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Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D_f^{\epsilon}=zf_{z}-\epsilon\overline{z}f_{\overline{z}}~(|\epsilon|=1)$ and $F_{\lambda}(z)=(1-\lambda)f+\lambda D_f^{\epsilon}~(0\leq\lambda\leq 1)$ when the coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of $h$ and $g$ satisfy the corresponding necessary conditions of the harmonic convex function $f=h+\overline{g}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. \cite{Kalaj2014}.
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참고문헌 (15)
참고문헌 신청
J. W. Alexander / 1915 / Functions which map the interior of the unit circle upon simple regions / Ann. of Math. 17(1):12~22
M. Chuaqui / 2004 / Curvature properties of planar harmonic mappings / Comput. Methods Funct. Theory 4(1):127~142
J. Clunie / 1984 / Harmonic univalent functions / Ann. Acad. Sci. Fenn. Ser. A I Math. 9:3~25
L. de Branges / 1985 / A proof of the Bieberbach conjecture / Acta Math. 154(1-2):137~152
P. Duren / 2004 / Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156 / Cambridge University Press
M. Chuaqui / 2004 / Curvature properties of planar harmonic mappings / Comput. Methods Funct. Theory 4(1):127~142
J. Clunie / 1984 / Harmonic univalent functions / Ann. Acad. Sci. Fenn. Ser. A I Math. 9:3~25
L. de Branges / 1985 / A proof of the Bieberbach conjecture / Acta Math. 154(1-2):137~152
P. Duren / 2004 / Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156 / Cambridge University Press
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