Sufficient conditions and applications for Carathéodory functions (Q355832)

Summary: Let \(P\) be the class of functions of the form \(p(z) = 1 + \sum^\infty_{n=1} p_nz^n\) which are analytic in \(U\). By using the method of differential subordinations, we give some sufficient conditions for Carathéodory function, that is, if \(p(z) \in P\) satisfies \(p(z) \neq 0\) (\(0 < |z| < 1\)) and \[ p(z)^m + \lambda(zp'(z)/(p(z))^\mu) \prec h(z) \] where \[ h(z) = ((1 + az)/(1 + bz))^m + \lambda(a - b)z/(1 + bz)^{2-\mu}(1 + az)^\mu, \] \(-1 \leq b < a \leq 1\), \(\lambda > 0\), \(m, \mu \in \{0, 1, 2\}\), \(m + \mu \leq 2\), then \[ p(z) \prec (1 + az)/(1 + bz). \] Some useful consequences of this result are also given.