Some classes of harmonic mappings with analytic part defined by subordination

Abstract

Let S H role=presentation> S H S H S_{H} be the class of functions f = h + g ¯ role=presentation> f = h + g ¯ f = h + g ¯ f=h+\bar{g} that are harmonic univalent and sense-preserving in the open unit disk U = { z ∈ C : | z | 𝕌 = { z ∈ ℂ : | z | h , h , h, g role=presentation> g g g are analytic and f ( 0 ) = f z ′ ( 0 ) − 1 = 0 role=presentation> f ( 0 ) = f ′ z ( 0 ) − 1 = 0 f ( 0 ) = f z ′ ( 0 ) − 1 = 0 f(0)=f_{z}'(0)-1=0 in U . role=presentation> 𝕌 . U . \mathbb{U}. In this paper, we investigate the properties of some subclasses of S H role=presentation> S H S H S_{H} such that h ( z ) role=presentation> h ( z ) h ( z ) h(z) is a starlike (or convex) function defined by subordination. We provide coefficient estimates, extremal function, distortion and growth estimates of g role=presentation> g g g , growth, and Jacobian estimates of f role=presentation> f f f . We also obtain area estimates and covering theorems of the classes. The results presented here generalize some known results.