Convexity properties related to extremal functions

Abstract

In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Pleśniak in the circle of papers (most important joint with W. Pawłucki) showed deep connections between behavior of Siciak extremal function near compact $K$ and bounds for derivatives of polynomials. In 1990 W. Pleśniak introduced condition (P) which is equivalent to Markov property of compact $K$. In the same paper there was stated a problem which property of Siciak's extremal function are necessary to Markov's property. A stronger question is on Hölder continuity of the logarithm of the Siciak extremal function. This problem can be formulate in more general case of arbitrary norms $q$ on the space of polynomials. In the present paper we investigate the connection between behavior of generalizations of Siciak's function and the behavior of norms of derivatives of polynomials. In particular we get some deep properties of Markov factors $M_n(q,k)$ related to the main problems. One of the main result is the Kolmogorov-Landau type property of $M_n(q,k{)}^{1/k}$ which is a condition on the triangle sequence of family of derivatives of polynomials not for particular polynomials as for direct analogon of the Kolmogorov-Landau remarkable inequalities: $\log M_n(q,k{)}^{1/k}\le \log const.+(1-\frac{\log k}{\log n})\log M_n(q,1)+\frac{\log k}{\log n}{M}_n(q,n{)}^{1/n},1\le k\le n$.It seems that this condition is satisfied for arbitrary norm $q$. Separately this condition (a weaker version is sufficient) gives nothing. But if we assume that $q$ has A. Markov's property with respect to $q$ and satisfies a condition $C(q)>0$ then $q$ posseses Vladimir Markov property. In the case $q(P)=||P|{|}_E$ this means that non pluripolar Markov sets possese Hölder continuous pluricomplex Green function.