Inequalities for the Coefficients of Schwarz Functions

Abstract

AbstractThe relation between a considered family of analytic functions and the class $${\mathcal {P}}$$ P of functions with a positive real part is one of the main tools used in solving various extremal problems, among others coefficient problems. Another approach can be useful in solving such tasks. This approach is to exploit the correspondence between a considered family and the family $${\mathcal {B}}_0$$ B 0 of bounded analytic functions $$\omega $$ ω such that $$\omega (0)=0$$ ω ( 0 ) = 0 . Such functions appear in the well-known Schwarz lemma, so they are called Schwarz functions. In the literature, there are numerous coefficient functionals discussed for functions in $${\mathcal {P}}$$ P . On the other hand, relative functionals for functions in $${\mathcal {B}}_0$$ B 0 are not so commonly studied. Consequently, we do not know so much about coefficient inequalities for Schwarz functions. We shall fill the gap to some extent considering two types of functionals. The first one is a Zalcman-type functional $$c_{n}-c_{k}c_{n-k}$$ c n - c k c n - k ; the other one is the Hankel determinant $$c_{n-1}c_{n+1}-c_{n}{}^2$$ c n - 1 c n + 1 - c n 2 . For these functionals, bounds with respect to a fixed first coefficient $$c_1$$ c 1 (or a few initial coefficients) are obtained. Some generalizations of these functionals are also given. All results are sharp.