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 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. 1
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants. LINKS 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. Steven R. Finch, Errata and Addenda to Mathematical Constants, January 22, 2016. [Cached copy, with permission of the author] 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

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Last modified September 20 01:58 EDT 2019. Contains 327207 sequences. (Running on oeis4.)