## Landau-Kolmogorov Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let be the Supremum of , a real-valued function defined on . If is twice differentiable and both and are bounded, Landau (1913) showed that

 (1)

where the constant 2 is the best possible. Schoenberg (1973) extended the result to the th derivative of defined on if both and are bounded,
 (2)

An explicit Formula for is not known, but particular cases are
 (3) (4) (5) (6) (7)

Let be the Supremum of , a real-valued function defined on . If is twice differentiable and both and are bounded, Hadamard (1914) showed that

 (8)

where the constant is the best possible. Kolmogorov (1962) determined the best constants for
 (9)

in terms of the Favard Constants
 (10)

by
 (11)

Special cases derived by Shilov (1937) are
 (12) (13) (14) (15) (16) (17) (18)

For a real-valued function defined on , define

 (19)

If is differentiable and both and are bounded, Hardy et al. (1934) showed that
 (20)

where the constant is the best possible for all and .

For a real-valued function defined on , define

 (21)

If is twice differentiable and both and are bounded, Hardy et al. (1934) showed that
 (22)

where the constant is the best possible. This inequality was extended by Ljubic (1964) and Kupcov (1975) to
 (23)

where are given in terms of zeros of Polynomials. Special cases are
 (24) (25) (26) (27) (28) (29)

where is the least Positive Root of
 (30)

and is the least Positive Root of
 (31)

(Franco et al. 1985, Neta 1980). The constants are given by
 (32)

where is the least Positive Root of
 (33)

An explicit Formula of this type is not known for .

The cases , 2, are the only ones for which the best constants have exact expressions (Kwong and Zettl 1992, Franco et al. 1983).

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lk/lk.html

Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. Bounds for the Best Constants in Landau's Inequality on the Line.'' Proc. Roy. Soc. Edinburgh 95A, 257-262, 1983.

Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. Best Constants in Norm Inequalities for Derivatives on a Half Line.'' Proc. Roy. Soc. Edinburgh 100A, 67-84, 1985.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities. Cambridge, England: Cambridge University Press, 1934.

Kolmogorov, A. On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Integral.'' Amer. Math. Soc. Translations, Ser. 1 2, 233-243, 1962.

Kupcov, N. P. Kolmogorov Estimates for Derivatives in .'' Proc. Steklov Inst. Math. 138, 101-125, 1975.

Kwong, M. K. and Zettl, A. Norm Inequalities for Derivatives and Differences. New York: Springer-Verlag, 1992.

Landau, E. Einige Ungleichungen für zweimal differentzierbare Funktionen.'' Proc. London Math. Soc. Ser. 2 13, 43-49, 1913.

Landau, E. Die Ungleichungen für zweimal differentzierbare Funktionen.'' Danske Vid. Selsk. Math. Fys. Medd. 6, 1-49, 1925.

Ljubic, J. I. On Inequalities Between the Powers of a Linear Operator.'' Amer. Math. Soc. Trans. Ser. 2 40, 39-84, 1964.

Neta, B. On Determinations of Best Possible Constants in Integral Inequalities Involving Derivatives.'' Math. Comput. 35, 1191-1193, 1980.

Schoenberg, I. J. The Elementary Case of Landau's Problem of Inequalities Between Derivatives.'' Amer. Math. Monthly 80, 121-158, 1973.