Generalization properties of Bernardi integral operator

Abstract

Let $A\left( n\right)$ be the class of analytic functions $f\left(z\right)$ of the form \begin{equation*} f\left( z\right) =z+\sum_{k=n}^{\infty }a_{k}z^{k},\;\;\left( n=2,3,4,...\right) \end{equation*} in the open unit disk $U.$ We introduce the integral operator $B_{j}f\left( z\right) $$=B\left( B_{j-1}f\left( z\right) \right) $, $B_{1}f\left( z\right) =Bf\left( z\right)$ and $B_{0}f\left( z\right) =f\left( z\right) $. In the present paper, we define the subclass $M_{j}\left( n,\gamma ,\alpha \right) $ and discuss some interesting properties of $f\left( z\right) \in A\left( n\right)$ concerning with the class $M_{j}\left( n,\gamma ,\alpha \right) .$

Keywords

• Analytic function
• Bernardi integral operator
• coefficient problem
• argument problem
• subordination

References

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