The number of real zeros of polynomials with constrained coefficients

We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most $cn^{1/2}(1+\log M)^{1/2}$ zeros in the interval $[-1,1]$. This result is sharp up to the multiplicative constant $c > 0$ and extends an earlier result of Borwein, Erd\'elyi, and K\'os from the case $M=1$ to the case $M \geq$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 $a \in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erd\'elyi by proving that there is a constant $\eta > 0$ such that every polynomial $P$ of the above form with $M = 1$ has at most $\eta n^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the multiplicative constant $\eta > 0$ depends only on the polygon.