  Planar functions are of great importance in the constructions of DES-like
iterated ciphers, error-correcting codes, signal sets and the area of
mathematics. They are defined over finite fields of odd characteristic
originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In
2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar
functions over $\F_{2^n}$. Very recently, D. Bartoli and M. Timpanella
\cite{Bartoli} characterized the condition on coefficients $a,b$ such that the
function $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$ is a planar
function over $\F_{2^{3m}}$ by the Hasse-Weil bound.
  In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil
bound, and the new approach introduced in \cite{Q}, we completely characterize
the necessary and sufficient conditions on coefficients of four classes of
planar functions over $\F_{q^k}$, where $q=2^m$ with $m$ sufficiently large
(see Theorem \ref{main}). The first and last classes of them are over
$\F_{q^2}$ and $\F_{q^4}$ respectively, while the other two classes are over
$\F_{q^3}$. One class over $\F_{q^3}$ is an extension of $f_{a,b}(x)$
investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In
addition, although the planar binomial over $\F_{q^2}$ of our results is
finally a known planar monomial, we also answer the necessity at the same time
and solve partially an open problem for the binomial case proposed in \cite{Q}.
