First, I should say I just know about Laplace transform through Wikipedia.
My question is,
Can you use Laplace transform to show a function is convex? If so, is there any link to an example?
I have a function $\frac{T(z)}{g(z)}$ that I want to show is convex. If it helps this is the specific question: normal distribution hazard rate increasing function
Let's suppose that $f:[0,\infty)\to\Bbb R$ is convex and of slower than exponential growth, so that it has a Laplace transform. Then $f'_+$ (the right-hand derivative of $f$) is right-continuous and non-decreasing. As such $f'_+$ is the "distribution function" of a measure $\mu$ on $[0,\infty)$: $\mu((a,b]) =f'_+(b)-f'_+(a)$ for $0\le a<b$. Integrate by parts to see that $$ \lambda^2 \hat f(\lambda)-\lambda f(0)=\int_0^\infty e^{-\lambda x}\mu(dx)=\hat\mu(\lambda), \qquad\lambda>0, $$ where the $\hat{}$ denotes Laplace transform. It is known that a function is the Laplace transform of a positive measure (like $\mu$) if and only if it is smooth and its derivatives (starting with the zeroth, which must be non-negative) alternate in sign. (Such a function is said to be completely monotone.) Thus it seems that for $f$ to be convex, the function $0<\lambda\mapsto \lambda^2\hat f(\lambda)-\lambda f(0)$ must be completely monotone.