MATHEMATICS ON WIKI

Let $$ \mathcal{H}(\mathrm{U})$$ denote a class of functions which are analytic in the open unit disk $$ \mathrm{U} = \{z \in \mathbb{C}: \; |z|<1\} $$. 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 $$ \delta < 1 $$, respectively, which are defined by

$$\mathcal{S}^*(\delta)= \left\{f \in \mathcal{A}: \; \Re \left( \frac{z f'(z)}{f(z)} \right) > \delta, \;\;\;z \in \mathrm{U} \right\},$$

$$\mathcal{K}(\delta)= \left\{f \in \mathcal{A}: \; \Re \left( 1+\frac{z f''(z)}{f'(z)} \right) > \delta, \;\;\; z \in \mathrm{U} \right\}. $$

If $$\,0 \leq \delta <1,$$ then a function in each of the classes $$ \mathcal{S}^*(\delta)$$ and $$ \mathcal{K}(\delta)$$ are univalent; if $$\delta <0$$ 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 $$\mathcal{B}(\mu,\nu), \; \mu \geq 0, \; 0 \leq \nu<1, $$ which satisfy the condition

$$ \left | \left (\frac {z}{f(z)}\right)^{\mu} f'(z) - 1 \right | < 1 - \nu, \;\;\; z \in \mathrm{U}.$$

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.

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