In a previous paper with N. Danikas the author proved that if f is an analytic function in the Hardy space \(H^ 1\) of the unit disk D then for each \(z_ 0\in D\) there exists an interval I on the unit circle such that \(f(z_ 0)\) equals the average value of f on I. In the present paper he refines this result when f is univalent by showing that I can be taken to have length at least \(\ell (1-| z_ 0|)\), where \(\ell >0\) is an absolute constant. The question of whether such a result is true for arbitrary \(H^ 1\) functions is left open. The proof for univalent functions uses the Koebe one-quarter theorem and elementary distortion theorems.