CONVEX ANALYSIS WITH APPLICATIONS IN DIFFERENTIATION OF CONVEX FUNCTIONS (Research Notes in Mathematics, 58)

Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

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J. R. Giles, Convex analysis with application in the differentiation of convex functions (Research Notes in Mathematics No. 58, Pitman, 1982), £9.95.

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028133 ◽

1983 ◽

Vol 26 (1) ◽

pp. 113-113

Author(s):

G. J. O. Jameson

Keyword(s):

Convex Functions ◽

Convex Analysis

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A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

Abstract and Applied Analysis ◽

10.1155/2012/836804 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15

Author(s):

Sangho Kum ◽

Yongdo Lim

Keyword(s):

Convex Functions ◽

Convex Analysis ◽

Geometric Mean ◽

Positive Semidefinite ◽

Dual Operator ◽

Positive Semidefinite Matrices ◽

Essential Properties ◽

Harmonic Means ◽

Further Development ◽

Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

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Algebraic hyper-structures associated to convex analysis and applications

Filomat ◽

10.2298/fil1201055k ◽

2012 ◽

Vol 26 (1) ◽

pp. 55-65 ◽

Cited By ~ 1

Author(s):

Delavar Khalafi ◽

Bijan Davvaz

Keyword(s):

Convex Programming ◽

Convex Functions ◽

Convex Analysis ◽

Linear Functions

In this paper, we generalize some concepts of convex analysis such as convex functions and linear functions on hyper-structures. Based on new definitions we obtain some important results in convex programming. A few suitable examples have been given for better understanding.

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