On a first order strong differential subordination and application to univalent functions
Commun. Korean Math. Soc. 2022 Vol. 37, No. 2, 445-454
https://doi.org/10.4134/CKMS.c210070
Published online March 31, 2022
Printed April 30, 2022
Rasoul Aghalary, Parviz Arjomandinia
Urmia University; Urmia University
Abstract : Using the concept of the 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, some of which extend and improve the results obtained in \cite{1.a}.
Keywords : Convex, close-to-convex function, starlike function, strong differential subordination
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