Close-to-convexity of convolutions of classes of harmonic functions (Q1751313)

Summary: For \(j = 1,2\) and for positive integers \(m\) and \(n\), we consider classes of harmonic functions \(f_j = h_j + \overline{g_j}\), where \(g_1(z) = z^n h_1(z)\) and \(g_2'(z) = z^n h_2'(z)\) or \(g_1'(z) = z^n h_1'(z)\) and \(g_2'(z) = z^m h_2'(z)\), and we prove that their convolution \(f_1 \ast f_2 = h_1 \ast h_2 + \overline{g_1 \ast g_2}\) is locally one-to-one, sense-preserving, and close-to-convex harmonic in \(|z|<1\).