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%I #18 Jul 17 2014 10:41:18
%S 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,
%T 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,
%U 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
%N Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).
%C 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)|.
%C 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.
%C 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.
%C [All comments made after Steven R. Finch].
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Landau-KolmogorovConstants.html">Landau-Kolmogorov Constants</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/FavardConstants.html">Favard Constants</a>
%F 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).
%F C(3,1) = 3^(2/3)/2 = (9/8)^(1/3).
%e 1.0400419115259520572650284121789426931689026701866310548487955401...
%t 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
%t (* or, directly: *) RealDigits[3^(2/3)/2, 10, 104] // First
%Y Cf. A050970, A050971, A244091.
%K nonn,cons,easy
%O 1,3
%A _Jean-François Alcover_, Jul 17 2014
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