On the maximum modulus of polynomials (Q5902922)

The reviewed paper contains some extensions of the inequalities concerning the modulus of the polynomial taken on different circles. For example: if \(p(z)=\sum^{n}_{k=1}a_ kz^ k\) is a polynomial of degree n, having no zeros in \(| z| <1\), then for \(0\leq r\leq \rho \leq 1\) \[ (*)\quad M(p,r)\geq ((1+r)/(1+\rho))^ nM(p,\rho),\quad (M(p,r):=\max_{| z| =r}| p(z)|) \] and the result is best possible. It is also proved a sharp inequality of the type (*) under the additional condition \(p'(0)=0\).