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%I #35 Jul 01 2023 14:34:05
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C The decimal expansions of M(4) and M(6) are A219245 and A219336, respectively.
%D 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.
%H Steven R. Finch, <a href="/A219245/a219245.pdf">Carleman's inequality</a>, 2013. [Cached copy, with permission of the author]
%H Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, <a href="http://www.emis.de/journals/AMAPN/vol24_2/7.html">The Best Constant for Carleman's Inequality of Finite Type</a>, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008
%e 1.486353228963....
%t 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 *)
%Y Cf. A219245, A219336, A249403.
%K nonn,cons
%O 1,2
%A _Roman Witula_, Nov 16 2012
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