A new proof and a generalization of a theorem of de Bruijn

Aziz, Abdul

doi:10.1090/S0002-9939-1989-0933511-6

A new proof and a generalization of a theorem of de Bruijn
HTML articles powered by AMS MathViewer

by Abdul Aziz

Proc. Amer. Math. Soc. 106 (1989), 345-350

DOI: https://doi.org/10.1090/S0002-9939-1989-0933511-6

PDF | Request permission

Abstract:

Using a recently developed interpolation formula, we present elementary new and simple proofs of De Bruijn’s theorem and Zygmund’s inequality concerning the integral mean estimates for polynomials. We also present a generalization of De Bruijn’s theorem which leads to a refinement of a theorem of Erdös and Lax.

References

Abdul Aziz and Q. G. Mohammad, Simple proof of a theorem of Erdős and Lax, Proc. Amer. Math. Soc. 80 (1980), no. 1, 119–122. MR 574519, DOI 10.1090/S0002-9939-1980-0574519-4
N. G. de Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch., Proc. 50 (1947), 1265–1272 = Indagationes Math. 9, 591–598 (1947). MR 23380
K. K. Dewan and N. K. Govil, An inequality for self-inversive polynomials, J. Math. Anal. Appl. 95 (1983), no. 2, 490. MR 716098, DOI 10.1016/0022-247X(83)90122-1
Peter D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513. MR 10731, DOI 10.1090/S0002-9904-1944-08177-9
Q. I. Rahman, Functions of exponential type, Trans. Amer. Math. Soc. 135 (1969), 295–309. MR 232938, DOI 10.1090/S0002-9947-1969-0232938-X
A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565–579. MR 5163, DOI 10.1090/S0002-9904-1941-07510-5

A remark on conjugate series

34