A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

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

Offset: 1

Author

Roman Witula, Nov 16 2012

Keywords

• nonn
• cons

Comments

We note that 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 equal to (1+sqrt(2))/2 for n=2 and 4/3 for n=3. Moreover it can be proved that M(n) < (1 + 1/n)^(n-1) - it is a finite version of Carleman's inequality (see the paper of Witula et al. for the proof). The sequence M(n), n=2,3,..., is increasing.

The decimal expansions of M(5) and M(6) are A219246 and A219336, respectively.

Examples

			M(4) = 1.42084438540961...
		

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. A219246, A219336, A249403.

Programs

Mathematica

RealDigits[N[Root[387420489 + 22039921152*#1 + 373658292864*#1^2 + 12841816536576*#1^3 + 274965186525696*#1^4 - 201976270848000*#1^5 + 42624005978423296*#1^6 + 342213608420278272*#1^7 + 660475521813381120*#1^8 - 2629784260986273792*#1^9 + 41447678188009291776*#1^10 + 427447433656163893248*#1^11 - 198705178996352483328*#1^12 - 2098418839125516877824*#1^13 + 16905530303693690241024*#1^14 + 14417509185682352898048*#1^15 - 20033038006659651207168*#1^16 - 149735761790067869220864*#1^17 + 18738444188050884919296*#1^18 + 361130725214496730644480*#1^19 + 220843507713085418766336*#1^20 - 1387347813563214701002752*#1^21 + 1472163837099830446915584*#1^22 - 654295038711035754184704*#1^23 + 109049173118505959030784*#1^24 & , 4], 105]][[1]] (* Vaclav Kotesovec, Oct 26 2014 *)