Partial sums of the normalized Dini functions

TL;DR

This paper investigates bounds for ratios of normalized Dini functions and their partial sums, providing geometric insights into their image domains and extending understanding of their analytic properties.

Contribution

It derives lower bounds for ratios involving normalized Dini functions and their partial sums, and explores the geometric image domains of these functions.

Findings

Lower bounds for ratios of Dini functions and partial sums.

Geometric descriptions of the image domains of these functions.

Analytic properties of normalized Dini functions and their partial sums.

Abstract

Let $\left( w_{\alpha ,v}\right) _{m}(z)=z+\sum\limits_{n=1}^{m}a_{n}z^{n+1} $ be the sequence of partial sums of normalized Dini functions $w_{\alpha ,v}(z)=z+\sum\limits_{n=1}^{\infty }a_{n}z^{n+1}$ where $a_{n}=\frac{\left( -1\right) ^{n}\left( 2n+\alpha \right) }{\alpha 4^{n}n!\left( v+1\right) _{n}% }$. The aim of the present paper is to obtain lower bounds for $\mathcal{R} \left\{ \frac{w_{\alpha ,v}(z)}{\left( w_{\alpha ,v}\right) _{m}(z)}\right\} ,$ $\mathcal{R}\left\{ \frac{\left( w_{\alpha ,v}\right) _{m}(z)}{w_{\alpha ,v}(z)}\right\} ,$ $\mathcal{R}\left\{ \frac{w_{\alpha ,v}^{^{\prime }}(z)}{ \left( w_{\alpha ,v}\right) _{m}^{^{\prime }}(z)}\right\}$ and $\mathcal{R} \left\{ \frac{\left( w_{\alpha ,v}\right) _{m}^{^{\prime }}(z)}{w_{\alpha ,v}^{^{\prime }}(z)}\right\}$ . Also we give a few geometric description regarding image domains of some functions.