An inequality for derivatives of polynomials with positive coefficients (Q1344178)

A Bernstein type inequality is proved for the class \(P^ +_ N\) of polynomials of degree at most \(N\) whose coefficients are nonnegative. Let \(n(r,f)\) count the number of zeros with multiplicity of the polynomial \(f\) in the disk \(| z|<r\) and let \(M(r,f)= \max_{| z|=r} | f(z)|\). Then, if \(f\in P^ +_ N\) for \(N=1,2,3,4\), the bound \(M(r, f')\leq [(N+ n(r,f))/ 2r] M(r,f)\) holds for \(r>0\). Furthermore, let \(5m\leq N< 5(m+1)\), \(m=1,2,3,\dots\;\). Then there is a polynomial \(f\in P^ +_ N\) for which \(M(3,f')= [(N+ n(3,f)+ 2m/7) /6] M(3,f)> [(N+ n(3,f)) /6] M(3,f)\).