An application of a subordination chain (Q1811949)

Summary: Let \(K\) denote the class of functions \(g(z)=z+a_{2}z^{2}+\cdots\) which are regular and univalently convex in the unit disc \(E\). In the present note, we prove that if \(f\) is regular in \(E\), \(f(0) = 0\), then for \(g\in K\), \(f(z)+\alpha z f'(z)\prec g(z)+\alpha zg'(z)\) in \(E\) implies that \(f(z)\prec g(z)\) in \(E\), where \(\alpha > 0\) is a real number and the symbol `` \(\prec\)'' stands for subordination.