On a Coefficient Inequality for Carathéodory Functions

Abstract

In the paper we consider a general inequality $$|p_{n-1}p_{n+1}-p_n{}^2|\le 4-|p_1|^2$$ | p n - 1 p n + 1 - p n 2 | ≤ 4 - | p 1 | 2 involving coefficients of functions with a positive real part. We prove this inequality for $$n=2$$ n = 2 and $$n=3$$ n = 3 . Consequently, the relative inequalities involving coefficients of Schwarz functions are obtained. As an application, the two sharp estimates of the Hankel determinants $$H_{3,1}$$ H 3 , 1 and $$H_{2,3}$$ H 2 , 3 are proved for functions in $${\mathcal S}^*(1/2)$$ S ∗ ( 1 / 2 ) and $${\mathcal {M}}$$ M , respectively.