During my research I came across the following problem:


I need to find a root of the following function:
$$\Gamma\_{N}(x) = \sum\limits\_{i=0}^{M}\left(\frac{\sum\limits\_{n=0}^{n\_F}n\ \alpha\_{i,n} x^n}{\sum\limits\_{n=0}^{n\_F}\alpha\_{i,n} x^n}\right)- N,$$
where $\alpha\_{i,n}$ are positive coefficients such that in general, $\alpha\_{i,n} \in [0,1]$. Parameter $M$ and $n\_F$ are integer numbers and $N$ is real. I want to consider only positive $x$, so I am searching for root in the domain $x \in [0, +\infty]$. I will do all calculations numerically, but I was wondering if this function possesses some properties?


Question: Is this function monotonic, such that there exists only a single root? 



Root of a special rational function with positive coefficients
More elementary argument than in the answer by Alexandre Eremenko:


it suffices to prove that if $a\_k\geqslant 0$ for all $k$ and $x>y>0$, then $$\frac{\sum na\_nx^n}{\sum a\_n x^n}\geqslant \frac{\sum na\_ny^n}{\sum a\_n y^n}.$$
Multiplying by common denominator this reduces to $$\sum\_{n,k}a\_na\_k(n-k)(x^ny^k-x^ky^n)\geqslant 0,$$
each summand is non-negative.


