The titled Carathodory class and its applications

Abstract

This survey mainly deals with the tilted Carath{\'e}odory class by angle $\lambda$ (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. In section 3, various estimates of functionals over $\mathcal{P}_{\lambda}$ are deduced from the known corresponding estimates of $\mathcal{P}_{0}$ or elementary functional analysis. Finally some subsets of analytic functions related to $\mathcal{P}_{\lambda}$ including close-to-convex functions with argument $\lambda$, $\lambda$-spirallike functions, $\lambda$-Robertson functions and analytic functions whose derivative is in $\mathcal{P}_{\lambda}$ are also considered as applications