zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Starlike harmonic functions in parabolic region associated with a convolution structure. (English) Zbl 1238.30014

Denote by the family of functions

f=h+g ¯(1)

which are harmonic, univalent and orientation preserving in the open unit disc 𝕌={z:|z|<1}, and assume that f is normalized by f(0)=f ' (0)-1=0. Thus, for f=h+g ¯, the functions h and g analytic 𝕌 can be expressed in the following forms:

h(z)=z+ n=2 a n z n ,g(z)= n=1 b n z n (0b 1 <1),

and f is then given by

f(z)=z+ n=2 a n z n + n=1 b n z n ¯(0b 1 <1)·

For functions f given by (1) and 𝔽 given by

𝔽(z)=(z)+𝔾(z) ¯=z+ n=2 𝔸 n z n + n=1 𝔹 n z n ¯,

we recall that the Hadamard product (or convolution) of f and 𝔽 is defined to be

(f*F)(z)=z+ n=2 a n 𝔸 n z n + n=1 b n 𝔹 n z n ¯(z𝕌)·

For the purpose of this paper, the authors introduce a subclass of denoted by H (F;λ,γ) which involves convolution and consist of all functions of the form (1) satisfying the inequality:

Re (1+e iψ )z(f(z)*F(z)) ' (1-λ)z+λ(f(z)*F(z))-e iψ γ·

Also they denote 𝕋 (F;λ,γ)= H (F;λ,γ)𝕋 , where 𝕋 is the subfamily of consisting of harmonic functions f=h+g ¯ of the form

f(z)=z- n=2 a n z n + n=1 b n z n ¯(0b 1 <1)·

The authors obtain a sufficient coefficient condition for functions f given by (1) to be in the class (F;λ,γ). It is shown that this coefficient condition is necessary also for functions belonging to the class 𝕋 (F;λ,γ).

Further, distortion results and extreme points for functions in 𝕋 (F;λ,γ) are also obtained.

MSC:
30C45Special classes of univalent and multivalent functions
30C50Coefficient problems for univalent and multivalent functions