Simple proofs of Bernstein-type inequalities

Mohapatra, R. N.; O’Hara, P. J.; Rodriguez, R. S.

doi:10.1090/S0002-9939-1988-0928994-0

Simple proofs of Bernstein-type inequalities
HTML articles powered by AMS MathViewer

by R. N. Mohapatra, P. J. O’Hara and R. S. Rodriguez PDF

Proc. Amer. Math. Soc. 102 (1988), 629-632 Request permission

Abstract:

A polynomial identity is established by the use of Lagrange interpolation. This identity is used to obtain simple proofs of Bernstein-type inequalities, one of which is an improvement of a recent result of Frappier, Rahman, and Ruscheweyh.

References

C. Frappier, Q. I. Rahman, and St. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc. 288 (1985), no. 1, 69–99. MR 773048, DOI 10.1090/S0002-9947-1985-0773048-1
N. K. Govil and Q. I. Rahman, Functions of exponential type not vanishing in a half-plane and related polynomials, Trans. Amer. Math. Soc. 137 (1969), 501–517. MR 236385, DOI 10.1090/S0002-9947-1969-0236385-6
N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543–546. MR 325932, DOI 10.1090/S0002-9939-1973-0325932-8
M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. (2) 1 (1969), 57–60. MR 249583, DOI 10.1112/jlms/s2-1.1.57
M. A. Malik, An integral mean estimate for polynomials, Proc. Amer. Math. Soc. 91 (1984), no. 2, 281–284. MR 740186, DOI 10.1090/S0002-9939-1984-0740186-3
P. J. O’Hara, Another proof of Bernstein’s theorem, Amer. Math. Monthly 80 (1973), 673–674. MR 322143, DOI 10.2307/2319171
P. J. O’Hara and R. S. Rodriguez, Some properties of self-inversive polynomials, Proc. Amer. Math. Soc. 44 (1974), 331–335. MR 349967, DOI 10.1090/S0002-9939-1974-0349967-5
Qazi Ibadur Rahman, Applications of functional analysis to extremal problems for polynomials, Séminaire de Mathématiques Supérieures, No. 29 (Été, vol. 1967, Les Presses de l’Université de Montréal, Montreal, Que., 1968. MR 0251195
W. W. Rogosinski, Extremum problems for polynomials and trigonometrical polynomials, J. London Math. Soc. 29 (1954), 259–275. MR 62859, DOI 10.1112/jlms/s1-29.3.259
E. B. Saff and T. Sheil-Small, Coefficient and integral mean estimates for algebraic and trigonometric polynomials with restricted zeros, J. London Math. Soc. (2) 9 (1974/75), 16–22. MR 382609, DOI 10.1112/jlms/s2-9.1.16
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

Über einen Satz des Herrn Serge Bernstein

5