Refined harmonic analogue with |f(z)|^p and higher powers of S_r/π for the class P^0_{H(M)}

Determine whether refined Bohr–Rogosinski-type inequalities can be established for the class P^0_{H(M)} of close-to-convex harmonic mappings f=h+overline{g} in H_0 (defined by (z h''(z)) > -M + |z g''(z)| for z∈D and M>0), by incorporating a general power |f(z)|^p for p∈N and adding higher power terms of the area functional S_r/π; and ascertain whether harmonic analogues of Liu–Liu–Ponnusamy’s refined Bohr inequalities (Theorem 2.3) and of the inequalities in Theorems 2.5 through 1.21 for bounded analytic functions can be obtained for P^0_{H(M)}.

Background

Earlier sections survey refined and improved Bohr-type inequalities for bounded analytic functions, including Bohr–Rogosinski inequalities involving partial sums and the area functional S_r/π. Remark 2.1 observes a limitation: adding an arbitrary positive term F(S_r) to certain improved inequalities fails for the analytic class B.

Motivated by these refinements and limitations, the authors pose a problem to investigate whether analogous refined inequalities can be developed for the close-to-convex harmonic class P^0_{H(M)}, specifically allowing powers |f(z)|^p and higher powers of S_r/π, and whether sharp harmonic analogues of previously known analytic results (Theorems 2.3, 2.5, …, 1.21) can be obtained in this setting.