It seems clear that for $A, B, C$ infinite cardinals with $A > B $ one could define an injection from $B^C \to A^C$ and so $A > B \Rightarrow A^C \ge B^C$, but is the inequality strict and what is the proof ?
After reading the counter example my question is extended and becomes "Are there clear circumstances in which $ A^C = B^C$ and in which $ A^C > B^C$ ?"
Counterexample:
$$\aleph_0<2^{\aleph_0}\qquad\text{and}\qquad \aleph_0^{\aleph_0}=2^{\aleph_0}=(2^{\aleph_0})^{\aleph_0}$$
Additionally, if $C$ is sufficiently large, then $A^C$ and $B^C$ are both equal to $2^C$.