Geometric analysis of a class of harmonic mappings defined by a differential inequality

In this paper, we introduces and undertake as a systematical investigation of the class ${\mathcal{P}}_{\mathcal{H}}^0(\alpha ,M)$ of normalized harmonic mappings $f=h+\bar{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality $$\text{Re}((1-\alpha )h^{\mathrm{\prime }}(z)+\alpha z{h}^{\mathrm{\prime }\mathrm{\prime }}(z))>-M+\mid (1-\alpha )g^{\mathrm{\prime }}(z)+\alpha z{g}^{\mathrm{\prime }\mathrm{\prime }}(z)\mid \text{for}z\in \mathbb{D},$$ where $M>0$, $\alpha \in (0,1]$, and $g^{\mathrm{\prime }}(0)=0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that ${\mathcal{P}}_{\mathcal{H}}^0(\alpha ,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.