Mathematics

The asymptotic behaviour, with respect to the large order, of the radii of starlikeness of two types of normalised Bessel functions is considered. We derive complete asymptotic expansions for the radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums and asymptotic inversion. The techniques employed in the paper could be useful to treat similar problems where inversion of asymptotic expansions is involved.

In 1955, Lehto showed that, for every measurable function ? on the unit circle T, there is a function f holomorphic in the unit disc, having ? as radial limit a.e. on T. We consider an analogous problem for solutions f of homogenous elliptic equations Pf=0 and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path γ from 0 to infinity such that f(z)?�a tends to 0 as z tends to infinity along γ . The Denjoy--Carleman--Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2 , mean type. A long-standing problem asks whether this conclusion holds for entire functions having n distinct asymptotic (entire) functions, each of growth at most order 1/2 , minimal type. In this paper conditions on the function f and associated asymptotic paths are obtained that are sufficient to guarantee that f satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In addition, for each positive integer n , an example is given of an entire function of order n having n distinct, prescribed asymptotic functions, each of order less than 1/2 .

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′′ +P(z)f= 0 , where P(z) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E.~Hille's 1969 book \emph{Lectures on Ordinary Differential Equations} are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille's book.

We show that the orthogonal speed of semigroups of holomorphic self-maps of the unit disc is asymptotically monotone in most cases. Such a theorem allows to generalize previous results of D. Betsakos and D. Betsakos, M. D. Contreras and S. Díaz-Madrigal and to obtain new estimates for the rate of convergence of orbits of semigroups.

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of C m , in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets Ω⊂ C m whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.

We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces F p W , whose weight W is not necessarily radial. We show that in the spaces F p W which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form P n , i.e., finite dimensional subspaces consisting of polynomials of degree at most n . In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type) we establish an analogue of de Branges' Ordering Theorem. We then construct examples which show that the result fails for general Fock-type spaces of larger growth.

Balayage of measures with respect to classes of all subharmonic or harmonic functions on an open set of a plane or finite-dimensional Euclidean space is one of the main objects of potential theory and its applications to the complex analysis. For a class H of functions on O , a measure ω on O is a balayage of a measure δ on O with respect to this class H if ∫ O hdδ≤ ∫ O hdω for each h∈H . In our previous works we used this concept to study envelopes relative to classes of subharmonic and harmonic functions and apply them to describe zero sets of holomorphic functions on O with growth restrictions near the boundary of O . In this article, we consider the complex plane C as O , and instead of the classes of all (sub)harmonic functions on C , we use only the classes of harmonic polynomials of degree at most p , often together with the logarithmic functions-kernels z↦ln|w−z| , w∈C . Our research has show that this case has both many similarities and features compared to previous situations. The following issues are considered: the sensitivity of balayage of measures to polar sets; the duality between balayage of measures and their logarithmic potentials, together with a complete internal description of such potentials; extension/prolongation of balayage with respect to polynomials and logarithmic kernels to balayage with respect to subharmonic functions of finite order p . The planned applications of these results to the theory of entire and meromorphic functions of finite order are not discussed here and will be presented later.

We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc.

Every real Bank-Laine function of finite order, whose zeros are all real but neither bounded above nor bounded below, either has an explicit representation in terms of trigonometric functions or has zeros with exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.