Treffer: On Some Generalization of the Well-known Class of Bounded Univalent Functions: On some generalization of the well-known class of bounded univalent functions

Let \(D=\{w\): \(| w|0\), \(\alpha\in(0,1)\) are any fixed numbers. Let \(S\) denote the well-known class of functions \(f(z)=z+a_ 2 z^ 2+\dots\) holomorphic and univalent in the disc \(\Delta=\{z\): \(| z|0\). In the paper the authors introduce and investigate the basic properties of the class \(S(M,\alpha)=\{f\in S\): \(f\prec F\}\) where \(f\prec F\) means that the function \(f\) is subordinate to the function \(F\) in \(\Delta\). In the proofs of the theorems they make use of the definition of subordination, the properties of the function \(F\) and the properties of the class \(S(R)=\{f\in S\): \(| f(z)