On Invex Sets and Preinvex Functions

doi:10.1006/JMAA.1995.1057

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On Invex Sets and Preinvex Functions

Journal Article•DOI•

On Invex Sets and Preinvex Functions

01 Feb 1995-Journal of Mathematical Analysis and Applications (Academic Press)-Vol. 189, Iss: 3, pp 901-908

TL;DR: In this paper, the authors consider the class of the invex functions introduced by Hanson and show that under certain conditions, a function defined on a set of non-invex sets A is pre-quasi-inverse on A.

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About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 1995-02-01 and is currently open access. It has received 365 citations till now. The article focuses on the topics: Invex function.

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Journal Article•DOI•

Generalized invexity and generalized invariant monotonicity

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Xinmin Yang¹, Xiaoqi Yang², Kok Lay Teo²•Institutions (2)

Chongqing Normal University¹, Hong Kong Polytechnic University²

01 Jun 2003-Journal of Optimization Theory and Applications

TL;DR: In this article, several kinds of invariant monotone maps and generalized invariant maps are introduced, and relations between generalized monotonicity and generalized invexity are established, which are generalizations of those presented by Karamardian and Schaible.

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Abstract: In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.

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128 citations

Journal Article•DOI•

On Properties of Preinvex Functions

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Xinmin Yang¹, Duan Li²•Institutions (2)

Chongqing Normal University¹, The Chinese University of Hong Kong²

01 Apr 2001-Journal of Mathematical Analysis and Applications

TL;DR: In this article, a characterization of a preinvex function in terms of its relationship with an intermediate-point pre-vexity and prequasi-inveXity is provided.

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124 citations

Journal Article•DOI•

Some quantum integral inequalities via preinvex functions

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Muhammad Aslam Noor¹, Khalida Inayat Noor¹, Muhammad Uzair Awan¹•Institutions (1)

COMSATS Institute of Information Technology¹

15 Oct 2015-Applied Mathematics and Computation

TL;DR: Some new quantum analogues of Hermite-Hadamard and Iyengar type inequalities for some classes of preinvex functions are obtained.

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109 citations

Cites background or result from "On Invex Sets and Preinvex Function..."

...Thus, it is true that every convex set is also an invex set with respect to η(v,u) = v− u, but the converse is not necessarily true, see [15,37] and the references therein....
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...To prove some of our results, we need the well-known Condition C introduced by Mohan and Neogy [15]....
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...For more details, see [1–3,15,17–22,25,29–32,36,37]....
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...9 ([15])....
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Journal Article•DOI•

Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex

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A. Barani¹, A. G. Ghazanfari¹, Sever S Dragomir², Sever S Dragomir³•Institutions (3)

Lorestan University¹, Victoria University, Australia², University of the Witwatersrand³

29 Oct 2012-Journal of Inequalities and Applications

TL;DR: In this article, the Hermite-Hadamard-type inequality for preinvex functions is extended to functions of several variables on invex subsets of a function.

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Abstract: In this article, we extend some estimates of the right-hand side of a Hermite-Hadamard-type inequality for preinvex functions. Then, a generalization to functions of several variables on invex subsets of is introduced.

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93 citations

Cites background from "On Invex Sets and Preinvex Function..."

...For properties and applications of preinvex functions see [16, 13] and references therein....
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Journal Article•DOI•

Invex equilibrium problems

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Muhammad Aslam Noor

15 Feb 2005-Journal of Mathematical Analysis and Applications

TL;DR: In this paper, a new class of equilibrium problems known as invex equilibrium problems was introduced, which includes equilibrium problems, variational inequalities and variational-like inequalities as special cases.

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91 citations

Cites background or methods from "On Invex Sets and Preinvex Function..."

...Mohan and Neogy [24] have shown that a differentable function which is invex on an invex set K, is also a preinvex function provided the following condition holds....
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...For the recent applications, numericals methods and formulations of variational inequalities, variational-like inequalities and equilibrium problems, see [1–41] and the references therein....
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