The tilted Carathéodory class and its applications (Q2898891)

Let \(\mathcal {A}_0\) be the family of all functions \(f\) analytic in the unit disk \(\mathbb D\) and normalized by \(f(0) = 1\). The author investigates the tilted Carathéodory class denoted by \(\mathcal P_{\lambda}\) (tilted by angle \(\lambda\)) and defined as NEWLINE\[NEWLINE{\mathcal P_{\lambda}} = \{p \in \mathcal A_0: \text{Re} \, e^{i\lambda}p(z) > 0, \;z \in \mathbb D\},\quad -\pi/2 < \lambda < \pi/2.NEWLINE\]NEWLINE The author proves some equivalent conditions for a function \(p \in {\mathcal A_0}\) to be in the class \(\mathcal P_{\lambda}\). Moreover, convolution properties, extreme points, and sharp estimates of some functionals in \(\mathcal P_{\lambda}\) are obtained. Finally, some applications of the results to close-to-convex functions, spirallike functions and analytic functions with the derivative in \(\mathcal P_\lambda\) are presented.