Markov inequality for polynomials of degree n with m distinct zeros

✍ Scribed by David Benko; Tamás Erdélyi

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 128 KB
• Volume: 122
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1016/s0021-9045(03)00074-1

✦ Synopsis

Let P m n be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove that max xA½0;1 jP 0 ðxÞjp32 Á 8 m n max xA½0;1 jPðxÞj for every PAP m n : This is far away from what we expect. We conjecture that the Markov factor 32 Á 8 m n above may be replaced by cmn with an absolute constant c40: We are not able to prove this conjecture at the moment. However, we think that our result above gives the bestknown Markov-type inequality for P m n on a finite interval when mpc log n: