Search Results (316)

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed $(v,w)$–s–convex response-curve certification. The response curve is treated as an element of the Banach space of continuous functions under the supremum norm, while derivative-based certificates are handled in a Lipschitz and Sobolev-type norm when required. Generalized convexity is represented through a bounded structural operator, whose order condition defines a closed convex acceptance set. The violation score is measured by the positive part of the operator residual, and the Hermite–Hadamard, Fejér, and Ostrowski quantities are interpreted as bounded certificate functionals. The auxiliary profiles are constructed from validation-curve residuals through a split-calibrated procedure and then tested on held-out triples. The framework certifies only scalar response-curve summaries under explicit structural and empirical assumptions; it does not certify a full learning system, guarantee generalization, or replace dense sampling when the structural gate fails. Full article

In this study, we obtain new Hermite–Hadamard type inequalities involving an extended form of the Atangana–Baleanu fractional integral operator having Mittag-Leffler kernels. The approach is based on a suitable integral identity for differentiable functions together with the convexity of the absolute value of the first derivative. Within this framework, we extend the classical Hermite–Hadamard inequality to a fractional setting governed by the parameters $\alpha \in (0,1)$, $\beta \in (0,1]$, and $\lambda >0$. Full article

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly $\delta$-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly $\delta$-geometric convex functions. Some new examples and graphical representations are provided in order to illustrate the validity of our results. Full article

In this paper, we introduce, for the first time, a novel class of (center-radius order $(P,\mathrm{m})$-superquadratic interval-valued functions) $\mathfrak{cr}$-$(P,\mathrm{m})$-superquadratic $\mathcal{IVF}$s, and systematically investigate their fundamental structural properties. Building upon these properties, we establish new Jensen and Hermite–Hadamard ($\mathcal{HH}$) type inequalities, together with their fractional extensions formulated via Riemann–Liouville ($\mathcal{RL}$) fractional integral operators within the setting of interval calculus. The validity and sharpness of the derived results are illustrated through numerical examples and graphical representations. Moreover, the theoretical developments are further enriched by applications in information theory, leading to meaningful generalizations and notable improvements over several existing results reported in the literature. Full article

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type ($H$-type) inclusions are established for fuzzy-number-valued convex functions ($F$·$N\cdot V$-functions) and $F$·$N\cdot V$-coordinated convex functions. These results are obtained by employing $F$·$N\cdot V$-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known $H\cdot H$-type inequalities previously established for real-valued and interval-valued functions ($I\cdot V$-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted $H\cdot H$-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity. Full article

We introduce a new class of fractional remainder operators, denoted by $R_{\alpha ,n}^{\xi +}$ and $R_{\alpha ,n}^{\eta -}$, which generalize and unify various classical integral identities. Using these operators, we formulate refined Hermite–Hadamard and trapezoidal inequalities for differentiable functions. The novelty of our approach lies in its symmetric kernel structure, which facilitates tighter error bounds. Numerical examples are given, including applications to non-differentiable convex functions, to demonstrate the applicability and limitations of the derived results. Full article

This paper develops a unified fractional version of the Hermite–Hadamard inequality and Bullen-type inequalities for convex functions defined on discrete time scales. By employing generalized fractional difference operators, the obtained result encompasses and extends previously known discrete formulations, including both the classical case and higher-order variants. Furthermore, we investigate the approximation accuracy of the introduced fractional mean operator. Specifically, we establish explicit error bounds for Lipschitz functions and for functions with convex differences, providing a more comprehensive analysis of the discrete fractional setting. Full article

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to the right-hand side of the Hermite–Hadamard–Mercer-type inequality. Moreover, several new results regarding Young’s inequality, bounded function and L-Lipschitzian function are deduced. The paper presents additional remarks and comments on the results to make sense of them. To illustrate the key findings, graphical representations are provided, and applications involving special means, midpoint formula, q-digamma function and modified Bessel function are presented to demonstrate the practical utility of the derived inequalities. Full article

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-$(p,s)$-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET-$(p,s)$-convex functions and logarithmic $(p,s)$-convex functions. Moreover, we obtain enhancements of inequalities like the Hermite–Hadamard, midpoint, and Fejér types for two extended convex functions by employing the Caputo fractional derivative. The research presents a numerical example with graphical representations to confirm the correctness of the obtained results. Full article

The main goal of this study is to explain the idea of generalized interval-valued ($\mathbb{I}.\mathbb{V}$) convexity on a fractal set. We first define the basic operations for a generalized interval of ${\mathbb{R}}^s$ with $0<s\le 1$. Then, we expand the idea of ($\mathbb{I}.\mathbb{V}$) Riemann integration to ($\mathbb{I}.\mathbb{V}$) local fractal integration, which sets the stage for further research. This is followed by the proof of new Jensen, Hermite, Hadamard, Pachpatte, and Fejer inequalities that are ($\mathbb{I}.\mathbb{V}$) and have to do with the generalized class of ($\mathbb{I}.\mathbb{V}$) convexity defined over the fractal domain. We furnish validation through visual and comparative approaches. Our outcomes are the refinement of many existing results, indicating that they are fruitful. In fractal settings, this is the first paper to work on ($\mathbb{I}.\mathbb{V}$) convexity and some set-valued versions of Hermite–Hadamard-type containments. Full article

Strongly convex functions form a central subclass of convex functions and have gained considerable attention due to their structural advantages and broad applicability, particularly in optimization and information theory. In this paper, we investigate the class of strongly F-convex functions, which generalizes the classical notion of strong convexity by introducing an auxiliary convex control function F. We establish several fundamental structural characterizations of this class and provide a variety of nontrivial examples such as power, logarithmic, and exponential functions. In addition, we derive refined Jensen-type and Hermite–Hadamard-type inequalities adapted to the strongly F-convex concept, thereby extending and sharpening their classical forms. As applications, we obtain new analytical inequalities and improved error bounds for entropy-related quantities, including Shannon, Tsallis, and Rényi entropies, demonstrating that the concept of strong F-convexity naturally yields strengthened divergence and uncertainty estimates. Full article

This study introduces and analyzes several new functionals defined on the interval $[0,1]$, which are associated with weighted integral inequalities for geometrically–arithmetically ($GA$) convex functions. Building upon the classical Hermite–Hadamard and Fejér inequalities, we define mappings such as $\mathcal{G}\left(u\right)$, ${\mathcal{H}}_y\left(u\right)$, ${\mathcal{K}}_y\left(u\right)$, $\mathcal{N}\left(u\right)$, $\mathcal{L}\left(u\right)$, ${\mathcal{L}}_y\left(u\right)$, and ${\mathcal{S}}_y\left(u\right)$, which incorporate a $GA$-convex function x and a non-negative, integrable weight function y that is symmetric about the geometric mean $\sqrt{s_1{s}_2}$. Under these conditions, we establish novel Fejér-type inequalities that connect these functionals. Furthermore, we investigate essential properties of these mappings, including their $GA$-convexity, monotonicity, and symmetry. The validity of our main results is demonstrated through detailed examples. The findings presented herein provide significant refinements and weighted generalizations of known results in the literature. Full article

In this study, we introduce and rigorously formalize the notion of (P, m)-superquadratic stochastic processes, representing a novel and far-reaching generalization of classical convex stochastic processes. By exploring their intrinsic structural characteristics, we establish advanced Jensen and Hermite–Hadamard ($\mathbb{H}.\mathbb{H}$)-type inequalities within the mean-square stochastic calculus framework. Furthermore, we extend these inequalities to their fractional counterparts via stochastic Riemann–Liouville ($\mathbb{RL}$) fractional integrals, thereby enriching the analytical machinery available for fractional stochastic analysis. The theoretical findings are comprehensively validated through graphical visualizations and detailed tabular illustrations, constructed from diverse numerical examples to highlight the behavior and accuracy of the proposed results. Beyond their theoretical depth, the developed framework is applied to information theory, where we introduce new classes of stochastic divergence measures. The proposed results significantly refine the approximation of stochastic and fractional stochastic differential equations governed by convex stochastic processes, thereby enhancing the precision, stability, and applicability of existing stochastic models. To ensure reproducibility and computational transparency, all graph-generation commands, numerical procedures, and execution times are provided, offering a complete and verifiable reference for future research in stochastic and fractional inequality theory. Full article

By leveraging the concept of k-Caputo fractional derivatives for stochastic processes, in this paper, we derive a generalized Hermite–Hadamard inequality tailored to n-times differentiable convex stochastic processes, providing a powerful tool for analyzing systems governed by fractional dynamics in probabilistic settings. Additionally, we establish two new integral identities that serve as the foundation for developing midpoint- and trapezium-type inequalities for $(n+1)$-times differentiable convex stochastic processes. These results not only enrich the theoretical underpinnings of fractional calculus, but also offer practical implications for modeling and understanding complex systems with memory and randomness. The proposed framework opens new avenues for future research in stochastic analysis and fractional calculus, with potential applications in fields such as financial mathematics, engineering, and physics. Full article

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex stochastic. We first provide the Hermite–Hadamard inequality, then by introducing a novel identity involving mean-square stochastic Riemann–Liouville fractional integral operators, we derive several midpoint-type inequalities for harmonically s-convex stochastic processes. Illustrative example with graphical depiction and a practical application are provided. Full article