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A219246
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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.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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1.486353228963....
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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