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We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain
$C(\log H)^{\unicode[STIX]{x1D702}}$
bounds for the number of algebraic points of height at most
$H$
on certain subsets of the graphs of such functions. The constant
$C$
and exponent
$\unicode[STIX]{x1D702}$
depend on data associated with the functions and can be effectively computed from them.
Let
$B$
be a rational function of degree at least two that is neither a Lattès map nor conjugate to
$z^{\pm n}$
or
$\pm T_{n}$
. We provide a method for describing the set
$C_{B}$
consisting of all rational functions commuting with
$B$
. Specifically, we define an equivalence relation
$\underset{B}{{\sim}}$
on
$C_{B}$
such that the quotient
$C_{B}/\underset{B}{{\sim}}$
possesses the structure of a finite group
$G_{B}$
, and describe generators of
$G_{B}$
in terms of the fundamental group of a special graph associated with
$B$
.
In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.
This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.
We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.
We give an upper estimate for the order of the entire functions in the Nevanlinna parameterization of the solutions of an indeterminate Hamburger moment problem. Under a regularity condition this estimate becomes explicit and takes the form of a convergence exponent. Proofs are based on transformations of canonical systems and I.S.Kac' formula for the spectral asymptotics of a string. Combining with a lower estimate from previous work, we obtain a class of moment problems for which order can be computed. This generalizes a theorem of Yu.M.Berezanskii about spectral asymptotics of a Jacobi matrix (in the case that order is ⩽ 1/2).
We prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that
${\rm dist} (p_n, U_n)\to 0$
as
$n\to \infty $
. We also prove that if
$U_n \cap P(f)=\emptyset $
and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.
We investigate several quantitative properties of entire and meromorphic solutions to some differential-difference equations and generalised delay differential-difference equations. Our results are sharp in a certain sense as illustrated by several examples.
Let
$\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$
and
$\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$
be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set
${\mathcal{N}}$
of natural numbers such that their ratio
$\mathbf{F}(n)/\mathbf{G}(n)$
is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences
$F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$
and
$G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$
, where the
$f_{i}$
and
$g_{j}$
are nonconstant entire functions and the
$a_{i}$
and
$b_{j}$
are non-zero constants except that
$a_{0}$
can be zero. We will show that the set
${\mathcal{N}}$
of natural numbers such that
$F(n)/G(n)$
is an entire function is finite under the assumption that
$f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$
is not constant for any non-trivial index set
$(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$
.
We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function
$f(x)$
in a quasianalytic Denjoy–Carleman class
${\mathcal{Q}}_{M}$
, which is formally composite with a generically submersive mapping
$y=\unicode[STIX]{x1D711}(x)$
of class
${\mathcal{Q}}_{M}$
, at a single given point in the source (or in the target) of
$\unicode[STIX]{x1D711}$
can be written locally as
$f=g\circ \unicode[STIX]{x1D711}$
, where
$g(y)$
belongs to a shifted Denjoy–Carleman class
${\mathcal{Q}}_{M^{(p)}}$
; (2) a statement on a similar loss of regularity for functions definable in the
$o$
-minimal structure given by expansion of the real field by restricted functions of quasianalytic class
${\mathcal{Q}}_{M}$
. Both results depend on an estimate for the regularity of a
${\mathcal{C}}^{\infty }$
solution
$g$
of the equation
$f=g\circ \unicode[STIX]{x1D711}$
, with
$f$
and
$\unicode[STIX]{x1D711}$
as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.
We show that the family of all holomorphic functions
$f$
in a domain
$D$
satisfying
$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$
(where
$k$
is a natural number and
$C>0$
) is quasi-normal. Furthermore, we give a general counterexample to show that for
$\unicode[STIX]{x1D6FC}>1$
and
$k\geq 2$
the condition
$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70C}\in (0,\infty ]$
be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of
$\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$
, which is closed under complex conjugation and contains
$0$
, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence
$\unicode[STIX]{x1D70C}$
. This solves the question posed by K. Mahler completely.
We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].
This paper concerns the problem of algebraic differential independence of the gamma function and
${\mathcal{L}}$
-functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the
${\mathcal{L}}$
-function in a domain
$D:=\{z:0<\text{Re}\,z<\unicode[STIX]{x1D70E}_{0}\}$
for a positive constant
$\unicode[STIX]{x1D70E}_{0}$
.
In this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.
In this paper, we prove some value distribution results which lead to normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family, and improve some recent results. In particular, the main result of this paper leads to a counterexample to the converse of Bloch’s principle.
Let be a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set of f equals 2q/(q+1), where q is the maximal multiplicity of poles of f. We also consider the escaping parameters in the family fβ = βf, i.e. the parameters β for which the orbit of one critical value of fβ escapes to infinity. Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters ε in the family fβ is greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.
We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree
$d$
or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree
$d$
(or less) analytic cover of
$\mathbb{C}$
.