On the maximum modulus of polynomials (Q5906516)

The following refinement involving the first coefficients of one result of \textit{T. J. Rivlin} [Am. Math. Month. 67, 251-253 (1960)] is proved: if \(p(z)= \sum^ n_ 0 c_ k z^ k\) is a polynomial of degree \(n\), having no zeros in \(| z|< 1\) and \(| c_ 1|\leq \rho n| c_ 0|\), then for \(0\leq r\leq \rho\leq 1\), \[ M(p,r)\geq \left({1+ r\over 1+ \rho}\right)^ n\cdot \left[1- {(1- \rho)( \rho n| c_ 0|- | c_ 1|) n\over \rho(1+ \rho^ 2) n| c_ 0|+ 2\rho| c_ 1|}\cdot \left({\rho- r\over 1+ \rho}\right) \left({1+ r\over 1+ \rho}\right)^{n- 1}\right]^{-1}\cdot M(p,\rho), \] where \(M(p,r)= \max_{| z|= r} | p(z)|\).