A219336 - OEIS

%I #35 Jul 01 2023 14:34:02

%S 1,5,3,7,9,3,7,5,5,6,5,2,0,0,3,4,9,3,1,3,6,8,1,5,8,7,1,6,0,2,6,3,2,6,

%T 8,1,5,6,0,8,6,4,5,0,8,9,8,6,3,2,1,9,6,3,3,3,2,4,6,4,3,1,1,6,3,0,0,9,

%U 2,7,6,4,1,4,2,6,1,2,9,3,4,2,5,2,3,7,7,9,3,8,0,1,3,1,4,4,2,2,9,9,5,1,9

%N Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 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(5) are A219245 and A219246, 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 1.537937556520034931368158716...

%t RealDigits[c6/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6*x2^2, x3/3 + x4/4 + x5/5 + x6/6 == c6*x3^3/x2^2, x4/4 + x5/5 + x6/6 == c6*x4^4/x3^3, x5/5 + x6/6 == c6*x5^5/x4^4, x6/6 == c6*x6^6/x5^5},{{c6,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2},{x6,1/2}},WorkingPrecision->120],10,105][[1]] (* _Vaclav Kotesovec_, Oct 27 2014 *)

%Y Cf. A219245, A219246, A249403.

%K nonn,cons

%O 1,2

%A _Roman Witula_, Nov 18 2012