Necessary and sufficient conditions for two subclasses of analytic functions associated with Pascal distribution series - Frasin, Basem Aref (Sarajevo Journal of Mathematics, 2022)
In the present paper, we determine necessary and sufficient condition for \begin{equation*} Φ _{q}^{m}(z):=z-\sum \limits_{n=2}^{∞ }\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 $\mathcal{H}_{\mathcal{T}}(β _{1},β _{1},… ,β _{k};α )$ of analytic functions with negative coefficients defined in the open unit disk. Further, we give necessary and sufficient condition for the integral operator $\mathcal{G}_{q}^{m}f(z)=\int_{0}^{z}\frac{Φ _{q}^{m}(t)}{t}dt$ \ to be in the class $\mathcal{G}_{\mathcal{T}}(β _{1},β _{1},… ,β _{k};α )$.