Communications of the
Korean Mathematical Society CKMS

Using the concept of the strong differential subordination introduced in cite{4.a}, we find conditions on the functions $ heta, varphi, G, F$ such that the first order strong subordination $$ heta(p(z))+frac{G(xi)}{xi} zp'(z)varphi(p(z))prec prec heta(q(z))+F(z)q'(z)varphi(q(z),$$ implies $$p(z)prec q(z),$$ where $p(z), q(z)$ are analytic functions in the open unit disk $mathbb{D}$ with $p(0)=q(0)$. Corollaries and examples of the main results are also considered, some of which extend and improve the results obtained in cite{1.a}.

Keywords: Convex, close-to-convex function, starlike function, strong differential subordination

MSC numbers: Primary 30C45; Secondary 30C80