  For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in
(m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called
an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that
almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many
expansions. In this paper we study the existence of expansions $(d_i)$ of $x$
satisfying the inequalities $\sum_{i=1}^n d_iq^{-i} \geq \sum_{i=1}^n c_i
q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.
