Sum-free sets which are closed under multiplicative inverses

Let \( A \) be a subset of a finite field \( \mathbb{F} \). When \( \mathbb{F} \) has prime order, we show that there is an absolute constant \( c > 0 \) such that, if \( A \) is both sum-free and equal to the set of its multiplicative inverses, then \( |A| < (0.25 – c)|\mathbb{F}| + o(|\mathbb{F}|) \) as \( |\mathbb{F}| \to \infty \). We contrast this with the result that such sets exist with size at least \( 0.25|\mathbb{F}| – o(|\mathbb{F}|) \) when \( \mathbb{F} \) has characteristic 2.