Partial sums of generalized Rabotnov function

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 $\mathfrak{R}\left \{ \frac{\mathbb{R}_{α,β,γ}(z)% }{(\mathbb{R}_{α,β,γ})_{m}(z)}\right \} ,\mathfrak{R}% \left \{ \frac{(\mathbb{R}_{α,β,γ})_{m}(z)}{\mathbb{R}% _{α,β,γ}(z)}\right \} ,$ $\mathfrak{R}\left \{ \frac{\mathbb{R}_{α,β,γ}(z)}{(\mathbb{% R}_{α,β,γ})_{m}^{\prime }(z)}\right \} ,\mathfrak{R}% \left \{ \frac{(\mathbb{R}_{α,β,γ})_{m}^{\prime }(z)}{% \mathbb{R}_{α,β,γ}(z)}\right \} .$ Furthermore, we give lower bounds for $\mathfrak{R}\left \{ \frac{\mathbb{I}\left[ \mathbb{R}% _{α,β,γ}\right] (z)}{(\mathbb{I}\left[ \mathbb{R}_{α,β,γ}\right] )_{m}(z)}\right \} $ and $\mathfrak{R}\left \{ \frac{% (\mathbb{I}\left[ \mathbb{R}_{α,β,γ}\right] )_{m}(z)}{% \mathbb{I}\left[ \mathbb{R}_{α,β,γ}\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.