State complexity of homomorphisms of regular languages
Here is a simple construction that disproves your conjecture.

Let $a,b$ two relatively prime integers. It is known that the largest integer that cannot be represented as a non-negative integer combination of $a,b$ is $ab-a-b$.

Consider now the language $L = (0+1)^*$ and the homomorphism $h$ given by $h(0) = 0^a$ and $h(1) = 0^b$. Then $L$ is accepted by a one-state DFA, while $h(L)$ requires $ab-a-b+1$ states.