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.
Categories: math.CV