On Certain Differential Subordination and its Dominant

Abstract

Denote by $\cal A'$, the class of functions $f$, analytic in $E$ which satisfy $f(0)=1$. Let $\alpha >0, \beta \in (0,1]$ be real numbers and let $\gamma, {\rm Re} \gamma >0$, be a complex number. For $p, q \in \cal A'$, the authors study the differential subordination of the form $$(p(z))^\alpha \left[1+\frac {\gamma zp'(z)}{p(z)}\right]^\beta \prec(q(z))^\alpha \left[1+\frac {\gamma zq'(z)}{q(z)}\right]^\beta, z\in E,$$ and obtain its best dominant. Its applications to univalent functions are also given.