On Convex Functions with Complex Order Through Bounded Boundary Rotation

Abstract

Let F be the class of functions $f(z)=z+a_{2}z^{2}+\cdots$ which are analytic in ${\mathcal {D}}=\{z: |z|<1\}$ and satisfies the condition

$$\begin{aligned} 1+\frac{1}{b}z\frac{f''(z)}{f'(z)}=p_{t}(z), (b\ne 0, b\in {\mathcal {C}}, z\in {\mathcal {D}}) \end{aligned}$$

where $p_{t}(z)=\left( \frac{t}{4}+\frac{1}{2}\right) p_{1}(z)-\left( \frac{t}{4}-\frac{1}{2}\right) p_{2}(z)$, $t\ge 2, p_{1}(z),p_{2}(z)\in {\mathcal {P}}$. ${\mathcal {P}}$ is the class of analytic functions with the positive real part (Caratheodory class) then this function will be called convex function by means of bounded boundary rotation and denoted by K(t, b). In this present paper, we will introduce this class and its some properties.