Integral norm estimates for the polar derivative of Lacunary-type complex polynomials

In this paper, we prove some integral-norm inequalities for the polar derivative of Lacunary-type complex polynomials having zeros in closed exterior or closed interior of a circle. The results obtained besides derive polar derivative analogues of some classical Bernstein and Tur?n-type inequalities for the uniform-norm also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials as well.

Ricci curvature decay and upper bound of total mass

We obtain a positive upper bound of total mass from a relative volume comparison by a weighted integral norm of Ricci curvature.