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)
α-logarithmically convex functions. (English) Zbl 0924.30012

Let A denote the class of normalised analytic functions f defined by f(z)=z+a 2 z 2 + for zD={z:|z|<1}.

For α0 the authors introduce the class M α of normalised analytic α-logarithmically convex functions defined in the open unit disc D by

Re1+zf '' (z) f ' (z) α zf ' (z) f(z) 1-α >0·

For fM α , a best possible subordination theorem is obtained which implies that M α forms a subset of the starlike functions S * . Some extreme coefficient problems are also solved.

This definition and some properties are inspired by the well known α-convex functions, introduced in 1969 by P. T. Mocanu. The principal proofs of this paper are based on the so called “admissible functions method” defined and developed by P. T. Mocanu and S. S. Miller, method which used differential subordinations.

30C45Special classes of univalent and multivalent functions