Landau-Type Theorems for Certain Biharmonic Mappings

Ming-Sheng Liu; Zhen-Xing Liu; Jun-Feng Xu

doi:10.1155/2014/925947

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2014 Landau-Type Theorems for Certain Biharmonic Mappings

Ming-Sheng Liu, Zhen-Xing Liu, Jun-Feng Xu

Abstr. Appl. Anal. 2014(SI07): 1-7 (2014). DOI: 10.1155/2014/925947

Abstract

Let $F(z)=|z{|}^{2}g(z)+h(z) (|z|<1)$ be a biharmonic mapping of the unit disk $\mathrm{\Bbb D}$ , where $g$ and $h$ are harmonic in $\mathrm{\Bbb D}$ . In this paper, the Landau-type theorems for biharmonic mappings of the form $L(F)$ are provided. Here $L$ represents the linear complex operator $L=\mathrm{(z}\partial /\partial \mathrm{z)}-\overline{\mathrm{(z}}\partial /\partial \overline{\mathrm{z)}}$ defined on the class of complex-valued ${C}^{1}$ functions in the plane. The results, presented in this paper, improve the related results of earlier authors.

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Ming-Sheng Liu. Zhen-Xing Liu. Jun-Feng Xu. "Landau-Type Theorems for Certain Biharmonic Mappings." Abstr. Appl. Anal. 2014 (SI07) 1 - 7, 2014. https://doi.org/10.1155/2014/925947