Let $D$ denote the set of directions determined by the graph of a polynomial $f$ of $\mathbb{ F}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is contained in a multiplicative subgroup $M$ of $\mathbb{ F}_q^\times$, then by a result of Carlitz and McConnel it follows that $f(x)=ax^{ p^k}+b$ for some $k\in \mathbb{ N}$. Of course, if $D\subseteq M$, then $0\notin D$ and hence $f$ is a permutation. If we assume the weaker condition $D\subseteq M \cup \{ 0\}$, then $f$ is not necessarily a permutation, but Sziklai conjectured that $f(x)=ax^{ p^k}+b$ follows also in this case. When $q$ is odd, and the index of $M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and Sz\H onyi combined with a result of McGuire and G\"olo\u{ g}lu proves the conjecture. Assume $\deg f\geq 1$. We prove that if the size of $D^{ -1}D=\{ d^{ -1}d' : d\in D\setminus \{ 0\},\, d'\in D\}$ is less than $q-\deg f+2$, then $f$ is a permutation of $\mathbb{ F}_q$. We use this result to verify the conjecture of Sziklai.
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