Is there a rational function f ( x ) ∈ Q ( x ) such...

Arraryeldergox2

Arraryeldergox2

Answered

2022-06-26

Is there a rational function f ( x ) Q ( x ) such that 2 f ( x ) 2 x 3 for all x 2 ?
My thoughts : it is easy to find such an f if we relax the conditions to f ( x ) R ( x ) (take f constant equal to 2 ), 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 x + ( x 2 2 ) g ( x ) where g is another rational function. Then I am stuck.

Answer & Explanation

massetereqe

massetereqe

Expert

2022-06-27Added 21 answers

The simplest function I can find seems f ( x ) = 2 x + 2 x + 2 . It is easily verified this satisfies the double inequality for x 2 .
This was obtained by looking at linear approximants, and then setting the conditions f ( 2 ) = 2 , f ( x ) 2 and finally the RHS inequality.
Even among linear functions, there are of course many choices, many of which can be got from the form
f ( x ) = a x + 2 b b x + a , with
( a , b ) { ( 3 , 2 ) , ( 5 , 2 ) , ( 5 , 3 ) , ( 7 , 3 ) , ( 7 , 4 ) , . . . }
... which leads to the thought ( 2 n ± 1 ) x + 2 n n x + 2 n ± 1 could work for 1 < n N (not checked).

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