zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Characterization properties for starlikeness and convexity of some subclasses of analytic functions involving a class of fractional derivative operators. (English) Zbl 0952.30011

The paper is devoted to the properties of starlike and convex functions of order ρ defined on the unit disk U={z:z<1} [see P. L. Duren, Univalent functions (1983; Zbl 0514.30001)]. Some characterization properties are obtained and applied to fractional derivative operators on the mentioned function classes:

J 0,z λ,μ,η fz=d m dz m z λ-μ Γm-λ 0 z z-t 2 m-λ-1 F 1 μ-λ,m-η;m-λ;1-t zftdt,

where m-1λ<m, m, μ,η, 2 F 1 is the Gaussian hypergeometric function. The operator J 0,z λ,μ,η is a generalization of the Riemann-Liouville operator and is closely connected to the Erdélyi-Kober operator of the fractional calculus.

The characterization properties are formulated by using some conditions (inequalities) on Taylor’s coefficients of the function f. Some exactness of these conditions is illustrated by examples of functions for which the corresponding inequalities are attained.

Let f i (i=1,2) be given by

f i z=z+ n=2 a n,i z n ·

The Hadamard product of these functions is defined by

f 1 *f 2 z=z+ n=2 a n,1 a n,2 z n ·

The authors obtain several characterization properties of the Hadamard product on the classes of starlike and convex functions. The starlikeness and convexity properties of the following fractional derivative operator

P 0,z λ,μ,η fz=Γ2-μΓ2-λ+η Γ2-μ+ηz μ J 0,z λ,μ,η fz,

with λ0, μ<2, η>maxλ,μ-2 are investigated too. These results can be applied to obtaining corresponding properties for Riemann-Liouville and Erdélyi-Kober operators.

30C45Special classes of univalent and multivalent functions
26A33Fractional derivatives and integrals (real functions)