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)
On two extremal problems related to univalent functions. (English) Zbl 0818.30013

Let Λ:[0,1] be integrable over [0,1] and positive in (0,1) and S the class of functions univalent in the unit disk D and normalized as usual. The authors consider for fS,

L Λ (f)=inf 0 1 Λ(t)(Re (f(tz)/tz-1/(1+t) 2 ) d t z D

and L Λ (S)=inf{L Λ (f)fS} resp. L Λ (C)=inf{L Λ (f)fC}, where CS denotes the subclass of closed-to-convex functions.

They ask whether there are functions Λ such that L Λ (S)=0 and show that for

Λ(t)/(1-t 2 ) decreasing on (0,1), L Λ (C)=0. Furthermore they consider the class P β of functions f holomorphic in D normalized in the origin as usual for which f ' (D)-β lies in a halfplane bounded by a straight line through the origin and functions

λ:[0,1], 0 1 λ(t)dt=1,λ0·

They determine numbers β=β(λ) such that the conclusion

fP β V λ (f)(z)= 0 1 λ(t)f(tz)/tdtS

holds and for some special λ they find β=β(λ) for which V λ (P β )S * , where S * is the class of starlike functions. For λ(t)=(c+1)t c , c>-1, this solves a problem discussed before by many authors.


MSC:
30C55General theory of univalent and multivalent functions