Let \(S\) be the class of normalized functions \(f(z)=z+a_2 z^2+a_3z^3 + \cdots\) which are holomorphic and univalent on the open unit disk \({\mathbf D} = \{z:|z|< 1\}\). For any real number \(\alpha\) the author considers the problem of maximizing the functional \(|a_3-\alpha a^2_2|+ \alpha|a_2 |^2\) over the class \(S\). For \(\alpha\geq 1\) work of [\textit{J. A. Jenkins}, Princeton Math. Ser. 24, 159-194 (1960; Zbl 0103.30002)] implies that the maximal value is attained by the Koebe function \(k(z) = z/(1-z)^2\) and its rotations. Prior work of the author and \textit{Z. J. Jakubowski} [Math. Bohem. 118, No. 3, 281-296 (1993; Zbl 0783.30013)] using the Valiron-Landau estimate implies that the Koebe function and its rotations are still extremal for \(\alpha\leq 1/2\). The author employs the variational method to treat the remaining case \(\alpha \in(1/2,1)\). Interestingly, the Koebe function and its rotations remain extremal for \(e/(2(e-1)) < \alpha<1\), but not for \(1/2< \alpha < 3/(2(e-1))\).