A strengthening of McConnel's theorem on permutations over finite fields

Let $p$ be a prime, $q=p^n$, and $D\subset {\mathbb{F}}_q^{\ast }$. A celebrated result of McConnel states that if $D$ is a proper subgroup of ${\mathbb{F}}_q^{\ast }$, and $f:{\mathbb{F}}_q\to {\mathbb{F}}_q$ is a function such that $(f(x)-f(y))\mathrm{/}(x-y)\in D$ whenever $x\mathrm{\ne }y$, then $f(x)$ necessarily has the form $a{x}^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel’s theorem extends if $D$ has small doubling.