Rainer Brück

On Entire Functions which Share One Value CM with their First Derivative

If ƒ is a non-constant entire function which is sharing two distinct values a, b ∈ ℂ with ƒ′, then a result of Mues and Steinmetz states that ƒ′ = ƒ. In this note we consider the case that ƒ and ƒ′ share only one value counting multiplicity, where appropriate restrictions on the growth of ƒ are assumed.

Connectedness and Stability of Julia Sets of the Composition of Polynomials of the Form z2+cn

For a sequence ( c n ) of complex numbers, the quadratic polynomials f c n ( z ) := z 2 + c n and the sequence ( F n ) of iterates F n := f c n ∘…∘ f c 1 are considered. The Fatou set [Fscr ] ( c n ) is by definition the set of all z ∈ [Copf ]ˆ such that ( F n ) is normal in some neighbourhood of z , while the complement of [Fscr ] ( c n ) is called the Julia set [Jscr ] ( c n ) . The aim of this article is to study the connectedness and stability of the Julia set [Jscr ] ( c n ) provided that the sequence ( c n ) is bounded.

Invertibility of Holomorphic Functions with Respect to the Hadamard Product

Let G 1 and G 2 be domains in containing the open unit disk D, and let H(Gj ) be the set of all holomorphic functions in Gj for j = 1,2. We say that f εH(G 1) is invertible with respect to H(G 2) if there exists g ε H(G 2) such that (f * g)(z)= - 1/(1 - z) for z ε D, where f*g denotes the Hadamard product of f and g. In (R. Brück and J. Müller (1992). Math. Ann., 294, 421–438) it was shown that for certain kinds of domains G 1 and G 2 the invertible elements of H(G 1) with respect to H(G 2) are necessarily of a very special form. The main purpose of this paper is to show that this result is true for broader classes of domains, where our method of proof is different to that in (R. Brück and J. Müller (1992). Math. Ann., 294, 421-438).

Summability of sequences of interpolatory polynomials in the roots of unity: rate of convergence

Let be a region containing the disk for some R>1, and let f be a function holomorphic in G. Furthermore, let L n(.;f) denote the Lagrange interpolatory polynomial of f in the (n + 1)st roots of unity. Then it is well-known that L n(z;f)→ f(z) (n→∞) locally uniformly in D R. In [2] and [3] we applied certain matrix summability methods to the sequence L n(.;f) in order to enlarge the set of convergence. This set is an open set, it depends on the summability method and the singularities of f, and it often contains the disk D R as a proper subset. The aim of this paper is to study the rate of convergence.

On locally univalent entire functions

All univalent entire functions are linear and map onto itself. In this note we give a characterization of the class of locally univalent entire functions mapping onto . In particular, the function belongs to this class.

Uniqueness theorems for functions regular and of exponential type in a half plane

We consider the following problem. Let f be a function regular and of exponential type τ in the half plane which satisfies for n = 0,1,2,… and j = 1,…, p, where kj are integers with 0 < … <k p Exponential type τ means that f satisfies the inequality for each positive e. The problem is to find the greatest τ that allows to prove that f is identically zero or a certain special function. Special cases of this problem were solved by Carlson [3], Korevaar [5] and Gervais and Rahman [4]. We extend and generalize their results and give a complete solution of the problem if kp ≤4. In [2] we have already considered this problem for entire functions, where f and the derivatives vanish at all integers.