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A219246
Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.
5
1, 4, 8, 6, 3, 5, 3, 2, 2, 8, 9, 6, 3, 0, 5, 0, 6, 4, 0, 5, 2, 0, 4, 8, 7, 1, 6, 4, 6, 1, 9, 8, 5, 1, 5, 6, 6, 4, 3, 5, 4, 6, 9, 5, 6, 4, 1, 0, 0, 9, 3, 7, 9, 4, 5, 3, 2, 5, 3, 3, 5, 5, 8, 8, 2, 3, 9, 8, 9, 3, 8, 1, 0, 1, 4, 8, 1, 5, 9, 8, 7, 5, 5, 6, 6, 2, 4, 1, 9, 0, 0, 7, 4, 6, 1, 1, 3, 2, 2, 4, 4, 7
OFFSET
1,2
COMMENTS
The maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is discussed in A219245 - see also the paper of Witula et al. for the proofs.
The decimal expansions of M(4) and M(6) are A219245 and A219336, respectively.
REFERENCES
R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.
LINKS
Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008
EXAMPLE
1.486353228963....
MATHEMATICA
RealDigits[c5/.FindRoot[{1+x2/2+x3/3+x4/4+x5/5==c5, x2/2+x3/3+x4/4+x5/5==c5*x2^2, x3/3+x4/4+x5/5==c5*x3^3/x2^2, x4/4+x5/5==c5*x4^4/x3^3, x5/5==c5*x5^5/x4^4}, {{c5, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}}, WorkingPrecision->120], 10, 105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Roman Witula, Nov 16 2012
STATUS
approved