Boundary Method for Differential Subordinations of the Geometric Mean

In this paper we introduce and examine the differential subordination related to the geometric mean of the form $$\begin{aligned} {[}p(z)]^{1-\gamma }\left[ p(z)+zp'(z)\varphi \left( p(z),zp'(z)\right) \right] ^\gamma \prec h(z),\quad z\in {\mathbb {D}}, \end{aligned}$$ where $${\mathbb {D}}:=\{z\in {\mathbb {C}}:|z|<1\}$$ and h is a convex univalent function with $$0\in \partial h({\mathbb {D}}).$$ In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. At the same time new type of ordinary differential equation has been proposed for study. The main results allow us to construct non-obvious subclasses in the class of holomorphic functions on $${\mathbb {D}}$$ .