Search

In this paper, we prove that the self-improving property of the weighted Gehring class G λ p with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class A q . By using the rearrangement (nonincreasing rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property. Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose. [ABSTRACT FROM AUTHOR]

In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized K -fractional integral in the frame of a new parameter K > 0 . This paper offers some new important inequalities of Grüss type using the generalized K -fractional integral and associated integral inequalities. Our results with this new integral operator have the abilities to be implemented for the evaluation of many mathematical problems related to the real world applications. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

Fractional calculus operators play a very important role in generalizing concepts of calculus used in diverse fields of science. In this paper, we use Riemann–Liouville fractional integrals to establish generalized identities, which are further applied to obtain midpoint and trapezoidal inequalities for convex function with respect to a strictly monotone function. These inequalities reproduce midpoint and trapezoidal inequalities for convex, harmonic convex, p-convex, and geometrically convex functions. Also, some new inequalities can be generated via specific strictly monotone functions. [ABSTRACT FROM AUTHOR]

The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work. [ABSTRACT FROM AUTHOR]

Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new Hermite–Hadamard-type inequalities for r-convex and geometrically convex functions and, additionally, some new Hermite–Hadamard-type inequalities by using the Hölder–İşcan integral inequality and an improved power-mean inequality. [ABSTRACT FROM AUTHOR]

This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various types of q-Bessel functions and some power series of special type. It has been obtained for basic q-generating series, q-exponential and q-trigonometric functions as well. Various results and corollaries are provided as an application to this theory. [ABSTRACT FROM AUTHOR]

The aim of this work is to obtain quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. Additionally, by taking the limit q → 1 − , our results give classical results on the Hardy inequality. [ABSTRACT FROM AUTHOR]

Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional (p , q) -calculus on finite intervals, particularly the fractional (p , q) -integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional (p , q) -integral on finite intervals. Then, the obtained results are used to derive some fractional (p , q) -trapezoid and (p , q) -midpoint type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new Opial-type dynamic inequalities on time scales. Our results are obtained in frame of convexity property and by using the chain rule and Jensen and Hölder inequalities. For illustration purpose, we obtain some particular Opial-type inequalities reported in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels ( a − 1 A B R ∇ δ , γ y) (η) of order 0 < δ < 0.5 , β = 1 , 0 < γ ≤ 1 starting at a − 1 . If ( a − 1 A B R ∇ δ , γ y) (η) ≥ 0 , then we deduce that y (η) is δ 2 γ -increasing. That is, y (η + 1) ≥ δ 2 γ y (η) for each η ∈ N a : = { a , a + 1 , ... } . Conversely, if y (η) is increasing with y (a) ≥ 0 , then we deduce that ( a − 1 A B R ∇ δ , γ y) (η) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we establish some new results on trapezium type inequalities of coordinated distance-disturbed (ℓ 1 , h 1) – (ℓ 2 , h 2) -convex functions of higher orders (σ 1 , σ 2) by using the Katugampola (k 1 , k 2) -fractional integrals. As special cases of our general results, we recapture some earlier proved results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hermite–Hadamard–Fejér type inclusions for the product of two co-ordinated convex interval-valued functions. These inclusions are generalizations of some results given in earlier works. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is establishing new Hermite–Hadamard-type inequalities for interval-valued coordinated functions via Riemann–Liouville-type fractional integrals. Moreover, we obtain some fractional Hermite–Hadamard-type inequalities for the product of two coordinated h-convex interval-valued functions. Our results generalize several well-known inequalities. [ABSTRACT FROM AUTHOR]

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev's functionals by utilizing a fractional generalized weighted fractional integral involving another function G in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev's functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev's functionals with certain choices of ω (θ) and G (θ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev's functionals with certain choices of ω (θ) and G (θ) . [ABSTRACT FROM AUTHOR]

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of map T and interval-valued T -convex, and give some basic properties. Further, we extend fractional Hermite–Hadamard inequalities in the case of T -convex and Ostrowski type inequalities for interval-valued functions. Several examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we present a new bound for the Jensen gap with the help of a Green function. Using the bound, we deduce a converse of the Hölder inequality as well. Finally, we present some applications of the main result in information theory. [ABSTRACT FROM AUTHOR]

New nonlinear integral inequalities (NII) are presented in this paper. Based on mathematical analysis technique, several estimation results are obtained, which not only complement the aforementioned results, but also generalize the inequalities to the more general nonlinearities. As an application, they can be employed to estimate the bound on the solutions of power integro-differential equations (IDE). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized k-fractional integral in terms of a new parameter k > 0 , present some new important inequalities of Pólya–Szegö and Čebyšev types by use of the generalized k-fractional integral. Our consequences with this new integral operator have the abilities to implement the evaluation of many mathematical problems related to real world applications. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences. [ABSTRACT FROM AUTHOR]

In this paper, we establish reverse Hölder's inequality on time scales via diamond integral, which is defined as an 'approximate' symmetric integral on time scales. Moreover, we give some generalizations of diamond integral Hölder's inequality which is due to Brito da Cruz et al. Several other related inequalities are also presented. [ABSTRACT FROM AUTHOR]