A245198 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).

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

Offset: 1

Author

Jean-François Alcover, Jul 17 2014

Keywords

• nonn
• cons
• easy

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 (-infinity, infinity), the involved norm being the supremum norm, defined by ||f|| = sup |f(x)|.

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

Kolmogorov determined best constants C(n,k), 1 <= k <= n, for the inequality between derivatives in terms of Favard constants (A050970/A050971). These formulas giving C(n,k) include special cases discovered by G. E. Shilov for small values of n and k.

[All comments made after Steven R. Finch].

Examples

			1.0400419115259520572650284121789426931689026701866310548487955401...
		

References

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

Links

• Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
• Eric Weisstein's MathWorld, Favard Constants

Crossrefs

Cf. A050970, A050971, A244091.

Programs

Mathematica

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[3, 1], 10, 104] // First
(* or, directly: *) RealDigits[3^(2/3)/2, 10, 104] // First

Formula

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(3,1) = 3^(2/3)/2 = (9/8)^(1/3).