The goal of this paper is to establish some results for the polar derivative of
a polynomial in the plane that are inspired by a classical result of Turan that relates the
sup-norm of the derivative on the unit circle to that of the polynomial itself under some
conditions. The obtained results sharpen as well as generalize some known estimates that
relate the sup-norm of the polar derivative and the polynomial. Moreover, some concrete
numerical examples are presented, showing that in some situations, the bounds obtained by
our results can be considerably sharper than the previous ones known in very rich literature
on this subject.