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April 2005 Geometric properties of certain linear integral transforms
S. Ponnusamy, P. Sahoo
Bull. Belg. Math. Soc. Simon Stevin 12(1): 95-108 (April 2005). DOI: 10.36045/bbms/1113318133


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.


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S. Ponnusamy. P. Sahoo. "Geometric properties of certain linear integral transforms." Bull. Belg. Math. Soc. Simon Stevin 12 (1) 95 - 108, April 2005.


Published: April 2005
First available in Project Euclid: 12 April 2005

zbMATH: 1080.30016
MathSciNet: MR2134860
Digital Object Identifier: 10.36045/bbms/1113318133

Primary: 30C45 , 30C55

Keywords: integral transform , starlike and convex functions , Subordination , Univalent

Rights: Copyright © 2005 The Belgian Mathematical Society


Vol.12 • No. 1 • April 2005
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