Briot--Bouquet differential superordinations and sandwich theorems
Given complex numbers \(\beta ,\gamma\) and \(\Omega\subset\mathbb{C}\), an analytic function \(p:U\rightarrow\mathbb{C}\) in the unit disk is a solution of the Briot--Bouquet differential superordination, if \(\Omega\subset\{\widetilde{p}(z):=p(z)+zp'(z)(\beta p(z)+\gamma)^{-1} | z\in U\}\). In this article, properties are determined for functions \(p\) satisfying this differential superordination, as well as related sandwich theorems. To give an idea of the results obtained, let \(q:U\rightarrow\mathbb{C}\) be univalent with \(q(0)=a\). Considering now \(\widetilde{q}(z)\), suppose that \(\text{Re}(\beta q(z)+\gamma )>0\) and that \(\widetilde{q}(z)-q(z)\) is starlike. If \(p(z)=a+a_{1}z+a_{2}z^{2}+\cdots\) is analytic and injective in \(\overline{U}\setminus\{\zeta\in\partial U | \lim_{z\rightarrow\zeta}f(z)=\infty\}\), and \(\widetilde{p}\) is univalent in \(U\), then \(\widetilde{q}\prec\widetilde{p}\) implies that \(q\prec p\). Moreover, \(q\) is the best subordinant.
scientific article; zbMATH DE number 5129648