I wrote up this inequality while formulating an optimization problem.
$$(y_{1}-1) \leq \left(K + \dfrac{y_{2}}{x_{2}+b \cdot y_{2}} \right) (x_{1}+b \cdot y_{1} ) \leq y_{1}$$
Some further conditions on the above variables include the following.
$y_{1} < y_{2}$
$K,~b,~y_{1},~y_{2},~x_{1},~x_{2} > 0$
$K$ and $b$ are constants.
$y_{1}$ and $y_{2}$ are constrained to be integers but I think this should not affect the procedure for checking convexity.
For the objective function, I was able to find the Hessian matrix and show that it was positive semi-definite over some region. However, for the above constraint, how do I go about showing whether it is convex (or not)? I think I can start by considering each side of the inequality separately, but what techniques can I use as a starting point?