Is this integral transform related to the Laplace transform?

Question

The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by

$${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$

Let $\varphi: {\mathbb R}_+ \to {\mathbb R}_+$ be an injective continuous increasing function (not necessarily linear). Is the function:

$${\displaystyle F_{\varphi}(s)=\int _{0}^{\infty }f(t)e^{- \varphi(st)}\,dt},$$

related to the Laplace Transform or another known integral transform?

After some calculations, I arrived to an integral of that kind, so I was wondering if some theoretical properties of that kind of functions are known already.

Answer 1

A generalization along these lines, but with the single function $\phi(st)$ replaced by the product of two invertible functions $\Phi(s)E(t)$, has been studied in The generalized Laplace transform and fractional differential equations of distributed order. The Mellin transform is a notable example of this type, with $\Phi(s)E(t)=(1-s)\ln t$.