On an Operator Preserving Inequalities Between Polynomials

Generalizations of some Bernstein-type Inequalities for the Polar Derivative of a Polynomial

“…Many of these generalizations involve the comparison of polar derivative D β P (z) with various choices of P (z), β and other parameters. The latest research and development on this topic can be found in the papers ( [5,8,10,11,[13][14][15][16]20]).…”

Estimates for the polar derivative of a constrained polynomial on a disk

“…In the literature, for proving the inverse theorems in approximation theory, many inequalities in both directions relating the norm of the derivative and the polynomial itself play a significant role and, of course, have their own intrinsic appeal. As shown by various recent studies, numerous research papers have been published on these inequalities for constrained polynomials (for example, see [11,13,17,19,20,21]). We begin with the well-known Bernstein inequality [4] for the uniform norm on the unit disk in the plane: namely, if P (z) is a polynomial of degree n, then…”

Inequalities concerning the rate of growth of polynomials involving the polar derivative

“…The obtained inequalities generalize some known results for ordinary derivatives and provide crucial tools in obtaining inverse theorems in approximation theory. For the latest research and development on this topic one can consult the papers [7,[12][13][14][15]17]. Recently, Somsuwan and Nakprasit [17] extended (1.6) for the case k ≥ 1 to the polar derivative of P (z) and proved the following result.…”