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)
Harmonic univalent functions with negative coefficients. (English) Zbl 0908.30013

Denote by S H the class of functions f of the form: (1) f=h+g ¯ that are harmonic univalent and sense-preserving in the unit disc Δ={z:|z|<1} for which f(0)=f z (0)-1=0 and by S H 0 the subclass of S H for which f z ¯ (0)=0.

Let: (2) h(z)=z+ n=2 a n z n , g(z)= n=2 b n z n , zΔ. Denote by S H *0 and K H 0 the subclasses of S H 0 consisting of functions f that map Δ onto starlike and convex domains, respectively. Let T H *0 and TK H 0 be the subclasses of S H *0 and K H 0 , respectively, whose coefficients f=h+g ¯ take the form: (3) h(z)=z- n=2 a n z n , a n 0; g(z)=- n=2 b n z n , b n 0, zΔ.

In the present paper mentioned above classes of harmonic functions are considered. The author proves among others: Theorem 1. If f of the form (1-2) satisfies n=2 n(|a n |+|b n |)1, then fS H *0 . Corollary 1. If f of the form (1-2) satisfies n=2 n 2 (|a n |+|b n |)1, then fK H 0 . Theorem 2. For f of the form (1), (3), fT H *0 if and only if n=2 n(a n +b n )1. Theorem 3. For f of the form (1), (3), fTK H 0 if and only if n=2 n 2 (a n +b n )1.

30C45Special classes of univalent and multivalent functions
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)