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.

Two well known definitions of the flow of a plane vector field around the boundary of a region $\Omega$ are compared. The definition (appropriately arranged) based on the constantness of the stream function on every profile is not only invariant under conformal mappings but more general than the definition based on the vanishing of the normal component of the field on $\partial \Omega$.

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.