The number of real zeros of polynomials with constrained coefficients

We use Jensen's formula to give an upper bound for 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.