Remarks on Bodenmiller's theorem

The original version ofBodenmiller's Theorem states that the three circles with the diagonals of a complete quadrilateral as diameters intersect in the same two points. We provide a simple analytic proof of this fact which yields a much more general result. Furthermore we discuss some special configurations appearing in this context.Oswald Giering zum 60. Geburtstag gewidmet     

The growth theorem and distortion theorem of spirallike mappings on B p

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain B_p. Moreover, the distortion theorem of spirallike mappings is obtained.     

Remarks on a theorem of Burkholder and Gundy

Let (Y _n) be a sequence of i.i.d. random variables with zero mean such thatP(Y ₁0)>0. Consider the random walkS _n=Y₁+...+Y_n and an a.s. finite stopping time with respect to the -fieldsF _n=(Y₁,..., Y_n). In this paper we present a number of remarks on a theorem of Burkholder and Gundy concerning an estimate involvingE(sup _n1|Svn|^p).Dedicated to the memory of Professor József Mogyoródi.     

Remarks on the functional central limit theorem for martingales

Summary It is shown that filling a gap in the proof of Lemma 3 in Rootzén (1977) and using an appropriate truncation procedure one obtains general necessary conditions for the functional CLT for martingale difference arrays leading simultaneously to a solution of a remaining problem posed by Rootzén (1977). As to sufficient conditions which are weaker than previous ones in the literature Rootzén and the authors arrived independently at the result stated as Theorem 1 in the present paper.Dedicated to Professor Leopold Schmetterer on the occasion of his 60th birthday     

On conformal distortion and Sullivan's sector theorem

Using general bounds on the conformal distortion of univalent maps, we prove a strong version of Sullivan's sector theorem, which gives certain sufficient conditions for an arbitrarily long composition of univalent Herglotz functions to map the upper half-plane into a proper sub-sector.

     

Remarks on a theorem of Taskinen on spaces of continuous functions

We clarify and prove in a simpler way a result of Taskinen about symmetric operators on C(Kⁿ), K an uncountable metrizable compact space. To do this we prove that, for any compact space K and any n ∈ ?, the symmetric injective n–tensor product of C(K), , is complemented in C(B_C(K)*), a result of independent interest. The techniques we develop allow us to extend the result in several directions. We also show that the hypothesis of metrizability and uncountability cannot be removed.     

Remarks on a multiplicity theorem for a perturbed Neumann problem

The aim of this article is to prove that, given two potential functionals Ψ₁, Ψ₂ on W1,2(Ω) which coincide on a set of the type , then under suitable summability conditions, certain local minima of Ψ₁ are local minima for Ψ₂ as well. An application of this result allows us to obtain a multiplicity theorem for a Neumann problem where we impose a less restrictive oscillating behavior on the nonlinearity than the one required in an analogous result recently established by B. Ricceri.     

Remarks and examples concerning the weak basis theorem

This work was undertaken with the support of the AvH.