Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals

Liu, Wenjun; Wen, Wangshu; Park, Jaekeun

doi:10.22436/jnsa.009.03.05

Cited by 84 publications

(75 citation statements)

References 13 publications

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“…At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.Key words: Hermite-Hadamard type inequality, MT-convex function, Hölder's inequality, power mean inequality, fractional integral, m-invex, P -function.AMS Mathematics Subject Classification: 26A33, 26A51, 33B15, 26B25, 26D07, 26D10, 26D15. …”

supporting

confidence: 89%

“…At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.…”

supporting

confidence: 89%

“…Due to the wide application of fractional integrals, some authors extended to study fractional Hermite-Hadamard type inequalities for functions of different classes (see [13]) and the references cited therein. Now, let us recall some definitions of various convex functions.…”

Section: Introductionmentioning

confidence: 99%

“…For other recent results concerning HermiteHadamard type inequalities through various classes of convex functions, (see [9], [13], [6], [21], [5], [11], [10], [4], [17], [3]) and the references cited therein. In (see [19], [14]) and the references cited therein, Tunç and Yildirim defined the following so-called MT-convex function: Definition 1.…”

Section: Introductionmentioning

confidence: 99%

“…Then the following inequality holds: Definition 1. A function f : I ⊆ R −→ R is said to belong to the class of MT(I), if it is nonnegative and for all x, y ∈ I and t ∈ (0, 1) satisfies the following inequality:Fractional calculus (see [13]) and the references cited therein, was introduced at the end of the nineteenth century by Liouville and Riemann, the subject of which has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics.where. In the case of α = 1, the fractional integral reduces to the classical integral.…”

mentioning

confidence: 99%

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Hermite-Hadamard Type Inequalities for Mtm-Preinvex Functions

Kashuri

Liko

2017

Fasciculi Mathematici

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Abstract. In the present paper, the notion of M T m -preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving M T m -preinvex functions along with beta function are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for M T m -preinvex functions via classical integrals and Riemann-Liouville fractional integrals are established. At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.Key words: Hermite-Hadamard type inequality, MT-convex function, Hölder's inequality, power mean inequality, fractional integral, m-invex, P -function.AMS Mathematics Subject Classification: 26A33, 26A51, 33B15, 26B25, 26D07, 26D10, 26D15. Introduction and preliminariesThe following notations are used throughout this paper. We use I to denote an interval on the real line R = (−∞, +∞) and I • to denote the interior of I. For any subset K ⊆ R n , K • is used to denote the interior of K. R n is used to denote a generic n-dimensional vector space. The nonnegative real numbers are denoted byThe following inequality, named Hermite-Hadamard inequality, is one of the most famous inequalities in the literature for convex functions. Theorem 1. Let f : I ⊆ R −→ R be a convex function on an interval I of real numbers and a, b ∈ I with a < b. Then the following inequality holds: Definition 1. A function f : I ⊆ R −→ R is said to belong to the class of MT(I), if it is nonnegative and for all x, y ∈ I and t ∈ (0, 1) satisfies the following inequality:Fractional calculus (see [13]) and the references cited therein, was introduced at the end of the nineteenth century by Liouville and Riemann, the subject of which has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics.where. In the case of α = 1, the fractional integral reduces to the classical integral.Due to the wide application of fractional integrals, some authors extended to study fractional Hermite-Hadamard type inequalities for functions of different classes (see [13]) and the references cited therein. Now, let us recall some definitions of various convex functions.Definition 3 (see [7]). A nonnegative function f : I ⊆ R −→ R • is said to be P -function or P -convex, ifDefinition 4 (see [1]). A set K ⊆ R n is said to be invex with respect to the mapping η : K × K −→ R n , if x + tη(y, x) ∈ K for every x, y ∈ K and t ∈ [0, 1]. 79Notice that every convex set is invex with respect to the mapping η(y, x) = y − x, but the converse is not necessarily true. For more details please see (see [1], [20]) and the references therein.Definition 5 (see [16]). The function f defined on the invex set K ⊆ R n is said to be preinvex with respect η, if for every x, y ∈ K and t ∈ [0, 1], we haveThe concept of preinvexity is more general than convexity since every convex funct...

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supporting

confidence: 89%

“…At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.…”

supporting

confidence: 89%

Section: Introductionmentioning

confidence: 99%