The Sharp upper bound for $\mathfrak{R}(A_{3}-\lambda A_{2})$ in $U'_{\alpha}$

In this note we determine the exact value of max $\mathfrak{R}(A_{3}-\lambda A_{2}), \lambda \in \mathbb{R}$, within the linearly invariant family $U'_{\alpha}$ introduced by V. V. Starkov in [4]. For $\lambda = 0$ the sharp estimate for $|A_{3}|$ follows. If $\alpha = 1$ the corresponding result is valid for convex univalent functions in the unit disk.