I am struggling to solve the following problem: For a convex set $\mathcal{X}\subseteq\mathbb{R}^n$, $$\min_{x\in\mathcal{X}}~\left\{\frac{f(x)}{g(x)} + h(x)\right\},\label{prob}\tag{1}$$ where $f$ and $h$ are nonnegative convex functions w.r.t. $x$, and $g$ is a positive concave function w.r.t. $x$. I know that $f/g$ is quasiconvex, but I am not sure whether $f/g+h$ is also quasiconvex.
In more detail, for my case, \begin{align} f(x) &= \sum_{i=1}^n a_i f_i(x),\\ g(x) &= \sum_{i=1}^n x_i,\\ h(x) &= \sum_{i=1}^n b_i f_i(x), \end{align} where $x_i$ represents the $i$th element of vector $x$, $a_i$ and $b_i$ are positive given constants, $f_i$ is positive and flipped-log-shaped decreasing convex function for all $i$.
If $h(x)$ disappeared, the problem could be solved using Dinkelbach's method, i.e., global optimal solution $x^*$ is obtained by alternatively updating until convergence as follows: \begin{align} y_{k+1} &\gets \frac{f(x_k)}{g(x_k)},\\ x_{k+1} &\gets \arg\max_{x\in\mathcal{X}}~\{f(x)-y_{k+1}g(x)\}. \end{align}
However, I am not sure that the global solution $x^*$ to Problem \eqref{prob} can be obtained by updating in a similar way as above, i.e., \begin{align} y_{k+1} &\gets \frac{f(x_k)}{g(x_k)},\\ x_{k+1} &\gets \arg\max_{x\in\mathcal{X}}~\{f(x)-y_{k+1}g(x)+h(x)\}. \end{align}
To sum up, is there any way to find a global solution to Problem \eqref{prob}? If there are some references, please recommend me. Thank you.