## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Geometric properties of certain linear integral transforms

#### Abstract

For $\lambda>0$ and $0<\mu<n$, let ${\mathcal U}_n(\lambda,\mu)$ denote the class of all normalized analytic functions $f$ in the unit disc $\Delta$ of the form $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ such that $$\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right|<\lambda, ~ z\in \Delta,$$ where $n\in \mathbb N$ is fixed. In addition to the discussion of the basic properties of the class ${\mathcal U}_n(\lambda,\mu)$, we find conditions so that ${\mathcal U}_n(\lambda,\mu)$ is included in ${\mathcal S}_{\gamma}$, the class of all strongly starlike functions of order $\gamma$ $(0<\gamma\leq 1)$. We also find necessary conditions so that $f\in {\mathcal U}_n(\lambda,\mu)$ implies that $$\left|\frac{zf'(z)}{f(z)}-\frac{1}{2\beta}\right|<\frac{1}{2\beta},\quad \mbox{for all z\in \Delta,}$$ or $$\left|1+\frac{zf''(z)}{f'(z)}-\frac{1}{2\beta}\right| <\frac{1}{2\beta},\quad \mbox{for |z|<r<1},$$ where $r=r(\lambda,\mu,n)$ will be specified.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 1 (2005), 95-108.

Dates
First available in Project Euclid: 12 April 2005

https://projecteuclid.org/euclid.bbms/1113318133

Digital Object Identifier
doi:10.36045/bbms/1113318133

Mathematical Reviews number (MathSciNet)
MR2134860

Zentralblatt MATH identifier
1080.30016

#### Citation

Ponnusamy, S.; Sahoo, P. Geometric properties of certain linear integral transforms. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 1, 95--108. doi:10.36045/bbms/1113318133. https://projecteuclid.org/euclid.bbms/1113318133