Inequalities involving weighted means in a disc of the complex plane (Q1974229)

For complex numbers \(z_j\), \(j=1,2,\dots,n\), such that \(|z_j-1|\leq r\), \(r\in(0,1)\), the author considers the weighted means: \(H:= (\sum^n_{j=1} \lambda_jz_j^{-1})\), \(G:=\prod^n_{j=1} z_j^{\lambda_j}\) and \(A: =\sum^n_{j=1} \lambda_jz_j\), where \(\lambda_j>0\), \(\lambda_1+ \cdots+ \lambda_n =1\). Among other results, he proves: Theorem 1. \((1-r^2)|A|\leq|H |\leq |A|/(1-r^2)\). Theorem 4. \((1-r^2)|G|(1-|G-1 |^2)^{-1}\leq|H|\leq|G|^2/(1-r)\). The results are sharp. In the proofs the appropriate properties of starlike and convex functions are used.