Partial sums of the normalized Dini functions

Abstract

Let ${\left(w_{\alpha ,v}\right)}_m(z)=z+\sum _{n=1}^m{a}_n{z}^{n+1}$ be the sequence of partial sums of normalized Dini functions $w_{\alpha ,v}(z)=z+\sum _{n=1}^{\infty }{a}_n{z}^{n+1}$ where $a_{n}=\frac{\left( -1\right) ^{n}\left( 2n+α\right) }{α4^{n}n!\left( v+1\right) _{n}% }$. 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.