

A245198


Decimal expansion of the LandauKolmogorov constant C(3,1) for derivatives in the case L_infinity(infinity, infinity).


7



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

The corresponding LandauKolmogorov 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 realvalued 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].


REFERENCES

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


LINKS

Table of n, a(n) for n=1..104.
Eric Weisstein's MathWorld, LandauKolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants


FORMULA

C(n,k) = a(nk)*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 nth Favard constant A050970(n)/A050971(n).
C(3,1) = 3^(2/3)/2 = (9/8)^(1/3).


EXAMPLE

1.0400419115259520572650284121789426931689026701866310548487955401...


MATHEMATICA

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


CROSSREFS

Cf. A050970, A050971, A244091.
Sequence in context: A035639 A284689 A037214 * A203285 A203542 A204301
Adjacent sequences: A245195 A245196 A245197 * A245199 A245200 A245201


KEYWORD

nonn,cons,easy


AUTHOR

JeanFrançois Alcover, Jul 17 2014


STATUS

approved



