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%I #33 Jul 01 2023 14:34:09
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C The decimal expansions of M(5) and M(6) are A219246 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 M(4) = 1.42084438540961...
%t 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 *)
%Y Cf. A219246, A219336, A249403.
%K nonn,cons
%O 1,2
%A _Roman Witula_, Nov 16 2012
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