Journal of the
Korean Mathematical Society JKMS

This paper mainly deals with the tilted Carath{\'e}odory class by angle $\lambda\in(-\pi/2,\pi/2)$ (denoted by $\mathcal{P}_{\lambda}$) an element of which maps the unit disc into the tilted right half-plane $\{w: \Re e^{i\lambda} w>0\}$. Firstly we will characterize $\mathcal{P}_{\lambda}$ from different aspects, for example by subordination and convolution. Then various estimates of functionals over $\mathcal{P}_{\lambda}$ are deduced by considering these over the extreme points of $\mathcal{P}_{\lambda}$ or the knowledge of functional analysis. Finally some subsets of analytic functions related to $\mathcal{P}_{\lambda}$ including close-to-convex functions with argument $\lambda$, $\lambda$-spirallike functions and analytic functions whose derivative is in $\mathcal{P}_{\lambda}$ are also considered as applications.

Keywords: the tilted Carath{\'e}odory class, $\lambda$-spirallike functions, close-to-convex functions with argument $\lambda$, convolution, subordination

MSC numbers: Primary 30C45; Secondary 30C70