Refined harmonic analogue with |f(z)|^p and higher powers of S_r/(π−S_r) 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−S_r) in place of S_r/π; and ascertain whether harmonic analogues of Ismagilov–Kayumov–Ponnusamy’s improved inequalities (Theorem 1.18) and of the substitution result with S_r/(π−S_r) (Theorem 2.11) can be obtained for P^0_{H(M)}.
References
Motivated from the work of Ismagilov et al. , it is natural to raise the following question also. What can be deduced about the refine harmonic counterpart of Theorem \ref{TH-2.3} by incorporating a general power of $ |f(z)|p $ for $ p\in\mathbb{N} $ and adding higher power terms of $S_r/(\pi-S_r)$ instead of the term $S_r/\pi$ for the class $\mathcal{P}{0}_{\mathcal{H}(M)?$ Moreover, can we obtain the harmonic analogue of the Theorems \ref{th-1.18} and \ref{th-2.11} for the class $\mathcal{P}{0}_{\ mathcal{H}(M)?$