Inequalities Concerning the Lp Norm of a Polynomial and Its Derivative
Let Pn(z) = an ∏nv = 1 (z - zv), an ≠ 0, be a polynomial of degree n. It has been proved that if |zv| ≥ Kv ≥ 1, 1 ≤ v ≤ n, then for p ≥ 1, [formula] where Fp = (2π/∫2π0 |t0 + eiΘ)pdΘ)1/p and t0 = (1 + n/∑nv=1 (1/(Kv - 1))). This result generalizes the well known Lp inequality due to De Bruijn for polynomials not vanishing in |z| < 1. On making p → ∞, it gives the L∞ inequality due to Govil and Labelle which as a special case includes the Erdo(combining double acute accent)s conjecture proved by Lax.
Gardner, Robert B.; and Govil, Narendra K.. 1993. Inequalities Concerning the Lp Norm of a Polynomial and Its Derivative. Journal of Mathematical Analysis and Applications. Vol.179(1). 208-213. https://doi.org/10.1006/jmaa.1993.1345 ISSN: 0022-247X