New Integral Inequalities via Generalized Preinvex Functions

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.

Q-Analogue of Holder’s and Minkowski’s Integral Inequalities on Finite Intervals and Generalization

In recent years, the topic on Holder’s and Minkowski’s inequalities has been studied by several researchers and variety of new results has been developed on their variants, extensions and generalizations. In this paper we give the extension to the generalized q- Holder’s integral inequality and by using it we also establish the generalization on q- Minkowski’s integral inequality on the finite interval [a, b]

Some integral inequalities for approximately h—convex functions and their applications

In this paper, by applying the new and improved form of Hölder’s integral inequality called Hölder—Íşcan integral inequality three inequalities of Hermite—Hadamard and Hadamard integral type for (h, d)—convex functions have been established. Various special cases including classes for instance, h—convex, s—convex function of Breckner and Godunova—Levin—Dragomir and strong versions of the aforementioned types of convex functions have been identified. Some applications to error estimations of presented results have been analyzed. At the end, a briefly conclusion is given.

On Generalized Opial's Integral Inequalities in q-Calculus

In this paper, we establish results for q-analogues of generalized Opial integral inequalities and also present some extensions of the analogues. Using the concepts of q-differentiability and continuity of functions and the application of the Holder's integral inequality we establish the results.

A refinement of Holder's integral inequality

The purpose of this note is to show that there is monotonic continuous function $ p(t)$ such that$$ \int_a^b \left(\prod_{i=1}^n f_i(x)\right) dx\le p(t)\le \prod_{i=1}^n \left(\int_a^b f_i^{r_i}(x)dx \right)^{1\over r_i},$$where $ f_1$, $ f_2,\ldots,f_n$ are real positive continuous functions on $ [a,b]$ and $ r_1$, $ r_2,\ldots,r_n$ are real positive numbers with $ \sum_{i=1}^n {1\over r_i}=1$.