Let $P(z)$ be a polynomial and $P'(z)$ its derivative. In this paper, we shall obtain certain compact generalizations and sharp refinements of some results of Govil, Malik, Turán and others concerning the maximum modulus of $P(z)$ and $P'(z)$ on the unit circle $|z|=1$, which also yields a number of other interesting results for various choices of parameters.