On harmonic convolutions involving a vertical strip mapping. (English) Zbl 1308.31002
Summary: Let \( f_\beta=h_\beta+\overline g_\beta\) and \(F_a=H_a+\overline G_a\) be harmonic mappings obtained by shearing of analytic mappings
\[ h_\beta+g_\beta=\frac{1}{2i\sin \beta}\log\frac{1+ze^{i\beta}}{1+ze^{-i\beta}},\quad 0<\beta<\pi, \]
and \(H_a+G_a={z}/(1-z)\), respectively. R. Kumar et al. [“An application of Cohn’s rule to convolutions of univalent harmonic mappings”, Preprint, arXiv:1306.5375] conjectured that if \(\omega(z)=e^{i \theta}z^n\) (\(\theta\in\mathbb R\), \(n\in\mathbb N\)) and \(\omega_a(z)=(a-z)/(1-az)\), \(a\in(-1,1)\),
are dilatations of \(f_\beta\) and \(F_a\), respectively, then \(F_a\tilde* f_\beta\in S_H^0\) and is convex in the direction of the real axis, provided \(a\in [(n-2)/(n+2),1)\). They claimed to have verified the result for \(n=1,2,3\) and \(4\) only. In the present paper, we settle the above conjecture, in the affirmative, for \(\beta=\pi/2\) and for all \(n\in\mathbb{N}\).
\[ h_\beta+g_\beta=\frac{1}{2i\sin \beta}\log\frac{1+ze^{i\beta}}{1+ze^{-i\beta}},\quad 0<\beta<\pi, \]
and \(H_a+G_a={z}/(1-z)\), respectively. R. Kumar et al. [“An application of Cohn’s rule to convolutions of univalent harmonic mappings”, Preprint, arXiv:1306.5375] conjectured that if \(\omega(z)=e^{i \theta}z^n\) (\(\theta\in\mathbb R\), \(n\in\mathbb N\)) and \(\omega_a(z)=(a-z)/(1-az)\), \(a\in(-1,1)\),
are dilatations of \(f_\beta\) and \(F_a\), respectively, then \(F_a\tilde* f_\beta\in S_H^0\) and is convex in the direction of the real axis, provided \(a\in [(n-2)/(n+2),1)\). They claimed to have verified the result for \(n=1,2,3\) and \(4\) only. In the present paper, we settle the above conjecture, in the affirmative, for \(\beta=\pi/2\) and for all \(n\in\mathbb{N}\).
MSC:
| 31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
