New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

In this paper, we introduce a new subclass of harmonic functions f = s + ¯ t f=s+t¯ in the open unit disk U { z ∈ C : | < 1 } U={z∈C:|z|<1} satisfying ${\text{Re}}\left[ \gamma \mathfrak{s}^{\prime }(z)+\delta z\mathfrak{s}^{\prime \prime }(z)+\left( \frac{\delta -\gamma }{2}\right) z^{2}\mathfrak{s}^{\prime }\left( z\right) -\lambda \right]>\left \vert \mathfrak{t}^{\prime z\mathfrak{t}^{\prime\prime z^{2}\mathfrak{t}^{\prime \right \vert,$ where 0 ≤ λ γ δ , . 0≤λ<γ≤δ,z∈U. We determine several properties class such as close-to-convexity, coefficient bounds, and growth estimates. also prove that is closed under convex combination convolution its members. Furthermore, investigate fully starlikeness convexity class.