Functions star-and convex-univalent of order \(\alpha \) with weight
In this paper the author continues his previous work regarding the subclasses of the class \(Q\) of functions \(f(z)\) with \(f(0)=1\) holomorphic on the disk \(|z|<1\) and univalent with weight \(z\) on this disk. He introduces and studies the class \(M(\alpha )\) of functions \(f(z)\) with \(f(0)=1\) holomorphic on the disk \(|z|<1\) and star-univalent of order \(\alpha\) \((0<\alpha <1)\) with weight \(z\) and the class \(N(\alpha )\) of functions \(f(z)\) with \(f(0)=1\) holomorphic on the disk \(|z|<1\) and convex-univalent of order \(\alpha\) with weight \(z\). The author gives criteria for a function \(f(z)\) with \(f(0)=1\) holomorphic on the disk \(|z|<1\) to belong to the classes \(M(\alpha)\) and \(N(\alpha)\), a relationship between the functions of the classes \(M(\alpha)\) and \(N(\alpha)\), sharp bounds for the Taylor coefficients of functions from the classes \(M(\alpha)\) and \(N(\alpha)\) and sharp bounds for the moduli of a function \(f(z)\) and of the operator \(L(f(z))=f(z)+zf^{\prime}(z)\) in these classes.
holomorphic and univalent functions with weight z on the unit disk
scientific article; zbMATH DE number 5375925