Help with a simple problem involving a functional inequality (trying to prove Gronwall's inequality)

Help with a simple problem involving a functional inequality (trying to
prove Gronwall's inequality)

So while trying to prove Grownwall's inequality, my proof led me to the
following statement: $h'(x) \le h(x)g(x)$.
Now when $h'(x)=h(x)g(x)$ the following holds: $h(x)=k \exp G(x)$, where
$k$ is a constant number. Can I conclude from this that $h(x) \le k \exp
G(x)$ (since $h'(x)<h(x)g(x)$)?