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 A219245 The 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. 6
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS We note that the maximum M(n) of the ratio (sum{k=1,2,...,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 the 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) in A219246 and A219336 respectively are given. 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 EXAMPLE M(4) = 1.42084438540961... 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 *) CROSSREFS Cf. A219246, A219336, A249403. Sequence in context: A200496 A058546 A196774 * A299769 A091435 A330472 Adjacent sequences:  A219242 A219243 A219244 * A219246 A219247 A219248 KEYWORD nonn,cons AUTHOR Roman Witula, Nov 16 2012 STATUS approved

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Last modified January 18 21:06 EST 2021. Contains 340262 sequences. (Running on oeis4.)