Partial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta Function

Abstract

Let $f_n\left(z\right)=z+{{\displaystyle \sum }}_{k=2}^n{a}_k{z}^k$ be the sequence of partial sums of the analytic function $f\left(z\right)=z+{{\displaystyle \sum }}_{k=2}^{\infty }{a}_k{z}^k$. In this paper, we determine sharp lower bounds for $\Re \{f\left(z\right)/f_n\left(z\right)\},\Re \{f_n\left(z\right)/f\left(z\right)\},\Re \{f^{\prime }\left(z\right)/f_n^{\prime }\left(z\right)\}$, and $\Re \{f_n^{\prime }\left(z\right)/f^{\prime }\left(z\right)\}$. The usefulness of the main result not only provides the unification of the results discussed in the literature but also generates certain new results.