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Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Khan Muhammad Bilal, Noor Muhammad Aslam, Macías-Díaz Jorge E., Soliman Mohamed S., Zaini Hatim Ghazi

Demonstratio Mathematica

2022

Vol. 55, nr 1

387--403

It is a well-known fact that inclusion and pseudo-order relations are two different concepts which are defined on the interval spaces, and we can define different types of convexities with the help of both relations. By means of pseudo-order relation, the present article deals with the new notions of convex functions which are known as left and right log- s -convex interval-valued functions (IVFs) in the second sense. The main motivation of this study is to present new inequalities for left and right log- s -convex-IVFs. Therefore, we establish some new Jensen-type, Hermite-Hadamard (HH)-type, and Hermite-Hadamard-Fejér (HH-Fejér)-type inequalities for this kind of IVF, which generalize some known results. To strengthen our main results, we provide nontrivial examples of left and right log- s -convex IVFs.

On some k-fractional integral inequalities of Hermite-Hadamard type for twice differentiable generalized beta (r, g)-preinvex functions

Kashuri Artion, Liko Rozana

Journal of Applied Analysis

2019

Vol. 25, nr 1

59--72

In the present paper, a new class of generalized beta (r, g)-preinvex functions is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized beta (r, g)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized beta (r, g)-preinvex functions that are twice differentiable via k-fractional integrals are established. These general inequalities give us some new estimates for Hermite-Hadamard type k-fractional integral inequalities and also extend some results appeared in the literature; see [A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s, m, φ)-preinvex functions, Aust. J. Math. Anal. Appl. 13 (2016), no. 1, Article ID 16]. At the end, some applications to special means are given.

Some new Hermite-Hadamard type inequalities via k-fractional integrals concerning differentiable generalized relative semi-(r; m, p, q, h1 , h2 ) -preinvex mappings

Kashuri A., Liko R.

Fasciculi Mathematici

2018

Nr 60

59--78

In this article, we first presented a new identity concerning differentiable mappings defined on m-invex set via k-fractional integrals. By using the notion of generalized relative semi-(r; m,p, q, h1, h2 )-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard type inequalities via k-fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.