Pott et al. studied this number for the case $m=n$. In [@Pott] they showed that this number is upper bounded by $2^{n}-2^{n/2}$, which is attained by the functions of the form $F(x)=x^{2^{i}}(x+x^{2^{k}})$, where $n=2k, 0 \leq i\leq n-1$. They also pointed out that vectorial functions with maximal number of bent components can produce new vectorial bent functions. More precisely, for any $(n,n)$-function $G$, if ${\rm Tr}^{n}_{1}(\alpha G(x))$ is bent for any $\alpha\in\mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{k}}$, then ${\rm Tr}^{n}_{k}(\alpha G(x))$ is a vectorial bent function for any $\alpha \in \mathbb{F}_{2^{n}}\setminus\mathbb{F}_{2^{k}}$. Therefore, ${\rm Tr}^{n}_{k}(\gamma F(x))$ is vectorial bent for any primitive element $\gamma$ in $\mathbb{F}^{\ast}_{2^{n}}$ and the functions $F$ listed above. Later, Mesnager et al. presented a class of $(n,n)$-functions with maximal number of bent components of the form $H(x)=x^{2^{i}}(x+x^{2^{k}}+\sum\limits^{\rho}_{j=1}\gamma^{(j)}(x^{2^{t_{j}}}+x^{2^{t_{j}+k}}))$ under some additional conditions, where $\gamma^{(j)}\in \mathbb{F}_{2^{k}}$ [@MZTZ]. In the same paper, the authors also proved that APN plateaued functions cannot have the maximum number of bent components. In some earlier works, the number of bent components of APN functions and differentially 4-uniform functions were discussed and some lower bounds were obtained (see [@Bcc; @Cp]).

Let $n=2k$, and $m\geq k$. In this paper, at first we show that the maximal number of bent components of 