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%I #11 Jan 17 2020 16:12:27
%S 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,
%T 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,
%U 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
%N 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.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants.
%H David W. Boyd, <a href="http://projecteuclid.org/euclid.pjm/1102978495">Best constants in a class of integral inequalities</a>, Pacific J. Math. Volume 30, Number 2 (1969), 367-383
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 30.
%F 24/(2*sqrt(3) + 3*sqrt(2)*arcsinh(sqrt(2)))^2.
%e 0.34611896560593345099609054206876591598395281385974864...
%t (* 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
%Y Cf. A242440, A244263, A244347, A244354, A245292.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Sep 05 2014
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