05/25/2011, 11:04 PM
i have been aware of this as well.
i always said : " ill consider this later " and thought " it is probably trivial and most regular posters can easily prove this "
apart from the actual proof , i came to conjecture - without the gut to post it nor the effort to check it (srr)- :
almost all real-analytic functions of type
exp(1) > a_n > 1
f(x) = sum a_n^x
with a single (real) fixpoint and fixpoint derivate and second derivate equal to eta's
(f(x) has a unique real inverse)
satisfy the same property as the eta^x mentioned in this thread.
but it sounded to weird and silly ... and i guess there had to be many counterexamples. ( laying doubt on " almost all " )
but what do you think are the conditions for this strange property of upper and lower superfunctions ?
i always said : " ill consider this later " and thought " it is probably trivial and most regular posters can easily prove this "
apart from the actual proof , i came to conjecture - without the gut to post it nor the effort to check it (srr)- :
almost all real-analytic functions of type
exp(1) > a_n > 1
f(x) = sum a_n^x
with a single (real) fixpoint and fixpoint derivate and second derivate equal to eta's
(f(x) has a unique real inverse)
satisfy the same property as the eta^x mentioned in this thread.
but it sounded to weird and silly ... and i guess there had to be many counterexamples. ( laying doubt on " almost all " )
but what do you think are the conditions for this strange property of upper and lower superfunctions ?