uniqueness by the absense of bending points with respect to z in

sexp(slog(z) + r) for positive z and r.

( i use tet for 'my' sexp further )

let

tet(slog(x)) = x

tet(0) = 0 = slog(0)

consider

tet(slog(z) + r)

take derivate with respect to z.

tet'(slog(z) + r) x 1/tet'(slog(z))

take derivate with respect to z.

tet''(slog(z)+r)/tet'(slog(z))^2 - (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

hence

tet''(slog(z)+r)/tet'(slog(z))^2 = (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

thus

tet''(slog(z)+r) = tet'(slog(z)+r) * tet''(slog(z))/tet'(slog(z))

hence solve for positive z :

tet''(z+r) = tet'(z+r) * tet''(z)/tet'(z)

make symmetric

tet''(z+r)/tet'(z+r) = tet''(z)/tet'(z)

notice that if a z exists , another one must exist.

thus if for some r , a z exists , there exist oo z solutions.

take integral on both sides ( this step may be a bit dubious ? )

log(tet'(z+r)) + A = log(tet'(z)) + B

hence bending points in

sexp(slog(z) + r) correspond to bending points in sexp(z).

thus

all ( analytic ) sexp(z) with sexp(0) = 0 and positive real to positive real ,

without bending points are identical !!

headscratch ...

regards

sexp(slog(z) + r) for positive z and r.

( i use tet for 'my' sexp further )

let

tet(slog(x)) = x

tet(0) = 0 = slog(0)

consider

tet(slog(z) + r)

take derivate with respect to z.

tet'(slog(z) + r) x 1/tet'(slog(z))

take derivate with respect to z.

tet''(slog(z)+r)/tet'(slog(z))^2 - (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

hence

tet''(slog(z)+r)/tet'(slog(z))^2 = (tet'(slog(z)+r)) * tet''(slog(z))/tet'(slog(z))^3

thus

tet''(slog(z)+r) = tet'(slog(z)+r) * tet''(slog(z))/tet'(slog(z))

hence solve for positive z :

tet''(z+r) = tet'(z+r) * tet''(z)/tet'(z)

make symmetric

tet''(z+r)/tet'(z+r) = tet''(z)/tet'(z)

notice that if a z exists , another one must exist.

thus if for some r , a z exists , there exist oo z solutions.

take integral on both sides ( this step may be a bit dubious ? )

log(tet'(z+r)) + A = log(tet'(z)) + B

hence bending points in

sexp(slog(z) + r) correspond to bending points in sexp(z).

thus

all ( analytic ) sexp(z) with sexp(0) = 0 and positive real to positive real ,

without bending points are identical !!

headscratch ...

regards