UNIVALENCE CONDITIONS FOR A NEW INTEGRAL OPERATOR

Sh. Najafzadeh, A. Ebadian and H. Rahmatan

  DOI:  https://doi.org/10.37418/jcsam.1.2.1

Full Text

In the present paper, we will obtain norm estimates of the pre-Schwarzian derivatives for
$F_{\lambda,\mu}(z)$, such that
\[ F_{\lambda,\mu}(z) = \int_0^z \prod_{i=1}^{n} (f’_i(t))^{\lambda_i}\left( \frac{f_i(t)}{t} \right)^{\mu_i}dt \quad (z\in D),\]
where $\lambda_i,\mu_i\in \mathbb{R}$, $\lambda_i=(\lambda_1,\lambda_2,\ldots,\lambda_n$, $\mu_i=(\mu_1,\mu_2,\ldots,\mu_n$
and $f_i$ belongs to the class of convex univalent functions $\mathcal{C}\subset \mathcal{S}$.

Keywords: Integral Operator, Univalency, pre-Schwarzian derivatives.