Volume 2,  Issue 2, 2001

Article 23

SOME DISTORTION INEQUALITIES ASSOCIATED WITH THE FRACTIONAL DERIVATIVES OF ANALYTIC AND UNIVALENT FUNCTIONS

H.M. SRIVASTAVA, YI LING AND GEJUN BAO

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

DEPARTMENT OF MATHEMATICS
UNIVERSITY OF TOLEDO
TOLEDO, OHIO 43606-3304, U.S.A.
E-Mail: lyi@math.utoledo.edu

DEPARTMENT OF MATHEMATICS
HARBIN INSTITUTE OF TECHNOLOGY
HARBIN 15001, 
PEOPLE'S REPUBLIC OF CHINA

Received 16 October, 2000; accepted 4 March, 2001.
Communicated by: P. Cerone

ABSTRACT.   For the classes S and K of (normalized) univalent and convex analytic functions, respectively, a number of authors conjectured interesting extensions of certain known distortion inequalities in terms of a fractional derivative operator. While examining and investigating the validity of these conjectures, many subsequent works considered various generalizations of the distortion inequalities relevant to each of these conjectures. The main object of this paper is to give a direct proof of one of the known facts that these conjectures are false. Several further distortion inequalities involving fractional derivatives are also presented.
Key words:
Distortion inequalities, analytic functions, fractional derivatives, univalent functions, convex functions, hypergeometric function.

2000 Mathematics Subject Classification:
30C45, 26A33, 33C05.

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