A219336 -id:A219336 - cp's OEIS frontend

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

Roman Witula, Nov 16 2012

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.

			M(4) = 1.42084438540961...

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.

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.

Cf. A219246, A219336, A249403.

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 *)

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 16 2012

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(6) are A219245 and A219336, respectively.

			1.486353228963....

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.

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

Cf. A219245, A219336, A249403.

RealDigits[c5/.FindRoot[{1+x2/2+x3/3+x4/4+x5/5==c5, x2/2+x3/3+x4/4+x5/5==c5*x2^2, x3/3+x4/4+x5/5==c5*x3^3/x2^2, x4/4+x5/5==c5*x4^4/x3^3, x5/5==c5*x5^5/x4^4},{{c5,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2}},WorkingPrecision->120],10,105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

A249403 Decimal expansion of the maximum M(7) of the ratio (Sum_{k=1..7} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(7)) taken over x(1), ..., x(7) > 0.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 27 2014

M(2) = (1+sqrt(2))/2, M(3) = 4/3.

M(n) = exp(1) - 2*Pi^2*exp(1)/(log(n))^2 + O(1/(log(n))^3), [de Bruijn, 1963].

			1.5800372106320523520840634981832644921128158059165961970174236920601537371...

N. G. de Bruijn, Carleman's inequality for finite series, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag, Math., 25:505-514, 1963.
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.

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.

Cf. A174968 = M(2), A219245 = M(4), A219246 = M(5), A219336 = M(6).

RealDigits[c7/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x2^2, x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x3^3/x2^2, x4/4 + x5/5 + x6/6 + x7/7 == c7*x4^4/x3^3, x5/5 + x6/6 + x7/7 == c7*x5^5/x4^4, x6/6 + x7/7 == c7*x6^6/x5^5, x7/7 == c7*x7^7/x6^6}, {{c7, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}, {x6, 1/2}, {x7, 1/2}}, WorkingPrecision->120], 10, 105][[1]]

A249276 Numerators of fractions appearing in a generalization of Carleman's inequality.

Original entry on oeis.org

-1, 1, 1, 1, 73, 11, 3625, 5525, 5233001, 1212281, 927777937, 772193, 43791735453787, 6889178449747, 158996102434867, 107876982981287, 782501215247703271, 6225541612992329, 235541803917995571502409

Offset: 0

Jean-François Alcover, Oct 24 2014

			Fractions begin -1, 1/2, 1/24, 1/48, 73/5760, 11/1280, 3625/580608, ...

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Carleman's Inequality
Wikipedia, Carleman's inequality

Cf. A249277 (denominators), A219245, A219246, A219336.

b[0] = -1; b[n_] := b[n] = (-1/n)*Sum[b[n - k]/(k + 1), {k, 1, n}]; Table[b[n] // Numerator, {n, 0, 20}]
CoefficientList[Series[-(1 - x)^(-(1 - x)/x)/E, {x, 0, 20}], x] // Numerator (* Eric W. Weisstein, Apr 13 2018 *)

sum_{k >= 1} (a_1 a_2 ... a_k)^(1/k) < e*sum_{k >= 1} (1-sum_{j=1..m} b_j/(k+1)^j)*a_k, where a_k >= 0 for all k and a_l > 0 for at least one l, m being any positive integer.

b(0) = -1, b(n) = (-1/n)*sum_{k=1..n} b(n-k)/(k+1).

A249277 Denominators of fractions appearing in a generalization of Carleman's inequality.

Original entry on oeis.org

1, 2, 24, 48, 5760, 1280, 580608, 1161216, 1393459200, 398131200, 367873228800, 363331584, 24103053950976000, 4382373445632000, 115694658964684800, 88995891511296000, 726206474732175360000, 6455168664286003200

Offset: 0

Jean-François Alcover, Oct 24 2014

			Fractions begin -1, 1/2, 1/24, 1/48, 73/5760, 11/1280, 3625/580608, ...

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Carleman's Inequality
Wikipedia, Carleman's inequality

Cf. A249276 (numerators), A219245, A219246, A219336.

b[0] = -1; b[n_] := b[n] = (-1/n)*Sum[b[n - k]/(k + 1), {k, 1, n}]; Table[b[n] // Denominator, {n, 0, 20}]
CoefficientList[Series[-(1 - x)^((x - 1)/x)/E, {x, 0, 20}], x] // Denominator (* communicated by Eric W. Weisstein, Apr 13 2018, based on result by Michael Trott *)

b(0) = -1, b(n) = (-1/n)*sum_{k=1..n} b(n-k)/(k+1).

cp's OEIS Frontend

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.

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A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

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A249403 Decimal expansion of the maximum M(7) of the ratio (Sum_{k=1..7} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(7)) taken over x(1), ..., x(7) > 0.

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Mathematica

A249276 Numerators of fractions appearing in a generalization of Carleman's inequality.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A249277 Denominators of fractions appearing in a generalization of Carleman's inequality.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

A249403 Decimal expansion of the maximum M(7) of the ratio (Sum_{k=1..7} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(7)) taken over x(1), ..., x(7) > 0.