Abstract
In this paper, we establish some new inequalities in the plane that are inspired by some classical Bernstein-type inequalities that relate the sup-norm of a polynomial to that of its derivative on the unit circle. The obtained results relate the $$L^\gamma $$ -norm of the polar derivative and the polynomial. Our results besides derive polar derivative analogues of some classical Bernstein-type inequalities also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials, as well.
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