The object of the present paper is to derive some inequalities involving multivalent functions in the unit disk. One of our results is an improvement and a generalization of a result due to R. M. Robinson [4].

The purpose of the present paper is to determine lower bounds for $\Re \left\}\frac{{ℰ}_{k}f\left(z\right)}{{\left({ℰ}_{k}f\right)}_m\left(z\right)}\right\{$, $\Re \left\}\frac{{\left({ℰ}_{k}f\right)}_m\left(z\right)}{{ℰ}_{k}f\left(z\right)}\right\{,\Re \left\}\frac{{ℰ}_k^{\text{'}}f\left(z\right)}{{\left({ℰ}_{k}f\right)}_m^{\text{'}}\left(z\right)}\right\{$ and $\Re \left\}\frac{{\left({ℰ}_{k}f\right)}_m^{\text{'}}\left(z\right)}{{ℰ}_k^{\text{'}}f\left(z\right)}\right\{$, where ${ℰ}_{k}f$ is the generalized normalized error function of the form ${ℰ}_{k}f\left(z\right)=z+{\sum }_{n=2}^{\infty }\frac{{\left(-1\right)}^{n-1}}{(\left(n-1\right)k+1)\left(n-1\right)!}{z}^n$ and ${\left({ℰ}_{k}f\right)}_m$ its partial sum. Furthermore, we give lower bounds for $\Re \left\}\frac{𝕀\left[{ℰ}_{k}f\right]\left(z\right)}{{\left(𝕀\left[{ℰ}_{k}f\right]\right)}_m\left(z\right)}\right\{$ and $\Re \left\}\frac{{\left(𝕀\left[{ℰ}_{k}f\right]\right)}_m\left(z\right)}{𝕀\left[{ℰ}_{k}f\right]\left(z\right)}\right\{$, where $𝕀\left[{ℰ}_{k}f\right]$ is the Alexander transform of ${ℰ}_{k}f$. Several examples of the main results are also considered.

MSC 2010: 30C45, 30C50The object of this paper is to obtain sharp results involving coefficient bounds, growth and distortion properties for some classes of analytic and univalent functions with negative coefficients.