This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $𝔸\left(R_0\right)$ share four distinct values regardless of multiplicity and have the of positive counting function, then $f_1=f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing...

We prove the existence of sequences ${\varrho ₙ}_{n=1}^{\infty }$, ϱₙ → 0⁺, and ${zₙ}_{n=1}^{\infty }$, |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F\left(z\right)=F_{G,\alpha }\left(z\right)$ on ℂ such that $\varrho {ₙ}^{\alpha }F(nzₙ+n\varrho ₙ\zeta )$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_m$-normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^n{\left(f^{\left(k\right)}\right)}^m-p$ and $g^n{\left(g^{\left(k\right)}\right)}^m-p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018).

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{♯}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus forcurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony,...

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and...

Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$ $(i=1,...,k,k\in \mathbb{N})$ are distinct nonzero complex constants. Secondly, we deal with the growth of solutions of differential-difference equation of the form $$\sum _{i=0}^n\sum _{j=0}^m{A}_{ij}\left(z\right)f^{\left(j\right)}(z+c_i)=F\left(z\right),$$ where $A_{ij}\left(z\right)$ $(i=0,1,...,n,j=0,1,...,m,n,m\in \mathbb{N})$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$ $(i=0,...,n)$ are distinct complex constants. We extend some previous results...

The authors introduce two new subclasses $F_{p,k}(\lambda ,A,B)$ and $G_{p,k}(\lambda ,A,B)$ of meromorphically multivalent functions. Distortion bounds and convolution properties for $F_{p,k}(\lambda ,A,B)$, $G_{p,k}(\lambda ,A,B)$ and their subclasses with positive coefficients are obtained. Some inclusion relations for these function classes are also given.

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^n\left(z\right)+P_d(z,f)=p_1\left(z\right){\mathrm{e}}^{{\alpha }_1\left(z\right)}+p_2\left(z\right){\mathrm{e}}^{{\alpha }_2\left(z\right)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f\left(z\right)$ of degree $d\le n-1$ with small functions of $f\left(z\right)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and ${\alpha }_1$, ${\alpha }_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

Let f be a transcendental meromorphic function and $g\left(z\right)=f(z+c₁)+\cdots +f(z+c_k)-kf\left(z\right)$ and $g_k\left(z\right)=f(z+c₁)\cdots f(z+c_k)-f^k\left(z\right)$. A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_k\left(z\right)$, g(z)/f(z), and $g_k\left(z\right)/f^k\left(z\right)$.

Let $k$ be a nonnegative integer or infinity. For $a\in ℂ\cup \left\{\infty \right\}$ we denote by $E_k(a;f)$ the set of all $a$-points of $f$ where an $a$-point of multiplicity $m$ is counted $m$ times if $m\le k$ and $k+1$ times if $m>k$. If $E_k(a;f)=E_k(a;g)$ then we say that $f$ and $g$ share the value $a$ with weight $k$. Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to...

Let Mₚ denote the class of functions f of the form $f\left(z\right)=1/z^p+{\sum }_{k=0}^{\infty }aₖz^k$, p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n,p}\left(\alpha \right)=f:f\in Mₚ,Re-(z^{p+1}/p){\left(Dⁿf\right)}^{\text{'}}>\alpha$, α < 1, where $Dⁿf={\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}/(z^{p}n!)$. Results on $L_{n,p}\left(\alpha \right)$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.

Let ℱ be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d ≠ b,0 and c ≠ a. If, for each f ∈ ℱ, $f\left(z\right)=a\iff f^{\left(k\right)}\left(z\right)=b$, and $f^{\left(k\right)}\left(z\right)=d\Rightarrow f\left(z\right)=c$, then ℱ is a normal family on D. The same result holds for k=1 so long as b≠(m+1)d, m=1,2,....

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\le 1/d$ of the $L$-functions of any degree $d\ge 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

Given a germ $h$ of holomorphic function on $({ℂ}^n,0)$, we study the condition: “the ideal ${\text{Ann}}_{𝒟}1/h$ is generated by operators of order1”. We obtain here full characterizations in the particular cases of Koszul-free germs and unreduced germs of plane curves. Moreover, we prove that this condition holds for a special type of hyperplane arrangements. These results allow us to link this condition to the comparison of de Rham complexes associated with $h$.

Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and $\phi =R{e}^P$, where R ≢ 0 is a rational function and P is a polynomial, and let $a₀,a₁,...,a_{k-1}$ be holomorphic functions on ℂ. If all zeros of f have multiplicity at least k except possibly finitely many, and $f=0\iff f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a₀f=0$, then $f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a₀f-\phi$ has infinitely many zeros.

We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${\left(Q\left(f\right)\right)}^{\left(k\right)}$ and ${\left(Q\left(g\right)\right)}^{\left(k\right)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{\text{'}}$. The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).

Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_{\rho \left(f\right)}$, ρ(f) < ∞. We investigate rational approximants $r_{n,mₙ}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $\rho {\left(f\right)}^{-n}$ on E implies uniform maximal convergence in m₁-measure inside $E_{\rho \left(f\right)}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_{\rho \left(f\right)}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue...

Let $f$ be a dominant rational map of ${\mathbb{P}}^k$ such that there exists $s\&lt;k$ with ${\lambda }_s\left(f\right)\&gt;{\lambda }_l\left(f\right)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut}\left({\mathbb{P}}^k\right)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $log{\lambda }_s\left(f\right)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to ${\mathbb{P}}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...

This paper studies the uniqueness of meromorphic functions $$f^n\prod _{i=1}^k{\left(f^{\left(i\right)}\right)}^{n_i}\text{and}{g}^n\prod _{i=1}^k{\left(g^{\left(i\right)}\right)}^{n_i}$$ that share two values, where $n,n_k,k\in \mathbb{N}$, $n_i\in \mathbb{N}\cup \left\{0\right\}$, $i=1,2,...,k-1$. The results significantly rectify, improve and generalize the results due to Cao and Zhang (2012).

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Let ${\overline{L}}_i\to X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\overline{L}}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\overline{L}}_i$ are equivariant ${\left({ℂ}^{*}\right)}^{n_i}$-bundles satisfying some additional conditions....