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 aligned p(z)[ 1+zp'(z)/p(z)φ ( p(z),zp'(z)) ] ^γ h(z), z∈ D, aligned p ( z ) 1 + z p ′ ( z ) p ( z ) φ p ( z ) , z p ′ ( z ) γ ≺ h ( z ) , z ∈ D , where D:=∈ C:|z|<1\ D : = { z ∈ C : | z | < 1 } , h is a convex univalent function with 0∈ h( D) 0 ∈ h ( D ) , and with appropriate assumptions on φ φ 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.