Polyharmonic functions with negative coefficients


K. Al-Shaqsi - Department of Information Technology, Nizwa College of Technology, Ministry of Manpower, Sultanate of Oman
R. Al-Khal - Department of Mathematics, Sciences College, University of Dammam, Dammam, Saudi Arabia


A 2p times continuously differentiable complex-valued mapping \(F=u+i v\) in a domain \( \mathcal D \subset \mathbb C\) is polyharmonic if \(F\) satisfies the polyharmonic equation \(\underbrace{\Delta\cdot\cdot\cdot\Delta}_\text{p} F= 0\), where \(p \in \mathbb N^{+}\) and \(\Delta\) represents the complex Laplacian operator. The main aim of this paper is to introduce a subclasses of polyharmonic mappings. Coefficient conditions, distortion bounds, extreme points, of the subclasses are obtained.