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

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