Necessary and sufficient conditions for two subclasses of analytic functions associated with Pascal distribution series

Abstract

In the present paper, we determine necessary and sufficient condition for
\begin{equation*}
\Phi _{q}^{m}(z):=z-\sum \limits_{n=2}^{\infty }\binom{n+m-2}{m-1}%
q^{n-1}(1-q)^{m}z^{n}
\end{equation*}
whose coefficients are probabilities of Pascal distribution to be in the class $\mathcal{H}_{\mathcal{T}}(\beta _{1},\beta _{1},\ldots ,\beta
_{k};\alpha )$ of analytic functions with negative coefficients defined in the open unit disk. Further, we give necessary and sufficient condition for the integral operator $\mathcal{G}_{q}^{m}f(z)=\int_{0}^{z}\frac{\Phi
_{q}^{m}(t)}{t}dt$ \ to be in the class $\mathcal{G}_{\mathcal{T}}(\beta
_{1},\beta _{1},\ldots ,\beta _{k};\alpha )$.