On maximum modulus of polar derivative of a polynomial

Abstract

Let p(z) be a polynomial of degree n and for any real or complex number α, Dα p(z) denotes the polar derivative of p(z) with respect to α, then D_αp(z) = np(z) + (α −z)p’(z). In this paper, we consider the more general class of polynomials p(z) = a₀ + ∑ a_νz^ν, 1 ≤ µ ≤ n, not vanishing in |z| < k, k > 0, to estimate max |D_αp(z)| in terms of max |p(z)| by involving some coefficients of p(z), where 0 < r ≤ ρ ≤ k. Interestingly, the results improve and extend other well known inequalities to polar derivative. Moreover, our results give several interesting results as special cases.