  Negabent functions as a class of generalized bent functions have attracted a
lot of attention recently due to their applications in cryptography and coding
theory. In this paper, we consider the constructions of negabent functions over
finite fields. First, by using the compositional inverses of certain binomial
and trinomial permutations, we present several classes of negabent functions of
the form $f(x)=\Tr_1^n(\lambda x^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$, where
$\lambda\in \F_{2^n}$, $2\leq k\leq n-1$, $(u,v)\in \F^*_{2^n}\times
\F^*_{2^n}$, and $\Tr_1^n(\cdot)$ is the trace function from $\F_{2^n}$ to
$\F_{2}$. Second, by using Kloosterman sum, we prove that the condition for the
cubic monomials given by Zhou and Qu (Cryptogr. Commun., to appear, DOI
10.1007/s12095-015-0167-0.) to be negabent is also necessary. In addition, a
conjecture on negabent monomials whose exponents are of Niho type is given.
