On The Studies Of The Properties Of Several Classes Of Biharmonic Mappings

Suppose that D denotes a domain in the complex plane C and that F=u+iv denotes a four times continuously differentiable complex-valued function defined in D. We say that F is biharmonic if the real-valued functions u and v satisfy the biharmonic equation Δ(Δu)=Δ(Δv)=0, where Δ represents the Laplacian operator andThe main aim of this dissertation is to discuss some properties of biharmon-ic mappings. It consists of three chapters, and the arrangement is as follows.In Chapter one, we mainly introduce the background of our research and state our main results.In Chapter two, we define two classes of biharmonic mappings SBH(n,λ,γ,δ, α) and TBH(η,λ,γ,δ,α) by using the Salagean operator. First, we give some co-efficient conditions to make sure that a biharmonic mapping belongs to SBH(n,λ, γ,δ,α)(resp. TBH(η,λ,γ,δ,α)). Then we apply these conditions to discuss the distortion theorem and existence of extreme points of TBH(η,γ,γ,δ,α). The obtained results are generalizations of the corresponding results in Acta Univ. Sapientiae Math. obtained by Murugusundardmoorthy and Vijaya in2010.In Chapter three, we introduce a class of biharmonic mappings BHS(α,β,η, m,λ) and discuss some properties of this class. We investigate the univalence, distortion theorem, existence of extreme points and convex combination of el-ements in this class. Our results are the generalizations of the corresponding results in Appl. Math. Sci. obtained by Yalcin, Joshi and Yasar in2010.