How to prove inequality with stricly increasing concave function?

Question

I have following inequality I need to prove: $$\frac{x_i +x_j}{x_i}>\frac{u(R)}{u(R\frac{x_i}{x_i +x_j})}$$We have that $x_i,x_j,R>0$ as well as $u$ being an strictly increasing concave function in the respective interval.

Answer 1

If $u\colon [0,+\infty) \to \mathbb{R}$ is a concave function, with $u(0) \geq 0$, then, setting $\mu := x_i/(x_i+x_j) \in (0,1)$, $$ u(\mu R) = u ((1-\mu) 0 + \mu R) \geq (1-\mu) u(0) + \mu u(R) \geq \mu \, u(R). $$ If $u$ is strictly concave, the inequality is strict, i.e. $u(\mu R) > \mu u(R)$.

In addition, if $u$ is positive, then the above inequality gives the required one.