A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2*f'(x)^2 dx <= K*integral_{0..1} f'(x)^4 dx.

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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants.

Links

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

David W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. Volume 30, Number 2 (1969), 367-383

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 30.

Formula

24/(2*sqrt(3) + 3*sqrt(2)*arcsinh(sqrt(2)))^2.

Example

0.34611896560593345099609054206876591598395281385974864...

Mathematica

(* Using Boyd's formula *) I0[p_, q_, r_] := Integrate[(((q - 1)*t + 1)*t^(1/p - 1))/ (((r*(q - 1))*t)/(r - q) + 1)^((r*p + p + q)/(r*p)), {t, 0, 1}]; K[p_, q_, r_] := (beta = (((r - 1)*p + (r - q))/((r - 1)*(p + q)))^(1/r); (((r - q)*p^p)*beta^(p + q - r))/(I0[p, q, r]^p*((r - 1)*(p + q)))); RealDigits[K[2, 2, 4], 10, 104] // First

Crossrefs

Cf. A242440, A244263, A244347, A244354, A245292.

Sequence in context: A085196 A021293 A275979 * A122601 A125776 A066977

Adjacent sequences: A246856 A246857 A246858 * A246860 A246861 A246862

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 05 2014

Status

approved