OFFSET
0,1
COMMENTS
Positive matrices do not always behave like positive numbers: for instance 0 <= A <= B does not imply e^A <= e^B [e^A stands here for matrix exponential of A, and 0 <= A <= B means that A and B-A are positive matrices, a positive matrix being a matrix which has nonnegative entries]. The counterexample given by Paul Halmos is A = e^{{1/2, 1/2}, {1/2, 1/2}} = {{(1+e)/2, (e-1)/2}, {(e-1)/2, (1+e)/2}} and B = {{e^x, 0}, {0, e^y}}, with x < log((1+e)/2).
REFERENCES
Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 9Q (Exponential inequality) p. 273.
FORMULA
Equals Integral_{x=0..1} 1/(exp(-x) + 1) dx. - Amiram Eldar, Aug 16 2020
EXAMPLE
0.620114506958277524631763373509679073839779951310093120598383505343801279539...
MATHEMATICA
RealDigits[Log[(1 + E)/2], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jul 22 2013
STATUS
approved