Partial sums of generalized Rabotnov function

TL;DR

This paper investigates lower bounds for the real parts of ratios involving partial sums of the generalized Rabotnov function and its Alexander transform, contributing new bounds and examples for these complex functions.

Contribution

It provides new lower bounds for ratios of partial sums and their derivatives of the generalized Rabotnov function and its Alexander transform.

Findings

Established lower bounds for ratios of partial sums and their derivatives.

Derived bounds for ratios involving the Alexander transform of the Rabotnov function.

Included several illustrative examples of the main results.

Abstract

Let $(\mathbb{R}_{\alpha ,\beta ,\gamma }(z))_{m}(z)=z+\sum_{n=1}^{m}A_{n}z^{n+1}$ be the sequence of partial sums of the normalized Rabotnov functions $\mathbb{R}_{\alpha ,\beta ,\gamma }(z)=z+\sum_{n=1}^{\infty }A_{n}z^{n+1}$ where $A_{n}=\frac{\beta ^{n}\Gamma \left( \gamma +\alpha \right) }{\Gamma \left( \left( \gamma +\alpha \right) (n+1)\right) }.$ The purpose of the present paper is to determine lower bounds for $\mathfrak{R}\left \{ \frac{\mathbb{R}_{\alpha ,\beta ,\gamma }(z)% }{(\mathbb{R}_{\alpha ,\beta ,\gamma })_{m}(z)}\right \} ,\mathfrak{R}% \left \{ \frac{(\mathbb{R}_{\alpha ,\beta ,\gamma })_{m}(z)}{\mathbb{R}% _{\alpha ,\beta ,\gamma }(z)}\right \} ,$ $\mathfrak{R}\left \{ \frac{\mathbb{R}_{\alpha ,\beta ,\gamma }(z)}{(\mathbb{% R}_{\alpha ,\beta ,\gamma })_{m}^{\prime }(z)}\right \} ,\mathfrak{R}% \left \{ \frac{(\mathbb{R}_{\alpha ,\beta ,\gamma })_{m}^{\prime }(z)}{% \mathbb{R}_{\alpha ,\beta ,\gamma }(z)}\right \} .$ Furthermore, we give lower bounds for $\mathfrak{R}\left \{ \frac{\mathbb{I}\left[ \mathbb{R}% _{\alpha ,\beta ,\gamma }\right] (z)}{(\mathbb{I}\left[ \mathbb{R}_{\alpha ,\beta ,\gamma }\right] )_{m}(z)}\right \} $ and $\mathfrak{R}\left \{ \frac{% (\mathbb{I}\left[ \mathbb{R}_{\alpha ,\beta ,\gamma }\right] )_{m}(z)}{% \mathbb{I}\left[ \mathbb{R}_{\alpha ,\beta ,\gamma }\right] (z)}\right \} $ where $\mathbb{I}\left[ \mathbb{R}_{\alpha ,\beta ,\gamma }\right] $ is the Alexander transform of $\mathbb{R}_{\alpha ,\beta ,\gamma }$. Several examples of the main results are also considered.