Inequalities for the Coefficients of Schwarz Functions

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