Techniques of the differential subordination for domains bounded by conic sections (Q1415093)

Summary: We solve the problem of finding the largest domain \(D\) for which, under given \(\psi\) and \(q\), the differential subordination \(\psi(p(z), zp^{\prime}(z)) \in D \Rightarrow p(z) \prec q(z)\), where \(D\) and \(q(\mathcal{U})\) are regions bounded by conic sections, is satisfied. The shape of the domain \(D\) is described by the shape of \(q(\mathcal{U})\). Also, we find the best dominant of the differential subordination \(p(z) +({zp^{\prime}(z)}/({\beta p(z) + \gamma})) \prec p_{k}(z)\), when the function \(p_k\) \((k\in [0,\infty))\) maps the unit disk onto a conical domain contained in a right half-plane. Various applications in the theory of univalent functions are also given.