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 Turán that relates the sup-norm of the
derivative on the unit circle to that of the polynomial itself (on
the unit circle) 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.
Key Words: Minimum modulus principle, Turán's
2010 Mathematics Subject Classification: Primary
30A10; Secondary 30C10, 26D10.
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