Mathematics – Classical Analysis and ODEs
Scientific paper
2005-04-20
Mathematics
Classical Analysis and ODEs
Scientific paper
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 Tur\'an, 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.
No associations
LandOfFree
Right order Turan-type converse Markov inequalities for convex domains on the plane does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Right order Turan-type converse Markov inequalities for convex domains on the plane, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Right order Turan-type converse Markov inequalities for convex domains on the plane will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-516384