I want to find an upperbound for the expectation $E[\frac{X}{Y}]$, where $X$ and $Y$ are both positive random variables and positively correlated. I have found that the following uses Taylor expansion to approximate the expectation:
$$ E\Big[\frac{X}{Y}\Big] \approx \frac{E[X]}{E[Y]} - \frac{\text{cov}(X,Y)}{E[Y]^2} + \frac{E[X]}{E[Y]^3}\text{var}(Y). $$
From simulations I've observed that $E\Big[\frac{X}{Y}\Big] < \frac{E[X]}{E[Y]} + \frac{E[X]}{E[Y]^3}\text{var}(Y)$, which makes sense since $X$ and $Y$ are positively correlated. However, is it possible to mathematically prove this or derive another upperbound?
You can't. Your inequality is not true in general. There's no guarantee even that $\mathbb E[1/Y]$ exists. Assuming $Y$ has a density, it will depend on the behaviour of the density near $y=0$, which has very little effect on $\mathbb E[Y]$ and the other terms on the right side of your inequality.