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)
Invexity and generalized convexity. (English) Zbl 0731.26009
A differentiable function $f:{\bbfR}\sp n\to {\bbfR}$ is said to be invex if there exists a function $\eta (y,x)\in {\bbfR}\sp n$ such that, for all $y,x\in {\bbfR}\sp n$ $$ f(y)-f(x)\ge \eta (y,x)\sp t\nabla f(x). $$ Various extensions of such functions including pseudo- and quasi-invex have been defined and their relationship to each other and other generalizations of convexity have been studied. For the non- differentiable case, f is said to be pre-invex if $$ f(x+t\eta (y,x))\le tf(y)+(1-t)f(x),\quad 0\le t\le 1. $$ This comprehensive paper brings together many of these scattered results which are then studied, compared, and extended. Some definitions that are introduced include $\eta$-invex subsets, pseudo and quasi-pre-invex functions. One very minor correction. Reference 2 should be to the Journal, not the Bulletin, of the Australian Mathematical Society.

26B25Convexity and generalizations (several real variables)
90C30Nonlinear programming
Full Text: DOI