We consider the class of all analytic and locally univalent functions f of the form f (z) = z + ∞ n=2 a 2n−1 z 2n−1 , |z| < 1, satisfying the condition Re 1 + z f (z) f (z) > − 1 2. We show that every section s 2n−1 (z) = z + n k=2 a 2k−1 z 2k−1 , of f , is convex in the disk |z| < √ 2/3. We also prove that the radius √ 2/3 is best possible, i.e. the number √ 2/3 cannot be replaced by a larger one.