A258042 Decimal expansion of least constant a0 such that, for all a >= a0, log(a + x) is submultiplicative on [1, +oo).

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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..73.

Jan Gustavsson, Lech Maligranda, Jaak Peetre, A submultiplicative function, Indagationes Mathematicae, 92:4 (1989), pp. 435-442.

Misha Zafran, The dichotomy problem for homogeneous Banach algebras, Annals of Mathematics Second Series, 108:1 (Jul., 1978), pp. 97-105. See p. 103.

Example

1.7550696583768241881099464713391062477942380830282683542357675831251401...

Prog

(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

Crossrefs

Sequence in context: A289033 A354644 A010510 * A340710 A289003 A295219

Adjacent sequences: A258039 A258040 A258041 * A258043 A258044 A258045

Keyword

nonn,cons,more

Author

Charles R Greathouse IV, May 22 2015

Status

approved