Kodai Mathematical Journal

Study of some subclasses of univalent functions and their radius properties

S. Ponnusamy and S. K. Sahoo

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An analytic function f(z)=z+a2z2 + … in the unit disk Δ = {z: |z| < 1} is said to be in $\mathcal{U}(\lambda, \mu)$ if

$\left|f'(z)\left(\frac{z}{f(z)} \right)^ {\mu +1}-1 \right|\le \lambda \quad (|z|<1)$

for some λ ≥ 0 and μ > –1. For –1 ≤ α ≤ 1, we introduce a geometrically motivated $\mathcal{S}_p(\alpha)$-class defined by

${\mathcal S}_p(\alpha) = \left \{f\in {\mathcal S}:\, \left |\frac{zf'(z)}{f(z)} -1\right |\leq {\rm Re}\, \frac{zf'(z)}{f(z)}-\alpha, \quad z\in \Delta  \right \},$

where ${\mathcal S}$ represents the class of all normalized univalent functions in Δ. In this paper, the authors determine necessary and sufficient coefficient conditions for certain class of functions to be in $\mathcal{S}_p(\alpha)$. Also, radius properties are considered for $\mathcal{S}_p(\alpha)$-class in the class ${\mathcal S}$. In addition, we also find disks |z| < r:= r(λ,μ) for which $\frac{1}{r}f(rz)\in \mathcal{U(\lambda,\mu)}$ whenever $f\in \mathcal{S}$. In addition to a number of new results, we also present several new sufficient conditions for f to be in the class $\mathcal{U}(\lambda, \mu)$.

Article information

Kodai Math. J., Volume 29, Number 3 (2006), 391-405.

First available in Project Euclid: 2 November 2006

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Ponnusamy, S.; Sahoo, S. K. Study of some subclasses of univalent functions and their radius properties. Kodai Math. J. 29 (2006), no. 3, 391--405. doi:10.2996/kmj/1162478770. https://projecteuclid.org/euclid.kmj/1162478770

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