On a first order strong differential subordination and application to univalent functions

Commun. Korean Math. Soc. Published online August 2, 2021

Rasoul Aghalary and Parviz Arjomandinia
Dept. of Math. Urmia Univ.; Dept. of Education

Abstract : Using the concept of strong differential subordination, introduced in \cite{4.a}, we find conditions on
the functions $\theta, \varphi, G, F$ such that the first order strong subordination
$$\theta(p(z))+\frac{G(\xi)}{\xi} zp'(z)\varphi(p(z))\prec \prec \theta(q(z))+F(z)q'(z)\varphi(q(z),$$
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,
which some of them extend and improve the results obtained in \cite{1.a}.