Inequalities concerning polynomials having zeros in closed exterior or closed interior of a circle (Q1879063)

The authors prove refinements for two previously known inequalities on the size of polynomials along circles of a fixed radius, under several assumptions. The first result (the other one has a longer and less pleasant statement but shares the same flavor) states that if \(p(z)\) is a polynomial of degree \(n\) having all its zeros in \(| z| \leq K\), \(K>1\), then for \(K<R<K^2\) \[ M(p,R) \geq R^s\bigl[\frac{R+K}{1+K}\bigr]^nM(p,1) + \frac{1}{K^n}\bigl\{R^n-R^s\bigl(\frac{R+K}{1+K}\bigr)\bigr\}m \] where \(M(p,R)\) is the maximum modulus of \(p\) for \(| z| =R\), \(m\) is the minimum modulus of \(p\) for \(| z| =K\), and \(s\) is the order of a possible zero of \(p(z)\) at \(z=0\). The novelty in this result lies in the second term of the sum on the right hand side. Without this term the inequality had been proved by V.K.Jain in 1999.