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final uniqueness condition ... probably
#1
d f^2 / d x^2 sexp(slog(x) + k) > 0 for all real x and all real k with 0 < k < Q where Q is any nonzero positive real.

assuming sexp resp slog to be C^2 of course , i 'believe' this condition implies analytic as well.

is this equivalent to d f^2 / d x^2 sexp(x) > 0 for all positive real x ?

i assume because of the substitution x = sexp(y)

sexp(slog(x) + k) = sexp(y + k) d sexp(y) ...

i believe the analogue uniqueness condition for other function as exp(x) that dont have a real fixpoint , map R to R and safisfy f ' (real) > 0 and f '' (real) > 0 to hold.

(this is an improved version of an earlier thread.)

regards

tommy1729
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