Between exp^[h] and elementary growth - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Between exp^[h] and elementary growth (/showthread.php?tid=1179) Between exp^[h] and elementary growth - tommy1729 - 09/04/2017 The context is asymptotics for real-analytic strictly rising f(x) , as x grows to + oo. Let f(x) grow (asymptoticly) much faster than any polynomial. So for large x f(x) >> exp(a ln(x)) for any fixed a > 0. But Also for large x we have f(x) << exp^[h](x) for any h > 0. Now if f(x) does grow faster than any elementary function can describe Then f(x) grows faster than Exp^[k]( a ln^[k](x) ) for any fixed a >1 , k > 0. Or equivalently  f(x) >> Exp^[k]( ln^[k](x) + a). For a,k > 0. Combining Exp^[k]( ln^[k](x) + a) << f(x) << Exp^[h](x) for any a,k,h >  0. Now you might have assumed that such an f(x) does not exist. But it does. This fascinates me. I think this deserves attention. And ofcourse a " fake ". Here is a possible solution I came up with while in kindergarten f(x) = sexp^[1/2]( slog^[1/2](x) + 1 ). So I encourage all investigations into this. Regards  tommy1729