The skewness-kurtosis curve of the family of lognormal random variables can be parametrized as \begin{eqnarray*} x(t) &=& (t+2)\sqrt{t-1} \newline y(t) &=& t^4+2t^3+3t^2-3 \end{eqnarray*} where $t>1$. When plotted (shown below), the "eyeball test" shows that this curve is convex, but I would like to prove this. How can I go about proving the convexity of $y$, viewed as a function of $x$? Rewriting $y$ as an explicit function of $x$ seems too tedious. I've also tried showing convexity of the epigraph, but so far have been unsuccessful. I'm hoping a short, elegant solution exists that uses a technique from convex analysis I've either overlooked or am not familiar with.
Enough is to prove that $t\mapsto \frac{y'(t)}{x'(t)}$ is increasing.