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A258042
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Decimal expansion of least constant a0 such that, for all a >= a0, log(a + x) is submultiplicative on [1, +oo).
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0
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1, 7, 5, 5, 0, 6, 9, 6, 5, 8, 3, 7, 6, 8, 2, 4, 1, 8, 8, 1, 0, 9, 9, 4, 6, 4, 7, 1, 3, 3, 9, 1, 0, 6, 2, 4, 7, 7, 9, 4, 2, 3, 8, 0, 8, 3, 0, 2, 8, 2, 6, 8, 3, 5, 4, 2, 3, 5, 7, 6, 7, 5, 8, 3, 1, 2, 5, 1, 4, 0, 1, 2, 3
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OFFSET
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1,2
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COMMENTS
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A function f is submultiplicative on S if, for any x and y in S, f(x)f(y) <= f(xy).
Gustavsson, Maligranda, & Peetre give as an example Zafran's use of log(e^2 + x) as a submultiplicative function. Since e^2 is greater than this constant it is submultiplicative on x >= 1. - Charles R Greathouse IV, Sep 13 2017
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LINKS
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EXAMPLE
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1.7550696583768241881099464713391062477942380830282683542357675831251401...
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PROG
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(PARI) solve(a=1.7, 1.8, my(x=solve(y=1.8, 2, y*(a+y)*log(a+y)-(a+y^2)*log(a+y^2))); log(a+x^2)/log(a+x)^2-1)
(PARI) F(x, a)=[x*(a+x)-(a+x^2)*log(a+x), log(a+x^2)-log(a+x)^2]~
dF(x, a)=my(L=log(x+a)); [((-2*L+1)*x^2+(-2*L+3)*a*x+(a^2-a))/(x+a), ((a-L)*x+(-L-1)*a)/(x+a); ((-2*L+2)*x^2+2*a*x-2*L*a)/(x^3+a*x^2+a*x+a^2), (-2*L*x^2+x+(-2*L+1)*a)/(x^3+a*x^2+a*x+a^2)]
\p1000
V=[1.8, 1.7]~; for(i=1, 11, V-=matsolve(dF(V[1], V[2]), F(V[1], V[2]))); V[2] \\ Bill Allombert, May 26 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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