In this paper we study sharp estimates of pre-Schwarzian derivatives functions belonging to the Nehari-type classes by using techniques from differential equations. sequel, also see that a solution complex equation has special form in terms ratio hypergeometric resulting an integral representation. Finally, attempt those univalent unit disk for which image domain is unbounded John domain.

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

In recent years, it has become clear that types in intensional MartinLöf Type Theory can be seen as spaces, alternatively to the traditional view as sets or propositions. This observation motivated Voevodsky's univalence axiom and the development of a whole branch of mathematics, known as Univalent Foundations and Homotopy Type Theory (HoTT). One of the most basic consequences of univalence is ...