# MATHEMATICS ON WIKI

Let $\mathcal{H}(\mathrm{U})$ denote a class of functions which are analytic in the open unit disk $\mathrm{U} = \{z \in \mathbb{C}: \; |z|\lt1\}$ . Let $\mathcal{A}$ the class of all functions $f \in \mathcal{H}(\mathrm{U})$ normalized by $f(0), f'(0)=1$ and having form

$f(z)= z+ a_2 z^2+ a_3 z^3+ \cdots,\; z \in \mathrm{U}.$

We denote by $\mathcal{S}$ the subclass of $\mathcal{A}$ consisting of functions which are also univalent in $\mathrm{U}$. In Robertson studied the classes $\mathcal{S}^*(\delta), \; \mathcal{K}(\delta)$ of starlike and convex of order $\delta \lt 1$, respectively, which are defined by
$\mathcal{S}^*(\delta)= \left\{f \in \mathcal{A}: \; \Re \left( \frac{z f'(z)}{f(z)} \right) \gt \delta, \;\;\;z \in \mathrm{U} \right\},$
$\mathcal{K}(\delta)= \left\{f \in \mathcal{A}: \; \Re \left( 1+\frac{z f''(z)}{f'(z)} \right) \gt \delta, \;\;\; z \in \mathrm{U} \right\}.$
If $\,0 \leq \delta \lt1,$ then a function in each of the classes $\mathcal{S}^*(\delta)$ and $\mathcal{K}(\delta)$ are univalent; if $\delta \lt0$ then function in the classes $\mathcal{S}^*(\delta)$ and $\mathcal{K}(\delta)$ may fail to be univalent. In particular we define $\mathcal{S}^*(0)=\mathcal{S}^*, \mathcal{K}(0)=\mathcal{K}.$

Recently, Frasin and Jahangiri, studied a subclass of analytic functions $f \in \mathcal{A},$ denoted by $\mathcal{B}(\mu,\nu), \; \mu \geq 0, \; 0 \leq \nu\lt1,$ which satisfy the condition
$\left | \left (\frac {z}{f(z)}\right)^{\mu} f'(z) - 1 \right | \lt 1 - \nu, \;\;\; z \in \mathrm{U}.$

Note that $\mathcal{B}(1,\nu) = \mathcal{S}^* (\nu)$. Also it was observed by Ozaki and Nunokawa that $\mathcal{B}(2, 0)=\mathcal{S}$. Furthermore $\mathcal{B}(2,\nu) =\mathcal{B}(\nu)$ is subclass of $\mathcal{A}$ which was studied by Frasin and Darus and further generalization of the class $\mathcal{B}(\mu,\nu)$ has further been studied by Prajapat.