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Let $X\subseteq\mathbb{R}^n$. I have the following function $f:X\rightarrow\mathbb{R}$: $$ f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i +\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}\enspace.$$

All the $a_i$, $b_i$, $c_i$, and $d_i$ are strictly greater than 0, and X is such that $$ c_2+\sum_{i=1}^n d_i x_i>0, \forall {\bf x}\in X\enspace.$$

Is $f$ convex or at least pseudo- or quasi-convex?

Note that $$f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i }{c_2+\sum_{i=1}^n d_i x_i}+\frac{\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}$$ and the first term is on the right side is pseudo-linear (hence pseudo-convex, hence quasi-convex) and third term is convex (hence pseudo-convex, hence quasi-convex). I know that the sum of quasi-convex functions is not in general quasi-convex, but I wonder whether something else was known that can help me show that $f$ is.

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Never mind: my original function was actually

$$ \frac{\sum_{i=1}^\ell \left(b_i + \sum_{j=1}^\ell a^{(i)}_j x_j\right)^2}{c_1+\sum_{i=1}^n d_ix_i}$$

(for different values of the constants from the original question)

so I can see it as a sum of $g_i^2/h$, where $g_i$ is affine (and actually non-negative in my domain) and $h$ is positive affine. Then each of the $g_i^2/h$ is convex, and so is their sum.

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