Some Lp Inequalities for the Polar Derivative of a Polynomial

✍ Scribed by N.K Govil; Griffith Nyuydinkong; Berhanu Tameru

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 95 KB
• Volume: 254
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7267

✦ Synopsis

Let p n z be a polynomial of degree n and D α p n z its polar derivative. It has been proved that if p n z has no zeros in z < 1, then for δ ≥ 1 and α ≥ 1, 2π 0 D α p n e iθ δ dθ 1/δ ≤ n α + 1 F δ 2π 0 p n e iθ δ dθ 1/δ where F δ = 2π/ 2π 0

1 + e iθ δ dθ 1/δ . We also obtain analogous inequalities for the class of polynomials having all their zeros in z ≤ 1 and for the class of polynomials satisfying p n z ≡ z n p n 1/ z .