Some Generalized Convolution Properties Associated with Certain Subclasses of Analytic Functions

Volume 3, Issue 3, 2002

Article 42

SOME GENERALIZED CONVOLUTION PROPERTIES ASSOCIATED WITH CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS

DEPARTMENT OF MATHEMATICS,
KINKI UNIVERSITY
HIGASHI-OSAKA,
OSAKA 577-8502, JAPAN
E-Mail: owa@math.kindai.ac.jp

DEPARTMENT OF MATHEMATICS AND STATISTICS,
UNIVERSITY OF VICTORIA,
VICTORIA, BRITISH COLUMBIA V8W 3P4,
CANADA
E-Mail: harimsri@math.uvic.ca

Received 5 March, 2002; Accepted 6 April, 2002.
Communicated by: G. Milovanovic

ABSTRACT. For functions belonging to each of the subclasses $\mathcal{M}_{n}^{\ast }\left( \alpha \right)$ and $\mathcal{N}_{n}^{\ast }\left( \alpha \right)$ of normalized analytic functions in open unit disk $\mathbb{U},$ which are introduced and investigated in this paper, the authors derive several properties involving their generalized convolution by applying certain techniques based especially upon the Cauchy-Schwarz and Hölder inequalities. A number of interesting consequences of these generalized convolution properties are also considered.

Key words:
Analytic functions, Hadamard product (or convolution), Generalized convolution, Cauchy-Schwarz inequality, Hölder inequality, Inclusion theorems.

2000 Mathematics Subject Classification:
Primary 30C45; Secondary 26D15, 30A10.

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