# Real analysis – squeezing between polynomial and exponential

To let $$mathbb {R} _ +$$ denote the set of positive reals. To let $$mathbb {R} _ + ^ { mathbb {R} _ +}$$ denote the set of all functions $$f: mathbb {R} _ + an mathbb {R} _ +$$, To the $$f, g: mathbb {R} _ + an mathbb {R} _ +$$ we say $$f leq ^ * g$$ If there is $$N in mathbb {N}$$ so that $$f (x) leq g (x)$$ for all $$x geq N$$,

To let $$A = {f in mathbb {R} _ + ^ { mathbb {R} _ +}: n in mathbb {N}: f (x) = x ^ n text {for all x} in mathbb {R} _ + }$$ and let it go $$B = {f in mathbb {R} _ + ^ { mathbb {R} _ +}: exists varepsilon in mathbb {R} _ +: f (x) = (1+ varepsilon )) ^ x text {for all x} in mathbb {R} _ + }$$,

Question. Is there $$s in mathbb {R} _ + ^ { mathbb {R} _ +}$$ so for all $$a in A$$ and for everyone $$b in B$$ we have $$a leq ^ * s leq ^ * b$$?

Soft question. (Only out of interest, not required to accept the answer.) If the above main question is answered "yes," there is a "natural" problem that is not polynomial but solvable in "time." $$s$$"