On the families of hyperbolic derivatives with the quasi-Löwner dynamics of pre-Schwarzians

The dynamics of the critical point set for the hyperbolic derivatives of the family of holomorphic functions in the unit disk with pre-Schwarzians satisfying the equation of the quasi-Löwner type is studied. In order to solve the corresponding Gakhov equation, we use the uniformization depending on the additional parameter and based on the Weierstraß preparation theorem and the Painlevé uniqueness theorem for the Cauchy problem. The action of two well-known (quasi-Löwner) families, level lines, and Hornich rays is demonstrated on the same generating function. The choice of the new form of the Gakhov equation leads to the new condition for (no more than) uniqueness of the critical point of the hyperbolic derivative of the holomorphic function – non-positivity of the Jacobian equation in terms of the pre-Schwarzian of the function. The resulting inequality is satisfied by the functions of the well-known Marx–Strohhacker class.