Volume 3,  Issue 3, 2002

Article 42

SOME GENERALIZED CONVOLUTION PROPERTIES ASSOCIATED WITH CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS

SHIGEYOSHI OWA AND H.M. SRIVASTAVA

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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