We obtain a generalization of Bernstein's result that if $p(z)$ and $q(z)$ are two polynomials with degree of $p(z)$ not exceeding that of $q(z)$ and $q(z)$ has all its zeros in $|z|\leq1$, with $|p(z)|\leq|q(z)|$, $|z|=1$, then $|p'(z)|\leq|q'(z)|$, $|z|=1$, and use the generalization so obtained to obtain two more generalizations. Three generalizations together turn out to be generalizations of many well known inequalities for polynomials, including Bernstein's inequality and inequality of the well known Erdös--Lax theorem.