Does the following inequality holds for every POSITIVE 'x' ?

[zetafunction]

does the following inequality holds for every POSITIVE 'x' ?

e^{-x}-1\le Cx^{1/4+e} here 'C' and e are positive constants

i think that for very very small 'e' the constant must be very BIG but no other hint i find

 

[disregardthat]

Certainly not without additional restrictions on e and C. (You should not use 'e' as a variable name to avoid confusion with e = 2.71...)

However, if e >= 1, the left hand side is non-positive, so in that case the equation words.