Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings

Let H(Ω) be the class of complex-valued functions harmonic in Ω and each f=h+ḡ∈ H(Ω), where h and g are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant rf∈ (0, 1) such that the inequality {align*} M_f(r):=r+∑ₙ₌₂∞(|a_n|+|b_n|)r^n≤ d(f(0), ∂Ω) \;for\;|z|=r≤ r_f, {align*} where d(f(0), ∂Ω) is the Euclidean distance between $ f(0) $ and the boundary of Ω:=f(D). The largest such radius rf is called the Bohr radius and the inequality Mf(r)≤ d(f(0), ∂Ω) is called the Bohr inequality for the class H(Ω). In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp.