I am trying to prove the following statement:
$$N\left(a f \left(\frac{x}{a}\right)\right) \geq b_1 f \left(\frac{x}{b_1}\right)+ b_2 f \left(\frac{x}{b_2}\right)+ \dots b_N f \left(\frac{x}{b_N}\right),$$ for $f$ concave function, where: $$\sum_{i=1}^N b_i =Na,$$ $$a \neq b_i. $$ In other words, uniformly distributed coefficients would result in greater value.
Since $f$ is concave, you have for $\lambda_i=b_i/Na$, and $x_i=x/b_i$ $$f(x/a)=f\left(\sum_{i=1}^N\lambda_ix_i\right)\geq\sum_{i=1}^N\lambda_if(x_i)=\frac{1}{Na}\sum_{i=1}^Nb_if(x/b_i)$$ which is what you want.