Concave function - proof

Question

Is there any analytical (if numerical it is obvious) way to prove that:

$$f(x) < 2f\left(\frac{x}{2}\right)$$

where $f(x)$ is any concave function?

Answer 1

For the concave function we obtain $$\frac{f\left(0\right)+f(x)}{2}\leq f\left(\frac{0+x}{2}\right).$$ Thus, for your inequality we need something like $f(0)>0.$

Answer 2

Take $f : x \mapsto \ln(x)$, $f$ is concave. For $x=1$

$$ \ln\left(1\right)=0 \text{ and }2\ln\left(\frac{1}{2}\right)=-2\ln\left(2\right) $$ You dont have $$ 0<-2\ln\left(2\right) $$