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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, 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).

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.

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.

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

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.

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.

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).

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 ].

Let $$X\subset {\mathbb {R}}^n$$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on X is a sum of 10 squares of meromorphic functions on X. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

In this paper, we prove some normality criteria involving product of spherical derivatives of a meromorphic function and that of its derivatives.

Recently, Q. Han [7] proved a Second Main Theorem for algebraically nondegenerate meromorphic maps over p-Parabolic manifolds intersecting with hypersurfaces in general position in smooth projective algebraic variety, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru and Philip P. W. Wong. In this paper, we will prove a general form of Second Main Theorem for meromorphic maps from p-Parabolic manifold into smooth projective variety intersecting with hypersurfaces in subgeneral position. As an application of that result, we get a Second Main Theorem for meromorphic maps on p-Parabolic manifold intersecting with hypersurfaces in l-subgeneral position, which extends the result of Q. Han [7].

In this paper we study some applications of second order differential subordination, superordination and sandwich-type results for certain admissible classes of meromorphically multivalent functions associated with the Liu-Srivastava operator.

In this paper, we investigate the zero distributions of $q$-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential polynomials to q-shift difference. We also investigate the uniqueness problem of $q$-shift difference-differential polynomials sharing a polynomial value with finite weight., 19 pages. arXiv admin note: substantial text overlap with arXiv:2103.03630

In this paper, one-parameter families $${\mathcal {F}}\equiv \left\{ f_{\lambda }(z)=\lambda \left( \cosh z+\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ and $${\mathcal {G}}\equiv \left\{ g_{\lambda }(z)=\lambda \left( \cosh z-\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ are considered and the dynamics of functions $$f_{\lambda }\in {\mathcal {F}}$$ and $$g_{\lambda }\in {\mathcal {G}}$$ are investigated. It is shown that both the functions $$f_{\lambda }$$ and $$g_{\lambda }$$ have finite number of singular values and the origin is always an attracting fixed point of $$g_{\lambda }(z)$$ . The dynamics of $$f_{\lambda }(z)$$ and $$g_{\lambda }(z)$$ on the extended complex plane are studied by investigating the nature of the real fixed points and the singular values of $$f_{\lambda }$$ and $$g_{\lambda }$$ . It is shown that a bifurcation and chaotic burst occur at a certain parameter value of $$\lambda $$ for the functions $$f_{\lambda }$$ in the family $${\mathcal {F}}$$ but there is no bifurcation in the family $${\mathcal {G}}$$ .

For entire or meromorphic function f, a value θ ∈ [0, 2π) is called a Julia limiting direction if there is an unbounded sequence {zn} in the Julia set satisfying $$\mathop {\lim }\limits_{n \to \infty } \;\arg {z_n} = \theta $$ . Our main result is on the entire solution f of P(z, f) + F(z)fs = 0, where P(z, f) is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F, with the integer s being no more than the minimum degree of all differential monomials in P(z, f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.

This paper is devoted to investigating the growth and zeros of meromorphic solutions of the generalized Fermat functional equation $${f^2}\left({z + 1} \right) + {A_1}\left(z \right)f\left({z + 1} \right)f\left({z + 1} \right)f\left(z \right) + {A_2}{f^2}\left(z \right) = {A_3}\left(z \right),$$ where A1(z), A2(z), A3(z) are polynomials with A3(z) ≢ 0. The corresponding homogeneous equation of the above equation is studied as well.

In this paper, we consider transcendental meromorphic solutions f of finite order $$\rho $$ and few poles in the sense that $$S_{\lambda }(r,f):=O(r^{\lambda +\varepsilon })$$ , where $$\lambda

We say that two meromorphic functions f and g share a small function $$\alpha $$ counting multiplicities if $$f-\alpha $$ and $$g-\alpha $$ admit the same zeros with the same multiplicities. Let Q be a polynomial of one variable. In [Comput. Methods Funct. Theory 17: 613-634, 2017, Theorem. 1.1], we proved that if $$(Q(f))^{(k)}$$ and $$(Q(g))^{(k)}$$ share $$\alpha $$ counting multiplicities then, with suitable conditions on the degree of Q and on the number of zeros and the multiplicities of the zeros of $$Q'$$ , there are explicit relations between Q(f) and Q(g). Unfortunately, there is a gap at the beginning of the proof of An and Phuong (Comput. Methods Funct. Theory 17:613–634, 2017, Theorem 1.1]. We will give a way to avoid the gap. This proof can also be used to fix the gaps of other authors’ published papers listed in Schweizer ( arXiv:1705.05048v2 , 2017).

We improve a normality result of Liu–Li–Pang [4] concerning shared values between two families. Let $$\mathcal F$$ and $$\mathcal G$$ be two families of meromorphic functions on D whose zeros are multiple. Suppose that $$\mathcal G$$ is normal on D, and no sequence contained in $$\mathcal G$$ $$\chi $$ -converges locally uniformly to $$\infty $$ or a function g satisfying $$g'\equiv 1$$ . If for every $$f\in \mathcal F$$ , there exists a function $$g\in \mathcal G$$ such that f and g share 0 and $$\infty $$ while $$f'$$ and $$g'$$ share 1, then $$\mathcal F$$ is also normal on D.

In this paper, we investigate the value distributions of linear q-difference polynomials $$f^{n}(z)+\sum ^{l}_{j=1}a_{j}(z)f(q_{j}z+c_j)$$ and $$f^{n}(z)\sum ^{l}_{j=1}a_{j}(z)f(q_{j}z+c_j)$$ when f is a transcendental meromorphic function of zero order. The uniqueness theorems of q-difference polynomials were also considered.

In this paper, we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations $$\begin{aligned} A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} 0, \\ A_{k}(z)f(z+c_{k})+\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} F, \end{aligned}$$ A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = 0 , A k ( z ) f ( z + c k ) + ⋯ + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = F , where $$A_{k}\left( z\right) ,\ldots ,A_{0}\left( z\right) ,$$ A k z , … , A 0 z , $$F\left( z\right) $$ F z are meromorphic functions and $$c_{j}$$ c j $$\left( 1,\ldots ,k\right) $$ 1 , … , k are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng, Belaïdi and Benkarouba.

In this paper, making use of the value distribution theory for meromorphic functions in several complex variables and its difference analogues, we mainly consider the uniqueness problem for meromorphic functions in several complex variables sharing values or small functions with their shifts or difference operators, and weaken the condition of growth of hyper-order of meromorphic functions, which generalize the corresponding results of one complex variable.

For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.

In this paper, we study the uniqueness problem of certain differential polynomials generated by two meromorphic functions. The results of the paper extend some recent results due to Meng and Li (J Anal 28:1–6, 2019).

Two meromorphic functions are said to share a set S ⊂ ℂ∪{∞} ignoring multiplicities (IM) if S has the same preimages 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” [RURSM (or URSM-IM)]. From the existing literature, it is known that there exists a RURSM with 17 elements. We reduce the cardinality of the existing RURSM and show that there exists a RURSM with 15 elements. Our result gives an affirmative answer to the question of L. Z. Yang [Int. Soc. Anal. Appl. Comput., 7, 551–564 (2000)].

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.

In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2\pi^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{\pi^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution., Comment: 20 pages

The purpose of the paper is to study the uniqueness problem of an L function in the Selberg class sharing one or two sets with an arbitrary meromorphic function having only finitely many poles. We manipulate the notion of weighted sharing of sets to improve a result of Q.Q. Yuan, X.M. Li, and H.X. Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2):249–262, 2018]. Most importantly, we have pointed out a number of logical shortcomings in the two results of P. Sahoo and S. Haldar [Results on L functions and certain uniqueness question of Gross, Lith. Math. J., 60(1):80–91, 2020]. As an attempt to rectify the results of Sahoo and Halder, we have improved them by presenting their accurate forms and proofs as far as practicable.

The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, $$f(z)^nf^{(k)}(z)$$ f ( z ) n f ( k ) ( z ) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case $$k=1$$ k = 1 . When $$k\ge 2$$ k ≥ 2 the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.

Let $$f$$ be a transcendental meromorphic function such that $$N_{1)}(r,0;f')+N(r,\infty ;f)=S(r,f)$$ . Let $$k\in \mathbb {N}\setminus \{1\}$$ and $$n\in \mathbb {N}$$ such that $$n\ge k+1$$ . If $$f^{n}$$ and $$(f^{n})^{(k)}$$ share 1 IM, then $$f^{n}\equiv (f^{n})^{(k)}$$ and f assumes the form $$f(z)=ce^{\frac{\lambda }{n}z}$$ , where $$c\in \mathbb {C}\setminus \{0\}$$ and $$\lambda ^{k}=1$$ .

Let Co(p), $$p\in (0,1]$$ be the class of all meromorphic univalent functions $$\varphi $$ defined in the open unit disc $${\mathbb {D}}$$ with normalizations $$\varphi (0)=0=\varphi '(0)-1$$ and having simple pole at $$z=p\in (0,1]$$ such that the complement of $$\varphi ({\mathbb {D}})$$ is a convex domain. The class Co(p) is called the class of concave univalent functions. Let $$S_{H}^{0}(p)$$ be the class of all sense preserving univalent harmonic mappings f defined on $${\mathbb {D}}$$ having simple pole at $$z=p\in (0,1)$$ with the normalizations $$f(0)=f_{z}(0)-1=0$$ and $$f_{\bar{z}}(0)=0$$ . We first derive the exact regions of variability for the second Taylor coefficients of h where $$f=h+\overline{g}\in S_{H}^{0}(p)$$ with $$h-g\in Co(p)$$ . Next we consider the class $$S_{H}^{0}(1)$$ of all sense preserving univalent harmonic mappings f in $${\mathbb {D}}$$ having simple pole at $$z=1$$ with the same normalizations as above. We derive exact regions of variability for the coefficients of h where $$f=h+\overline{g}\in S_{H}^{0}(1)$$ satisfying $$h-e^{2i\theta }g\in Co(1)$$ with dilatation $$g'(z)/h'(z)=e^{-2i\theta }z$$ , for some $$\theta $$ , $$0\le \theta

The aim of the paper is to study the uniqueness problems of meromorphic functions when certain differential polynomials generated by them share a small function. With the aid of weighted sharing we prove four theorems first two of which improve the results due to Boussaf et al. (Indagationes Mathematicae 24:15–41, 2013) and last two improve a result due to Bhoosnurmath and Pujari (Bull Korean Math Soc 52:13–33, 2015).

Due to the Nevanlinna theory, the paper gives the general structure of transcendental meromorphic solutions of a certain ordinary differential equation with rational coefficients. As an application, the meromorphic solutions of the FitzHugh–Nagumo system are obtained in explicit form.

Let f(z) be a transcendental meromorphic function, whose zeros have multiplicity at least 3. Set α(z): = β(z)exp (γ(z), where β(z) is a nonconstant elliptic function and γ(z) is an entire function. If σ(f(z)) > σ(α(z)), then f′(z) = α(z) has infinitely many solutions in the complex plane.

In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028 , 2019), of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values, and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563 , 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here, we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle are also accessible from the shift locus.

In this paper, we will consider normality and uniqueness property of a family $\mathcal {F}$ of meromorphic functions when [Q(f)](k) and [Q(g)](k) share α ignoring multiplicities, for any $f,g\in \mathcal {F}$ , where Q is a polynomial and α is a small function. Our results do not need all of zeros of Q have large order as other authors’ results.

In this paper, we give some results on $$\frac{zf^{\prime }(z)}{f(z)}$$ for the certain classes of f(z) meromorphic functions. For the function $$f(z)= \frac{1}{z}+c_{0}+c_{1}z+c_{2}z^{2}+\cdots$$ defined in the punctured disc $$U=\left\{ z:0

Meromorphic functions with a given growth of a spherical derivative on the complex plane are described in terms of the relative location of a-points of functions. The result obtained allows one to construct an example of a meromorphic function in ℂ with a slow growth of Nevanlinna characteristics and arbitrary growth of the spherical derivative. In addition, based on the universality property of the Riemann zeta-function, we estimate the growth of the spherical derivative of ζ(z).

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations., Comment: 32 pages. This is the final version for being published. arXiv admin note: text overlap with arXiv:1601.05716

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 $$j=1,2,3$$ .

In this paper, we make a very interesting study on the existence of unicity of entire and meromorphic functions which share a small function with a general differential-difference polynomial. The results obtained will greatly generalize the results due to Bhoosnurmath and Kabbur (International Journal of Analyse 8, 2013).

Let $$k, n \in {\mathbb {N}}, l \in {\mathbb {N}}\backslash \left\{ 1 \right\} , m\in {\mathbb {N}}\cup \left\{ 0 \right\} $$ , and let $$a(z)(\not \equiv 0)$$ be a holomorphic function, all zeros of a(z) have multiplicities at most m. Let $${\mathcal {F}}$$ be a family of meromorphic functions in D. If for each $$f \in {\mathcal {F}}$$ , the zeros of f have multiplicity at least $$k+m$$ , and for $$f\in {\mathcal {F}}$$ , $$f^{l}(f^{(k)})^{n}-a(z)$$ has at most one zero in D, then $${\mathcal {F}}$$ is normal in D.