Abstract
LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1,
where\(B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} \) andP (s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.
Similar content being viewed by others
References
Arestov V V, On integral inequalities for trigonometric polynomials and their derivatives.Izv. Akad. Nauk. SSSR. Ser. Mat. 45 (1981) 3–22
De Bruijn N G, Inequalities concerning polynomials in the complex domain,Nederl. Akad. Wetensch. Proc. Ser. A50 (1947) 1265–1272;Indag. Math. 9 (1947) 591–598
Dewan K K and Govil N K, Some integral inequalities for polynomialsIndian J. Pure Appl. Math. 14 (4) (1983) 440–443
Govil N K and Rahman Q I, Functions of exponential type not vanishing in a half plane and related polynomials,Trans. Am. Math. Soc. 137 (1969) 501–517
Rahman Q I and Schmeisser G,L p inequalities for polynomials. J. Approx. Theory53 (1988) 26–32
Zygmund A, A remark on conjugate series,Proc. London Math. Soc. 34 (2) (1932) 392–400
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aziz, A., Shah, W.M. L p inequalities for polynomials with restricted zeros. Proc. Indian Acad. Sci. (Math. Sci.) 108, 63–68 (1998). https://doi.org/10.1007/BF03161313
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03161313