Inner Method for Differential Subordinations of Geometric Mean

In this paper we introduce and examine the differential subordinations related to the geometric mean of the form \(\begin{aligned} p(z)\left[ 1+\frac{zp'(z)}{p(z)}\varphi \left( p(z),zp'(z)\right) \right] ^\gamma \prec h(z),\quad z\in \mathbb D, \end{aligned}\) $$\begin{array}{c}\hfill p(z){\left[1,+,\frac{z{p}^{\prime }(z)}{p(z)},\phi ,\left(p,(z),,,z,p^{\prime },(z)\right)\right]}^{\gamma }\prec h(z),z\in \mathbb{D},\end{array}$$ where \(\mathbb D:=\{z\in \mathbb C:|z|<1\}\) $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ , h is a convex univalent function with \(0\in h(\mathbb D)\) $0\in h(\mathbb{D})$ , and with appropriate assumptions on \(\varphi \) $\phi$ and p. Proofs of main results are based on the original results and offer a new approach in the theory. In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. Based on Loewner’s theory, the problem of the best dominant is discussed. Relevant applications within GFT are indicated.