Hadamard-type inequalities for generalized convex functions

Bessenyei, Mihály; Páles, Zsolt

doi:10.7153/mia-06-35

Cited by 50 publications

(51 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While the result is stated there for open intervals, it also holds for compact intervals. Note that what we call here a Haar pair (i.e., f 0 is strictly positive and f 1 /f 0 strictly increasing) is called in [5] a positive regular pair. …”

Section: Generalized Convexitymentioning

confidence: 99%

See 1 more Smart Citation

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

2009

View full text Add to dashboard Cite

Abstract. We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C[a, b] → U n with strictly increasing nodes, fixing f 0 , f 1 ∈ U n . If U n ⊂ U n+1 and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 , f 1 , ..., f n is a basis of U n such that the linear span of f 0 , .., f k is an extended Chebyshev space over [a, b] for each k = 0, ..., n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold

show abstract

Section: Generalized Convexitymentioning

confidence: 99%

“…Theorem 5, p. 388 of [5]). While the result is stated there for open intervals, it also holds for compact intervals.…”

Section: Generalized Convexitymentioning

confidence: 99%

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

2009

View full text Add to dashboard Cite

show abstract

“…Some refinements of the Hermite-Hadamard inequality for convex functions have been extensively investigated by a number of authors (e.g., [1], [2], [3], [4] and [5]). …”

Section: Theorem 16 ([9]mentioning

confidence: 99%

Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions

Chen¹,

Wu²

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

show abstract

“…Likewise, M. Bessenyei and Zs. Páles [2] proved a generalization of (1) for real-valued functions defined on an open interval I ⊆ R , which are convex with respect to a so-called positive regular pair over I . For an account on various results dealing with the Hermite-Hadamard inequality, the reader is referred to the monograph by S. S. Dragomir and C. E. M. Pearce [5].…”

Section: The Hermite-hadamard Inequalitymentioning

confidence: 99%

Characterizations of convex functions of a vector variable via Hermite-Hadamard's inequality

Триф¹

2008

Journal of Mathematical Inequalities

View full text Add to dashboard Cite

Abstract. The classical Hermite-Hadamard inequality characterizes the continuous convex functions of one real variable. The aim of the present paper is to give an analogous characterization for functions of a vector variable. The Hermite-Hadamard inequalityIn a letter sent on November 22, 1881, to the journal Mathesis (and published there two years later), Ch. Hermite [10] noted that every convex function f :The left-hand side inequality was rediscovered ten years later by J. Hadamard [7]. Nowadays, the double inequality (1) is called the Hermite-Hadamard inequality. The interested reader can find its complete story in the historical note by D. S. Mitrinović and I. B. Lacković [12]. The Hermite-Hadamard inequality has evoked the interest of many mathematicians. Especially in the last three decades it has been intensively investigated and generalized in several directions. For instance, a dual Hermite-Hadamard inequality was discussed by C. P. Niculescu [13], while a complete extension of (1) to the class of n -convex functions was obtained by M. Bessenyei and Zs. Páles [1]. Likewise, M. Bessenyei and Zs. Páles [2] proved a generalization of (1) for real-valued functions defined on an open interval I ⊆ R , which are convex with respect to a so-called positive regular pair over I . For an account on various results dealing with the Hermite-Hadamard inequality, the reader is referred to the monograph by S. S. Dragomir and C. E. M. Pearce [5].In what follows, we are concerned with Hermite-Hadamard-type inequalities for functions of a vector variable. As pointed out by C. P. Niculescu [15], the Choquet theory (see [17]) provides the framework for a natural extension of (1) to such functions. For the reader's convenience, we briefly present here this extension. Let E be a real locally

show abstract

Hadamard-type inequalities for generalized convex functions

Abstract: Abstract. In this paper we investigate (ω 1 , ω 2 ) -convex functions and obtain characterization theorems and Hadamard-type inequalities for them. (2000): 26A51, 26B25. Mathematics subject classification

Cited by 50 publications

References 8 publications

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions

Characterizations of convex functions of a vector variable via Hermite-Hadamard's inequality

Contact Info

Product

Resources

About