On a sufficient condition and an angular estimation for \(\Phi\)-like functions (Q1282129)

Let \(A\) denote the class of functions of the form \(f(z)=z+ \sum_{n=2}^\infty a_nz^n\), analytic in the unit disk \(U= \{z: | z|<1\}\), \(f\in A\) is called \(\varphi\)-like in \(U\) if \(\text{Re} \{\frac{zf'(z)} {\varphi(f(z))} \}>0\), \(z\in U\), where \(\varphi(\omega)\) is analytic in \(f(U)\), \(\varphi(0)= \varphi'(0)- 1=0\) and \(\varphi(\omega)\neq 0\) for \(\omega\in f(U)- \{0\}\). It is known that \(\varphi\)-like functions are univalent in \(U\). Starlike functions occur as a particular case when \(\varphi\) reduces to the identity function. In this paper a sufficient condition is derived for \(f\in A\) to be \(\varphi\)-like in \(U\). An angular estimate is also derived. Nunokawa's earlier results are deduced as corollaries. However, the author depends on a lemma of Nunokawa for proving his results.