On homogeneous planar functions

Let $p$ be an odd prime and ${\text{F}}_q$ be the finite field with $q=p^n$ elements. A planar function $f:{\text{F}}_q\to {\text{F}}_q$ is called homogenous if $f(\lambda x)=\lambda^df(x)$forall$\lambda\in\F_p$and$x\in\F_q$,where$d$ is some fixed positive integer. We characterize $x^2$ as the unique homogenous planar function over ${\text{F}}_{p^2}$ up to equivalence.