For an arbitrary entire functionf and anyr>0, letM(f,r):=max_|z|=r |f(z)|. It is known that ifp is a polynomial of degreen having no zeros in the open unit disc, andm:=min_|z|=1|p(z)|, then$$\begin{gathered} M(p',1) \leqslant \frac{n}{2}\{ M(p,1) - m), \hfill \\ M(p,R) \leqslant \left( {\frac{{R^n + 1}}{2}} \right)M(p,1) - m\left( {\frac{{R^n - 1}}{2}} \right),R > 1 \hfill \\ \end{gathered} $$ It is also known that ifp has all its zeros in the closed unit disc, then$$M(p',1) \geqslant \frac{n}{2}\{ M(p,1) - m\} $$. The present paper contains certain generalizations of these inequalities.

For a polynomial of degree 𝑛, we have obtained an upper bound involving coefficients of the polynomial, for moduli of its 𝑝 zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).