(Bohr 1935). A related inequality states that if is the class of functions such that
(2)
are absolutely continuous and , then
(3)
(Northcott 1939). Further, for each value of , there is always a function belonging to and not identically zero, for which the above inequality
becomes an equality (Favard 1936). These inequalities are discussed in Mitrinovic
et al. (1991).
Bohr, H. "Ein allgemeiner Satz über die Integration eines trigonometrischen Polynoms." Prace Matem.-Fiz.43, 1935.Favard,
J. "Application de la formule sommatoire d'Euler à la démonstration
de quelques propriétés extrémales des intégrale des fonctions
périodiques ou presquepériodiques." Mat. Tidsskr. B, 81-94,
1936. Reviewed in Zentralblatt f. Math.16, 58-59, 1939.Mitrinovic,
D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands:
Kluwer, pp. 71-72, 1991.Northcott, D. G. "Some Inequalities
Between Periodic Functions and Their Derivatives." J. London Math. Soc.14,
198-202, 1939.Tikhomirov, V. M. "Approximation Theory."
In Analysis
II. Convex Analysis and Approximation Theory (Ed. R. V. Gamkrelidze).
New York: Springer-Verlag, pp. 93-255, 1990.
has no spectrum in ![[-lambda,lambda]](/images/equations/Bohr-FavardInequalities/Inline2.svg)
, then
is the class of functions such that
, then
, there is always a function 
belonging to 
and not identically zero, for which the above inequality
becomes an equality (Favard 1936). These inequalities are discussed in Mitrinovic
et al. (1991).
