If f(x) = (1^)x+_e123x_e1234-1^, x0, then the least value of f(f(x))+f!(f!(4x)) is:

Question

If $f(x) = \dfrac{(	an1^{\circ})x+\log_e123}{x\log_e1234-	an1^{\circ}}$, $x>0$, then the least value of $f(f(x))+f\!\left(f\!\left(\dfrac4xight)ight)$ is:

Accepted Answer

Writing $f(x)=\dfrac{Ax+B}{Cx-A}$ (a Mbius involution), one finds $f(f(x))=x$ for all $x$.

$$f(f(x))+f\!\left(f\!\left(\frac4x\right)\right) = x+\frac4x \ge 2\sqrt{x\cdot\frac4x} = 4 \quad (\text{AM--GM}).$$

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