z) ≠ 0 and ∣Res(f, a) − j∣ ≥ δ for all j ∈{0, 1,…,k − 1} and all simple poles a of f in D. Then $${\cal F}$$ is quasi-normal on D of order 1.

This paper concerns the existence and precise expression form of entire solutions to a certain type of delay-differential equation. The significance of our results lie in that we generalize and supplement the related results obtained recently.

The generalized exponential function in a complex domain is called the Mittag-Leffler function (MLF). The implementations of MLF are significant in diverse areas of science. Over the past few decades, MLF and its analysis with generalizations have become an increasingly rich research area in mathematics and its allied fields. In the geometric theory of meromorphic functions, the main contribution to this discipline of study is to enrich areas of operator theory on complex punctured domains and differential complex inequalities, namely, subordination theory. This effort presents integrodifferential operator of meromorphic functions in the punctured unit disk. It is formulated by combining the differential operator and the integral operator correlating with the extended generalized Mittag-Leffler function. Furthermore, some interesting geometric features in terms of the subordination principle are investigated.

In this article, the ideas of post-quantum calculus and meromorphic multivalent functions are combined and some applications of these functions are discussed. We introduce a new subclass of meromorphic multivalent functions in association with Janowski domain. We investigate and study some useful geometric properties of this class of functions such as sufficiency criteria, distortion problem, growth theorem, radii of starlikeness and convexity, convex combination, and coefficient estimates for this class.

Let f be a transcendental meromorphic function on $\mathbb {C},$ k be a positive integer, and $Q_{0},Q_{1},\dots ,Q_{k}$ be polynomials in $\mathbb {C}[z]$ . In this paper, we will prove that the frequency of distinct poles of f is governed by the frequency of zeros of the differential polynomial form $Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})$ in f. We will also prove that the Nevanlinna defect of the differential polynomial form $Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})$ in f satisfies $$ \sum\limits_{a\in\mathbb{C}}\delta\left( a,Q_{0}(f)Q_{1}(f^{\prime}){\dots} Q_{k}\left( f^{(k)}\right)\right)\leq 1$$ with suitable conditions on k and the degree of the polynomials. Thus, our work is a generalization of Mues’s conjecture and Goldberg’s conjecture for the more general differential polynomials.

In this paper, we investigate the uniqueness problem of difference-differential polynomials of meromorphic functions sharing a small function IM. Consequently, we prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J. 71(7):1032–1042, 2019), and Zhang and Xu (Comput. Math. Appl. 61:722–730, 2011).

Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.

With the idea of normal family, we study the uniqueness of meromorphic functions f and g in the case where fn(ℒ(f))m − p and gn(ℒ(g))m − p share two values; here, ℒ(f) = akf(k) + ak−1f(k−1) + . . .+a1f’ +a0f, ak(6= 0),ak−1, . . . ,a1,a0 ∈ ℂ, and p(z)(6≢ 0) is a polynomial. The obtained result significantly improves and generalizes the result obtained by A. Banerjee and S. Majumder in [Bol. Soc. Mat. Mex. (2016); https://doi.org/10.1007/s40590-016-0156-0 ].

The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an L-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results in {W. J. Hao and J. F. Chen, Uniqueness of L-functions concerning certain differential polynomials, Discrete Dyn. Nat. Soc., 2018, DOI. 10.1155/2018/4673165}, {F. Liu, X. M. Li and H. X. Yi, Value distribution of L-functions concerning shared values and certain differential polynomials, Proc. Japan. Acad. Ser. A, 93 (2017), 41-46} and {P. Sahoo and S. Haldar, Uniqueness results related to L-functions and certain differential polynomials, Tbilisi Math. J., 11(4) (2018), 67-78}.