I am reading a research paper where it is mentioned that the following constraint is non-convex $$a\sum_{t=1}^{T}b_t\log \left(1+\frac{c}{(d^2+||q(t)-p_k||^2)^{m}}\right)\geq s$$ where $a>0, b_t>0, c>0, d>0, s>0, m>1$. $q(t),p_k$ denote the position in a two dimensional coordinate system. Further we have that $q(1)=\text{some starting position}$ and $q(M)=\text{some final position}$. It is written in the paper that the above constraint is non-convex. Can anybody explain why it is non-convex? Thanks in advance.
My Reasoning to show that the above constraint is non-convex:
If I replace $||q(t)-p_k||$ with some variable $x$ (which can have only positive values) then the first term on the left side becomes $$ab_1\log\left( 1+\frac{c}{(d^2+x^2)^m}\right)$$ then it can be shown that the double derivative of $\log\left( 1+\frac{c}{(d^2+x^2)^m}\right)$ is negative for $x=0$ while it becomes positive after certain value of $x$. Therefore it is not concave and not convex. Which, I think, also means that the upper set of the function is also not-convex. Since the right hand side is summation of such functions ($\sum_{t=1}^{T}b_t\log \left(1+\frac{c}{(d^2+||q(t)-p_k||^2)^{m}}\right)$) therefore the whole constraint is non-convex. Is this explanation right or wrong?