Bernstein type L_p inequalities for composition of polynomials - Malik, Shabir Ahmad (Sarajevo Journal of Mathematics, 2024)
Let P(z)=a_0+_v=μ^n a_v z^vP_n, μ and P(z) eq 0 for |z|<k, where k≥ 1, then for 0≤ p≤∞, Gardner and Weems (J. Math. Anal. Appl. 219 (1998), 472-478) proved that align* \|P'\|_p≤n\|k^μ+z\|_p \|P\|_p. align* In this note, we consider a more general class of polynomials P° QP_mn, μ defined by (P° Q)(z)=P(Q(z)), where QP_m and provide an extension of the above inequality and related results.