2024/09/14 by Erdélyi, Tamás
#11C08 #26C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.09553
We prove that there is an absolute constant c > 0 such that every polynomial P of the form P(z) = ∑j=0najzj , |a0| = 1 , |aj| ≤ M , aj ∈ ℂ , M ≥ 1 , has at most cn1/2(1+log M)1/2 zeros in the interval [-1,1]. This result is sharp up to the multiplicative constant c > 0 and extends an earlier result of Borwein, Erdélyi, and Kós from the case M=1 to the case M ≥ 1. This has also been proved recently with the factor (1+log M) rather than (1+log M)1/2 in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant c > 0 such that every polynomial P of the above form has at most (c/a)(1+log M) zeros in the interval [-1+a,1-a] with a ∈ (0,1). Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erdélyi by proving that there is a constant η> 0 such that every polynomial P of the above form with M = 1 has at most ηn1/2 zeros inside any polygon with vertices on the unit circle, where the multiplicative constant η> 0 depends only on the polygon.