On differential subordinations related to convex functions
Let \(S^c\) denote the class of all convex functions \(h\) in the unit disc \(U=\{z:| z|<1\}\) and normalized \(f(0)=f'(0)-1=0\) and let \(S^c_0\) denote the subclass of \(S^c\) of such functions \(h\) that there exists a boundary point \(Q\) of \(h(U)\) lying in \((-\infty,0)\) and the line going through \(Q\) and parallel to the imaginary axis is a supporting line of \(h(U)\). In the paper the following result is proved: Theorem. Let \(\alpha\in[0,1]\), \(\beta\in[\alpha,1]\), \(h\in S^c_0\) and let \(p\) be analytic in \(U,p(0)=0\), \(p(z)\not\equiv 0\) such that a function \(P(z)=p(z)[1+{zp'(z)\over p(z)}]^\beta\) is well defined in \(U\), where the power has the principal value. If \(P(z)\prec({3\over 2})^\alpha h(z)\) then \(p(z)\prec h(z)\) where ``\(\prec\)'' denotes the subordination in the unit disc. Some other similar problems are also investigated.
scientific article; zbMATH DE number 1332879
scientific article