I want to prove that the function $f_i(P)=f_i(P_1,..P_i,..,P_K)=log(1+(\frac{a_iP_i}{\eta+\sum\limits_{i'\neq i}a_{i'}P_{i'}}))$ is discetely concave, which means that I should prove: $\forall \lambda \in[0,1],\forall P \text{ and } P'\text{ } \lambda f_i(P)+(1-\lambda) f_i(P') \leq \max \limits_{z \in N(\lambda P+(1-\lambda)P')}(f(z))$ where : $N(\lambda P+(1-\lambda)P')=\{z, \left\|\lambda P+(1-\lambda)P'-z\right\|_1\leq 1 \}$ except that , up to now, I can not prove this inequality, is there another way to prove the descrete concavity of my function?
Thank you so much
