In an earlier paper, Dorff et al., considered the classes of harmonic univalent functions \(f_k=h_k+\overline{g_k}, k=1,2\), that are shears of \(h_k-g_k=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\) with dilatations \(\omega _k=e^{i\theta _k}z^{n_k}\) for \(n_k\) positive integers, and proved that if the convolution \(f_1*f_2= h_1*h_2+\overline{g_1*g_2}\) is locally one-to-one and sense-preserving, then \(f_1*f_2\) is convex in the direction of the real axis. Later, Jahangiri and Garg considered \(f_m=h_m+\overline{g_m}\), \(m=1,2\), where \(h_m+e^{i \theta _m}g_m=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\) and determined conditions on \(\omega _m\) such that the convolution \(f_1*f_2\) is convex in the direction of the imaginary axis provided \(f_1*f_2\) is locally one-to-one and sense-preserving. In this paper, we consider the harmonic univalent functions \(f_j=h_j+\overline{g_j}\), \(j=1,2\), where \(h_{1}-g_{1}=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\), and we determine conditions on \(f_2\) and on \(\omega _1 = g'_1/h'_1\) such that the convolution \(f_1*f_2\) is typically real and convex in the direction of the imaginary axis provided that \(f_1*f_2\) is locally one-to-one and sense-preserving.

在较早的一篇论文中，Dorff 等人考虑了谐波单价函数\(f_k=h_k+\overline{g_k}, k=1,2\) 的类，它们是\(h_k-g_k=\frac{1 }{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\)与膨胀\(\omega _k=e^{i\theta _k}z^{n_k} \)为\(n_k\) 个正整数，并证明如果卷积\(f_1*f_2= h_1*h_2+\overline{g_1*g_2}\)是局部一对一且意义保持的，则\( f_1*f_2\)在实轴方向是凸的。后来，Jahangiri 和 Garg 考虑了\(f_m=h_m+\overline{g_m}\) , \(m=1,2\)，其中\(h_m+e^{i \theta _m}g_m=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\)和确定的条件在\(\omega _m\) 上，使得卷积\(f_1*f_2\)在虚轴的方向上是凸的，条件是\(f_1*f_2\)是局部一对一的并且保持意义。在本文中，我们考虑调和单价函数\(f_j=h_j+\overline{g_j}\) , \(j=1,2\)，其中\(h_{1}-g_{1}=\frac{1 }{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\)，我们确定\(f_2\)和\(\omega _1 = g' 上的条件_1/h'_1\)使得卷积\(f_1*f_2\)如果\(f_1*f_2\)是局部一对一且保留意义的，则通常在虚轴的方向上为实数和凸面。