On Bernstein-Type Inequalities for the Polar Derivative of a Polynomial

If P(z) is a polynomial of degree n, and α a complex number, then polar derivative of P(z) with respect to the point α, denoted by D α P(z), is defined by D α P ( z ) = n P ( z ) + ( α − z ) P ′ ( z ) . $$\displaystyle{D_{\alpha }P(z) = nP(z) + (\alpha -z)P^{{\prime}}(z).}$$ Clearly, D α P(z) is a polynomial of degree n − 1, and it generalizes the ordinary derivative in the sense that lim α → ∞ D α P ( z ) α = P ′ ( z ) . $$\displaystyle{\lim _{\alpha \rightarrow \infty }\left [\frac{D_{\alpha }P(z)} {\alpha } \right ] = P^{{\prime}}(z).}$$ It is well known that if P(z) is a polynomial of degree n, then max | z | = 1 | P ′ (z) | ≤ nmax | z | = 1 | P(z) | . This inequality is known as Bernstein’s inequality (Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle. Gauthier-Villars, Paris, 1926), although this inequality was also proved by Riesz (Jahresber Dtsch Math-Verein 23:354–368, 1914) about 12 years before it was proved by Bernstein. The subject of inequalities for polynomials and related classes of functions plays an important and crucial role in obtaining inverse theorems in Approximation Theory. Frequently, the further progress in inverse theorems has depended upon first obtaining the corresponding analogue or generalization of Markov’s and Bernstein’s inequalities. These inequalities have been the starting point of a considerable literature in Mathematics, and was one of the areas in which Professor Q. I. Rahman worked for more than 50 years, and made some of the most important and significant contributions. Over a period, this Bernstein’s inequality and corresponding inequality concerning the growth of polynomials have been generalized in different domains, in different norms, and for different classes of functions, and in the literature one can find hundreds of papers on this topic. Here we study some of the research centered around Bernstein-type inequalities for polar derivatives of polynomials. The chapter is purely expository in nature and an attempt has been made here to provide results starting from the beginning to some of the most recent ones.