## On Convex Functions with Complex Order Through Bounded Boundary Rotation

Mathematics in Computer Science, vol.13, no.3, pp.433-439, 2019 (Refereed Journals of Other Institutions)

• Publication Type: Article / Article
• Volume: 13 Issue: 3
• Publication Date: 2019
• Doi Number: 10.1007/s11786-019-00405-8
• Title of Journal : Mathematics in Computer Science
• Page Numbers: pp.433-439

#### Abstract

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

$\begin{array}{r}1+\frac{1}{b}z\frac{{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}={p}_{t}\left(z\right),\left(b\ne 0,b\in \mathcal{C},z\in \mathcal{D}\right)\end{array}$

where ${p}_{t}\left(z\right)=\left(\frac{t}{4}+\frac{1}{2}\right){p}_{1}\left(z\right)-\left(\frac{t}{4}-\frac{1}{2}\right){p}_{2}\left(z\right)$$t\ge 2,{p}_{1}\left(z\right),{p}_{2}\left(z\right)\in \mathcal{P}$$\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(tb). In this present paper, we will introduce this class and its some properties.