Application of strong differential superordination to a general equation

Abstract

In this paper, we study the notion of strong differential superordination as a dual concept of strong differential subordination, introduced in~\cite {1.a}. The notion of strong differential superordination has recently been studied by many authors, see, for example, \cite {2.a, 3.a, 5.a}. Let $q(z)$ be an analytic function in $\mathbb {D}$ that satisfies the first order differential equation $$\theta (q(z))+F(z)q'(z)\varphi (q(z))=h(z).$$ \smallskip Suppose that $p(z)$ is analytic and univalent in the closure of the open unit disk $\overline {\mathbb {D}}$ with $p(0)=q(0)$. We shall find conditions on $h(z),G(z),\theta (z)$ and $\varphi (z)$ such that $$ h(z)\prec \prec \theta (p(z))+\frac {G(\xi )}{\xi }zp'(z)\varphi (p(z))\Longrightarrow q(z)\prec p(z). $$ Applications and examples of the main results are also considered.