Right order Turan-type converse Markov inequalities for convex domains on the plane
| dc.creator | Revesz, Szilard Gy. | |
| dc.date | 2005-04-20 | |
| dc.date.accessioned | 2026-07-07T05:19:17Z | |
| dc.date.available | 2026-07-07T05:19:17Z | |
| dc.description | For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Turán, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n ||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant. | |
| dc.identifier | https://arxiv.org/abs/math/0504416 | |
| dc.identifier | http://arxiv.org/abs/math/0504416 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74965 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Complex Variables | |
| dc.subject | 41A17 | |
| dc.title | Right order Turan-type converse Markov inequalities for convex domains on the plane | |
| dc.type | text |