The titled Carathodory class and its applications

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 $\{w:\mathrm{\Re }{e}^{i\lambda }w>0\}$. 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