In this paper we study the comparative growth properties of a composition of entire and meromorphic functions on the basis of the relative order (relative lower order) of Wronskians generated by entire and meromorphic functions.

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L1(R) such that its Λ translates φ(x - λ), λ ∈ Λ, span L1(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic clases. We also present examples of discrete spectra Λ ∈ R which do not admit a single generator while they admit a pair of generators.

In this paper, we investigate the relationship between small functions and differential polynomials $g_f\left(z\right)=d_2{f}^{\text{'}\text{'}}+d_1{f}^{\text{'}}+d_{0}f$, where $d_0\left(z\right)$, $d_1\left(z\right)$, $d_2\left(z\right)$ are entire functions that are not all equal to zero with $\rho \left(d_j\right)<1$$(j=0,1...$

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.

Let $N_{\alpha }$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{\beta }$ are Möbius invariant, but $N_{\alpha }$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{\varphi }f=f◦\varphi$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{\varphi }:N_{\alpha }\to B$, $B\to Q_{\beta }$, $N_{\alpha }\to Q_{\beta }$ which is bounded resp. compact.

A classic theorem of Pólya shows that the function $2^z$ is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

We give sufficient conditions for a diffeomorphism in the plane to be analytically conjugate to a shift in a complex neighborhood of a segment of an invariant curve. For a family of functions close to the identity uniform estimates are established. As a consequence an exponential upper estimate for splitting of separatrices is established for diffeomorphisms of the plane close to the identity. The constant in the exponent is related to the width of the analyticity domain of the limit flow separatrix....

Given a function $f\left(t\right)\ge 0$ on $ℝ$ with ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2\left)\right)dt\&lt;\infty$ and $\left|f\right(t)-f(t^{\text{'}}\left)\right|\le l|t-t^{\text{'}}|$, a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function$$F\left(z\right):\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|𝔍z\right|}{|z-t{|}^2}f\left(t\right)dt-Al\left|𝔍z\right|,$$where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $ℝ$, is such that ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2)dt\&lt;\infty$ and denote, for any constant $\alpha \&gt;0$ and each $x\in ℝ$, the unique...