Studies On The Properties Of Harmonic Mappings And Biharmonic Mappings

It is known that quasiconformal mappings are geheralizations of conformal mappings. Also harmonic mappings are generalizationsof analytic functions, and biharmonic mappings are generalizations of harmonic mappings.The main aim of this dissertation is to discuss some properties of harmonic mappings and biharmonic mappings. It is arranged as follows.In Chapter 1, we provide some backgrounds about our research and statements of our main results.In Chapter 2, we show the existence of Landau's and Bloch's constants for biharmonic mappings with the aid of L(F), whereL=(?) and L belongs to the class of biharmonicmappings of the form F(z) = |z|²G(z) + K(z) (|z|<1), where G and K are harmonic.In Chapter 3, our main aim is to obtain the Schwarz lemma for certain biharmonic mappings. By using the obtained results, we generalize M. Mateljevic, M. Arenovic and V. Manojlovic's corresponding result of harmonic mappings to the case of biharmonic mappings. Also we investigate the convexity of biharmonic mappings F(z) =λ₁|z|²f(z) +λ₂f(z) , where f are harmonic mappings, andλ₁ andλ₂ are constants. In Chapter 4, we study the harmonic Block space and obtain a result onα-Bloch spaces andβ-Bloch spaces. In the last chapter, we introduce the fundamental relationship between harmonic mappings and minimal surfaces, and state a famous conjecture about the curvatures of minimal surfaces.