Injectivity of sections of convex harmonic mappings and convolution theorems

We consider the class H 0 of sense-preserving harmonic functions defined in the unit disk | z | < 1 and normalized so that h (0) = 0 = h ′(0) − 1 and g (0) = 0 = g ′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes P H 0 ( α ) and G H 0 ( β ) of functions from H 0 and show that if f ∈ P H 0 ( α ) and F ∈ G H 0 ( β ), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums) where ∈ H 0 , s n ( h ) and s n ( g ) denote the n -th partial sums of h and g , respectively. We prove, among others, that if ∈ H 0 is a univalent harmonic convex mapping, then s n , n ( f ) is univalent and close-to-convex in the disk | z | < 1/4 for n ≥ 2, and s n , n ( f ) is also convex in the disk | z | < 1/4 for n ≥ 2 and n ≠ 3. Moreover, we show that the section s 3,3 ( f ) of f ∈ C H 0 is not convex in the disk | z | < 1/4 but it is convex in a smaller disk.