Essential singularities of quasimeromorphic mappings.
Let || · || be the uniform norm in the unit disk. We study the quantities Mn (α) := inf (||zP(z) + α|| - α) where the infimum is taken over all polynomials P of degree n - 1 with ||P(z)|| = 1 and α > 0. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that infα>0Mn (α) = 1/n. We find the exact values of Mn (α) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
The rate of growth of the energy integral of a quasiregular mapping is estimated in terms of a special isoperimetric condition on . The estimate leads to new Phragmén-Lindelöf type theorems.
The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space ; of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces...
We establish inequalities for Green functions on general bounded piecewise Dini-smooth Jordan domains in ℝ². This enables us to prove a new version of the 3G Theorem which generalizes its previous version given in [M. Selmi, Potential Anal. 13 (2000)]. Using these results, we give a comparison theorem for the Green kernel of Δ and the Green kernel of Δ - μ, where μ is a nonnegative and exact Radon measure.
2000 Mathematics Subject Classification: 12D10.In the present paper we consider degree 6 hyperbolic polynomials (HPs) in one variable (i.e. real and with all roots real). We are interested in such HPs whose number of equalities between roots of the polynomial and/or its derivatives is higher than expected. We give the complete study of the four families of such degree 6 even HPs and also of HPs which are primitives of degree 5 HPs.Research partially supported by research project 20682 for cooperation...