Special Classes of Univalent Functions with Missing Coefficients and Integral Transforms

Ponnusamy, S., and Sahoo, P., (2005) Special Classes of Univalent Functions with Missing Coefficients and Integral Transforms. Bulletin of the Malaysian Mathematical Sciences Society, 28 (2). pp. 141-156. ISSN 0126-6705

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Affiliations

Indian Institute of Technology, Dept. of Mathematics
Banaras Hindu University, Mahila Maha Vidyalaya (WMV), Dept. of Mathematics

Abstract

Let $\mathcal{A}_n$ be the class of all analytic functions $f$ of the form

$f(z)=z + \sum_{k=n+1}^{\infty} a_{k}z^{k}, z \in \Delta$ ,

where $n \in \mathbf{N}$ is fixed. For $\lambda > 0$ and $\alpha < 1$ , define

$\mathcal{U}_{n}(\lambda) = \{ f \in \mathcal{A}_{n} :|(\frac{z}{f(z)})^{n+1} f'(z)-1|<\lambda, z \in \Delta \}$

and

$\mathcal{S}_{\alpha}^{\ast} = \{ f \in \mathcal{S}^{\ast}(\alpha) : |\frac{zf'(z)}{f(z)} - 1| < 1 - \alpha, z \in \Delta \}$ .

In this paper, we find suitable conditions on $\lambda$ and $\alpha$ so that $\mathcal{U}_{n}(\lambda)$ is included in $\mathcal{S}_\alpha$ and $\mathcal{S}^\ast (\alpha)$ . Here $\mathcal{S}_\alpha$ and $\mathcal{S}^\ast (\alpha)$ denote the usual classes of strongly starlike and starlike of order $\alpha$ , respectively. We determine necessary conditions so that $f\in \mathcal{U}_n (\lambda)$ implies that

$|\frac{zf'(z)}{f(z)} - \frac{1}{2\beta}| < \frac{1}{2\beta}, z \in \Delta$

or

$|1 + \frac{zf''(z)}{f'(z)} - \frac{1}{2\beta}| < \frac{1}{2\beta}, |z| < r$ ,

where $r = r(\lambda, n)$ will be specified. For $c+1-n>0$ , define

$[I(f)](z) = F(z) = z[\frac{c+1-n}{z^{c+1-n}} \int_0^z (\frac{t}{f(t)})^n { } t^{c-n} dt]^{\frac{1}{n}}$ .

We also find conditions on $\lambda, \alpha$ and $c$ so that $I(\mathcal{U}_n (\lambda)) \subset \mathcal{S}_{\alpha}^{\ast}$ .

Item Type:

Journal

Keywords:

Univalent, starlike and convex functions, subordination, and integral transform.

Subjects:

Q Science, Computer Science

ID Code:

1406