Markov inequality for polynomials of degree n with m distinct zeros Original Research Article

Let Pnm be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatmaxx∈[0,1] |P′(x)|⩽32·8mn maxx∈[0,1] |P(x)|for every P∈Pnm. This is far away from what we expect. We conjecture that the Markov factor 32·8mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for Pnm on a finite interval when m⩽c log n.