Is there a rational function f ( x ) ∈ Q ( x ) such...
Is there a rational function such that for all ?
My thoughts : it is easy to find such an f if we relax the conditions to (take f constant equal to ), however no easy perturbation of this solution seems to solve the original problem. Clearly f must have zero degree in x and can be written in the form where g is another rational function. Then I am stuck.
Answer & Explanation
The simplest function I can find seems . It is easily verified this satisfies the double inequality for .
This was obtained by looking at linear approximants, and then setting the conditions and finally the RHS inequality.
Even among linear functions, there are of course many choices, many of which can be got from the form
... which leads to the thought could work for (not checked).