A219336 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.

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, 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, 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

Offset: 1

Author

Roman Witula, Nov 18 2012

Keywords

• nonn
• cons

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(5) are A219245 and A219246, respectively.

Examples

			1.537937556520034931368158716...
		

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.

Crossrefs

Cf. A219245, A219246, A249403.

Programs

Mathematica

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 *)