On certain differential subordination for sectors (Q2708126)

Let \(D\) denote the open unit disc of the complex plane. The main result of this paper is the following theorem. Let \(\gamma\neq 0\) be such that \(\rho =\text{Re} \gamma\geq 0\), \(\alpha\in(0,1]\). Let \(p(z)\) be an analytic function in \(D\) with \(p(0)=1\), \(p'(0)=\cdots=p^{(n-1)}(0)=0\). If \(p(z)\) satisfies the condition that \(p(z)+\gamma zp'(z)\) is subordinate to \(({1+z\over 1-z})^\alpha\) in \(D\) then there exists a constant \(\delta= \delta(\alpha, \gamma, n)>0\) such that for \(z\in D\) \(\text{Re} p(z)>\delta\). Indeed \(\delta= \int^1_0 ({1-t^{n\rho} \over 1+t^{n\rho}})^\alpha dt\) and the result is also shown to be sharp.