A245201 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(0, infinity).

3, 1, 2, 0, 1, 2, 5, 7, 3, 4, 5, 7, 7, 8, 5, 6, 1, 7, 1, 7, 9, 5, 0, 8, 5, 2, 3, 6, 5, 3, 6, 8, 2, 8, 0, 7, 9, 5, 0, 6, 7, 0, 8, 0, 1, 0, 5, 5, 9, 8, 9, 3, 1, 6, 4, 5, 4, 6, 3, 8, 6, 6, 2, 0, 3, 0, 0, 1, 5, 9, 4, 6, 7, 0, 9, 5, 9, 0, 3, 1, 6, 0, 4, 0, 9, 0, 8, 4, 9, 5, 5, 1, 8, 6, 1, 3, 6, 2, 3, 0, 0, 9, 3, 0

Offset

1,1

Comments

The corresponding Landau-Kolmogorov inequality for the first and third derivative is ||f'|| <= C(3,1) ||f||^(2/3) ||f'''||^(1/3) [see S. Finch ref. for C(n,k) and the general derivative inequalities], where the real-valued function f is defined on (0, infinity), the involved norm being the supremum norm, defined by ||f|| = sup |f(x)|.

Landau proved that if f is twice-differentiable and both f and f'' are bounded, then ||f'|| <= 2 ||f||^(1/2) ||f''||^(1/2), and the constant C(2,1) = 2 is the best possible.

Best constants C(n,k), 1 <= k <= n, can be explicitly computed only for particular cases.

[All comments made after Steven R. Finch].

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Links

Table of n, a(n) for n=1..104.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants

Formula

C(3,1) = (243/8)^(1/3).

Example

3.120125734577856171795085236536828079506708010559893164546386620300159467...

Mathematica

RealDigits[(243/8)^(1/3), 10, 104] // First

Crossrefs

Cf. A244091.

Sequence in context: A294402 A145738 A232802 * A211358 A211356 A176107

Adjacent sequences: A245198 A245199 A245200 * A245202 A245203 A245204

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 17 2014

Status

approved