MATHEMATICS ON WIKI
Let [math] \mathcal{H}(\mathrm{U})[/math] denote a class of functions which are analytic in the open unit disk [math] \mathrm{U} = \{z \in \mathbb{C}: \; |z|\lt1\} [/math] . Let [math] \mathcal{A} [/math] the class of all functions [math] f \in \mathcal{H}(\mathrm{U})[/math] normalized by [math] f(0), f'(0)=1 [/math] and having form
[math] f(z)= z+ a_2 z^2+ a_3 z^3+ \cdots,\; z \in \mathrm{U}. [/math]
We denote by [math]\mathcal{S} [/math] the subclass of [math] \mathcal{A} [/math] consisting of functions which are also univalent in [math] \mathrm{U} [/math]. In Robertson studied the classes [math]\mathcal{S}^*(\delta), \; \mathcal{K}(\delta) [/math] of starlike and convex of order [math] \delta \lt 1 [/math], respectively, which are defined by
[math]\mathcal{S}^*(\delta)= \left\{f \in \mathcal{A}: \; \Re \left( \frac{z f'(z)}{f(z)} \right) \gt \delta, \;\;\;z \in \mathrm{U} \right\},[/math]
[math]\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\}.
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If [math]\,0 \leq \delta \lt1,[/math] then a function in each of the classes [math] \mathcal{S}^*(\delta)[/math] and [math] \mathcal{K}(\delta)[/math] are univalent; if [math]\delta \lt0[/math] then function in the classes [math] \mathcal{S}^*(\delta) [/math] and [math] \mathcal{K}(\delta)[/math] may fail to be univalent. In particular we define [math]\mathcal{S}^*(0)=\mathcal{S}^*, \mathcal{K}(0)=\mathcal{K}.[/math]
Recently, Frasin and Jahangiri, studied a subclass of analytic functions [math]f \in \mathcal{A},[/math] denoted by [math]\mathcal{B}(\mu,\nu), \; \mu \geq 0, \; 0 \leq \nu\lt1, [/math] which satisfy the condition
[math] \left | \left (\frac {z}{f(z)}\right)^{\mu} f'(z) - 1 \right | \lt 1 - \nu, \;\;\; z \in \mathrm{U}.[/math]
Note that [math] \mathcal{B}(1,\nu) = \mathcal{S}^* (\nu)[/math]. Also it was observed by Ozaki and Nunokawa that [math]\mathcal{B}(2, 0)=\mathcal{S}[/math]. Furthermore [math]\mathcal{B}(2,\nu) =\mathcal{B}(\nu)[/math] is subclass of [math]\mathcal{A}[/math] which was studied by Frasin and Darus and further generalization of the class [math]\mathcal{B}(\mu,\nu)[/math] has further been studied by Prajapat.
