On certain univalent class associated with first order differential subordinations (Q2896923)

Let \(U\) be the complex unit disc and \(\mathcal{ H}[a,n]\) the class of all analytic functions \(f\) in \(U\) with \(f(z)=a+a_nz^n+a^{n+1}z^{n+1}+\cdots\). Also, let \(\mathcal A\) be the class of analytic functions \(f\) in \(U\), normalized with \(f(0)=f'(0)-1=0\). If \(\Phi :\mathbb C^2\to\mathbb C\) and (for \(f\in\mathcal A\)) \(\mathcal R\frac{zf'(z)}{\Phi(z)}>0\) in \(U\), then \(f\) is called \(\Phi\)-like. In the paper under review the author defines a new subclass of \(\mathcal A\), denoted by \(H(\alpha,\lambda;\Phi_1(f(z)),\Phi_2(f(z))\), involving two different types of \(\Phi\)-like functions (\(\Phi_1\) and \(\Phi_2\)), defined by NEWLINE\[NEWLINE\frac{zf'(z)}{\Phi_1(z)}\left[(1-\alpha)\frac{zf'(z)}{\Phi_2(z)}+\alpha(1+\frac{\lambda zf''(z)}{f'(z)})\right]\prec F(z),NEWLINE\]NEWLINE where \(\alpha\in[0,1]\), \(\lambda\in\mathbb R\) and \(F\) is a conformal mapping in \(U\) with \(F(0)=1\) and \(\Phi_1\), \(\Phi_2\) satisfy to some additional conditions. The author gives some sufficient conditions for a function \(f\) to be in this class and proves that every such function is univalent. Some corollaries are also given.