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Keywords:
partial sums; analytic functions; generalized error function
partial sums; analytic functions; generalized error function
Summary:
The purpose of the present paper is to determine lower bounds for $\mathfrak{R}\left\rbrace \frac{\mathcal{E}_{k}f(z)}{(\mathcal{E}_{k}f)_{m}(z)}\right\lbrace $, $\mathfrak{R}\left\rbrace \frac{(\mathcal{E}_{k}f)_{m}(z)}{\mathcal{E}_{k}f(z)}\right\lbrace , \mathfrak{R}\left\rbrace \frac{\mathcal{E}_{k}^{\prime }f(z)}{(\mathcal{E}_{k}f)_{m}^{\prime }(z)}\right\lbrace $ and $\mathfrak{R}\left\rbrace \frac{(\mathcal{E}_{k}f)_{m}^{\prime }(z)}{\mathcal{E}_{k}^{\prime }f(z)}\right\lbrace $, where $\mathcal{E}_{k}f$ is the generalized normalized error function of the form $\mathcal{E}_{k}f\left( z\right) =z+\sum _{n=2}^{\infty }\frac{\left( -1\right) ^{n-1}}{(\left( n-1\right) k+1)\left( n-1\right) !}z^{n}$ and $(\mathcal{E}_{k}f)_{m}$ its partial sum. Furthermore, we give lower bounds for $\mathfrak{R}\left\rbrace \frac{\mathbb{I}\left[ \mathcal{E}_{k}f\right] (z)}{(\mathbb{I}\left[ \mathcal{E}_{k}f\right] )_{m}(z)}\right\lbrace $ and $\mathfrak{R}\left\rbrace \frac{(\mathbb{I}\left[ \mathcal{E}_{k}f\right] )_{m}(z)}{\mathbb{I}\left[ \mathcal{E}_{k}f\right] (z)}\right\lbrace $, where $\mathbb{I}\left[ \mathcal{E}_{k}f\right] $ is the Alexander transform of $\mathcal{E}_{k}f$. Several examples of the main results are also considered.
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