Abstract

 In this paper we have obtained some necessary and sufficient conditions for the following classes:\\

1) $\mathbf{SVP_{\psi}(\nu,p)}$\\

A function $I(z)$ of the class $\mathcal{A}_p$ also contained in the subclass $SVP_{\psi}(\nu, p)$ if it satisfies the inequality

\begin{eqnarray*}

\bigg|\frac{I^{(p-1)}(z)}{(cos\psi+i sin\psi)z I^{(p)}(z)}-\frac{1}{3\nu}\bigg|< \frac{2}{3\nu}\;\;\; where\;\; \psi\in \mathbb{R}\;\; and\;\; 0<\nu<1.

\end{eqnarray*}

2) $\mathbf{CVP_{\psi}(\nu,p)}$\\

A function $I(z)\in \mathcal{A}_p$ is said to be in the class $CVP_{\psi}(\nu,p)$ if it satisfies the inequality

\begin{eqnarray*}

\bigg|\frac{I^{(p)}(z)}{(cos\psi+i sin\psi)z I^{(p+1)}(z)}-\frac{1}{3\nu}\bigg|< \frac{2}{3\nu}\;\;\; where \;\;\psi\in \mathbb{R} and\;\; 0<\nu<1.

\end{eqnarray*}

We have extended the previous results and derived some corollaries.

Keywords and Phrases

Differintegral operator, Pre-starlike functions, Spirallike functions, Starlike functions.

A.M.S. subject classification

30C10, 30C45.

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