The Schwarzian Derivative and Convex Functions

On maximal area integral problem for analytic functions in the starlike family

“…Marx [9] and Strohacker [20] proved that if f (z) maps the unit disk onto a convex domain, then f (z) is starlike of order 1/2. Gabriel [5] showed that the class of starlike functions of order 1/2 played an important role in the solution of differential equations. In 1968, Padmanabhan [13] discussed a different subfamily for the order of starlikeness.…”

A note on convex mappings

“…On taking real parts, and noting, in view of (2) and (10), that But, as shown before, this implies that the function /(z), normalized by (1), which is a solution of {/(z), z\ =2Xçi(z) does not map |z| <1 onto the complement of a convex region. This shows that the constant 2 in (2) is indeed the largest possible.…”

An application of the principal theorem of conformal mapping in the theory of linear differential equations of second order

“…Ąz'(x)J обозначает производную Шварца от функции г в точке х, позволяет нам отнести результаты теории дифференциальных уравнений (1) к мероморфным, в первую очередь одно-однозначным функциям в области Г([1], [2], [3], [4], [5]). С дру гой стороны, многие свойства дифференциального уравнения (1) можно вы вести из теорем об однозначных аналитических функциях ( [6], [7]).…”