Search Results (87)

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

This study demonstrates several novel dynamic inequalities of the Hardy and Littlewood types on time scales. As special cases, our studies include Hardy’s integral inequalities and Hardy and Littlewood’s discrete inequalities. The research findings are demonstrated using algebraic inequalities, Hölder’s inequality, and the chain rule on time scales. Full article

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces. Full article

This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition techniques. For a given domain space, we construct a suitable range space such that the operator remains bounded. Conversely, for a prescribed range space, we identify a corresponding domain space that guarantees boundedness. Illustrative examples are included to demonstrate the construction of such spaces. The main results hold when the essential infimum of the exponent function exceeds one, and we also establish weak-type estimates in the limiting case. Full article

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish several new biparametrized fractal integral inequalities for functions whose local fractional derivatives are of a generalized convex type. In addition to employing tools from local fractional calculus, our approach utilizes the Hölder inequality, the power mean inequality, and a refined version of the latter. Further results are also derived using the concept of generalized concavity. To support our theoretical findings, we provide a graphical example that illustrates the validity of the obtained results, along with some practical applications that demonstrate their effectiveness. Full article

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by $\alpha$, these operators offer a unique flexibility: setting $\alpha =1$ recovers the classical inequalities for harmonically convex functions, while setting $\alpha =0$ yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators. Full article

We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order $\eta \in (1,2)$. The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities. Full article

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results. Full article

In this paper, the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise is investigated with consideration of a new method for dealing with singular integrals. Firstly, the estimate on higher moments for the solution is given. Secondly, under some suitable assumptions, we prove the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise by using the Hölder inequality. Finally, a simulation example is given to verify the theoretical results. Full article

Let $p,q\ge 1$ be two real numbers such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, and let $a,b\in \mathbb{R}$ be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form $\left|f({\displaystyle \frac{a}{p}}+{\displaystyle \frac{b}{q}})\right|\le \left|f\left(a\right){|}^{{\displaystyle \frac{1}{p}}}\right|f\left(b\right){|}^{{\displaystyle \frac{1}{q}}},$ and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities. Full article

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales $\mathbb{T}$. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=\mathbb{R}$), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as $\mathbb{T}=q^{{\mathbb{N}}_0}$ for $q>1,$ which, to the best of the authors’ knowledge, is a largely novel conclusion. Full article

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., $s=1$, $\lambda =\alpha ,$ $\sigma \left(0\right)=1,$ and $w=0$. In conclusion, the methodology described in this article is expected to stimulate further research in this area. Full article

Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson’s type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson’s type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function. Full article