Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$, \begin{equation} \frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}\le\frac{v_{*}}{1-c_{*}x} \end{equation} for some $v_{*}$ and $c_{*}$? Of course, this is true for $v_{*}=n\text{max}_{i}v_{i}$ and $c_{*}=\text{max}_{i}c_{i}$.
Are there other $v_{*}$ and $c_{*}$ that can give me the above upper bound?