Refinements of inequalities on extremal problems of polynomials

Let H ( z ) be a polynomial of degree n , and for any complex number α , let D α H ( z ) = nH ( z ) + ( α − z ) H ′( z ) denote the polar derivative of H ( z ) with respect to α . In this paper, we establish refined versions of Bernstein and Turán-type inequalities, providing point-wise estimates for the modulus of both the derivative and the polar derivative of a polynomial. These results generalize and refine classical inequalities, offering broader implications and insights into related inequalities.