On certain harmonic mappings with some fixed coefficients

Abstract

For a fixed analytic function h an interesting problem arises to describe all function g such that f= h+ g¯ is complex valued harmonic mappings in the open unit dick D. In the present paper, we introduced two sub-classes FHδ(α) and GHδ(α) of harmonic mappings in D by restricting analytic part of every function in these classes as a member of Sδ(α),(δ≥0,0≤α<1), which is introduced by Kumar (J Math Anal Appl 126:70–77, 1987) (also see Mishra and Gochhayat (J Inequal Pure Appl Math 7(3), Art. 94:1–15, 2006)). Coefficient estimates, bounds for growth and area of the functions belonging to the these classes are established. Other geometric properties, such as the disk where the functions are fully starlike, fully convex, uniformly starlike and uniformly convex have been analyzed for some subclass of the newly defined classes. As an application, explicit representation of minimal surfaces and its conjugate surfaces are found corresponding to the harmonic mappings. Using Mathematica, the graphical illustration of some results are also presented. The article is concluded with some future aspects of the investigation. © 2018, Springer-Verlag GmbH Austria, part of Springer Nature.