The tilted Carath\'eodory function class and its practical applications

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for the tilted Carath\'eodory function of order $\beta$ and angle $\alpha$ ($\alpha \in (-\pi/2,\pi/2)$ and $\beta \in [0,\cos\alpha)$), which maps the unit disk $\mathbb{D}$ into the region $\{ w\in\mathbb{C}: {\rm{Re}}\{ {\rm e}^{{\rm i}\alpha} w \} > \beta \}$. Using these conditions, we also derive conditions for an analytic function that maps $\mathbb{D}$ into a sector defined by $\{ w\in\mathbb{C} : | \arg(w-\gamma) | < (\pi/2)\delta \}$, where $\gamma \in [0,1)$ and $\delta \in (0,1]$. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in $\mathbb{D}$. KCI Citation Count: 0