Search Results (161)

The present work is devoted to deriving Ostrowski-type inequalities for square modulus convex mappings $\mathcal{K}:[\alpha ,\beta ]\subset \mathbb{R}\to \mathfrak{B}\left(H\right)$ in complex Hilbert spaces. We show that the deviation of the weighted integral mean ${\int }_{\mathcal{X}}\rho {|\mathcal{K}\circ \phi |}^{2}d\mu$ from the pointwise value ${\left|\mathcal{K}\left(\xi \right)\right|}^2$ at any point $\xi \in [\alpha ,\beta ]$ is bounded in terms of the weighted standard deviation ${\sigma }_{\rho }\left(\phi \right)$ of $\phi$, giving the operator-valued analogue of the classical Ostrowski inequality. Taking $\xi ={\displaystyle \frac{\alpha +\beta }{2}}$ yields a midpoint inequality with an explicit constant. We also obtain a distribution-dependent estimate for the Jensen gap that vanishes whenever $\phi$ is constant. New two-point and multi-point Ostrowski inequalities and a superadditivity property of the Jensen gap are also derived. Full article

A conserved comparison ledger is an abstract system in which each state carries a positive ratio $r\left(s\right)$, every measurement factors through that ratio, and pairwise comparisons are scored by an admissible cost J. A separate classification fixes the admissible costs with polynomial combiner and, under a units normalization, selects the representative $J_{\mathrm{cost}}\left(x\right)=\frac{1}{2}(x+x^{-1})-1.$ This paper installs that classified cost in a ledger and determines what the ledger assumptions force. The argument has three structural steps and one neutrality axiom: H1 forces the scale ratio to $\sigma =\phi$; H2, using that classification, forces $J=J_{\mathrm{cost}}$; H3 reads $r\left(s\right)$ from a constrained cost minimizer of an internal positive vector, and A3 imposes zero total log-charge; together they force $r\equiv 1$, make the measurement map constant, and collapse the observational quotient to a singleton. The assumptions are also sharp in the following sense: removing H1, H2, H3, or A3 admits an explicit ledger satisfying the remaining meaningful assumptions while losing the corresponding conclusion. The examples verify joint satisfiability, include non-uniform internal data whose variational minimum gives $r=1$, and show nontrivial sector dynamics when A3 is not imposed. A final example realizes the full axiom package on the patch space of the Fibonacci quasicrystal, with the scale and the charge-sector structure supplied intrinsically by the tiling and the remaining installations stated explicitly. We frame this result as a coherence theorem for a deliberately chosen axiom system: each hypothesis is individually natural and, by design, controls a single output, so their joint forcing of $(\sigma ,J,r)$ exhibits the internal consistency of the axiom package rather than asserting that any independently given system must realize it. Full article

In this paper, we introduce Atangana–Baleanu–tempered–conformable (ABTC) generalized integral operators and establish associated Milne–Mercer-type inequalities for differentiable convex functions. The proposed kernel simultaneously incorporates AB-type local–nonlocal mixing, exponential attenuation, and conformable distance scaling. Using a kernel-dependent integral identity together with the Jensen–Mercer inequality, convexity arguments, and Hölder’s inequality, we derive estimates whose coefficients depend jointly on the independent parameters $\theta$, $\alpha$, $\beta$, and $\lambda$. None of the corresponding AB–conformable, tempered–conformable, or AB-type tempered specializations retains all these mechanisms simultaneously. Parameter reductions recover several previously known integral settings and provide consistency checks for the construction. Numerical examples are included to illustrate the validity, comparative behavior, and parameter sensitivity of the resulting bounds. Full article

This work constructs a family of weighted higher-order Opial-type estimates on an arbitrary time scale $\mathbb{T}$, with the weight structure generated by a non-decreasing auxiliary function $\omega$ and its delta derivative ${\omega }^{\Delta }$. For n-times delta differentiable functions whose delta derivatives of orders $0,1,\dots ,n-1$ vanish at the left endpoint, the mixed functional ${\int }_a^b{v\left(t\right)|u\left(t\right)|}^p{|u^{{\Delta }^n}\left(t\right)|}^q\Delta t$ is bounded by expressions involving only the highest-order delta derivative $u^{{\Delta }^n}$. Three forms are obtained: a Hölder-type product estimate, a Young-type two-term estimate with a free balancing parameter $\theta >0$, and an optimized single-integral estimate with the explicit constant $K={(r-1)}^{1-1/r}/r$, where $r=(p+q)/q$. The proofs rely on the Taylor representation on time scales, Hölder’s inequality, Jensen’s inequality, and Fubini’s theorem. The resulting bounds are governed by computable kernels that encode the interaction between the Taylor monomials, the auxiliary weight, and the external weight. Specializations to $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=\mathbb{Z}$, and the quantum lattice $q_0^{{\mathbb{N}}_0}$ recover classical, discrete, weighted, and quantum Opial inequalities with their standard constants and also yield several higher-order weighted estimates. As an application, a uniqueness criterion for higher-order dynamic initial value problems is established. Full article

Existing approaches for admissibility analysis of T-S fuzzy descriptor time delay systems fail to balance conservatism reduction and computational complexity. This paper proposes a low-conservatism analysis and stabilization method based on the symmetric Lyapunov–Krasovskii (L-K) functional. By exploiting the boundedness of membership function derivatives, and combining Jensen’s integral inequality with auxiliary slack matrices to achieve tight bounding of nonlinear terms, we derive an admissibility criterion for open-loop systems with significantly reduced conservatism. A well-suited L-K functional is constructed targeting the structural characteristics of fuzzy singular matrices $E_{\xi }$, a state feedback controller is designed via the parallel distributed compensation (PDC) strategy, and solvable sufficient conditions for the admissibility of closed-loop systems are established. Numerical examples demonstrate that the maximum allowable delay upper bound obtained by the proposed method outperforms that of existing state-of-the-art approaches while balancing conservatism and computation cost and verifying the superiority of the proposed method. Full article

By using recently established refinements of Jessen and converse Jessen-type inequalities, together with improved characterizations of strongly convex functions, we derive new Jensen–Mercer-type functional inequalities. The obtained results extend and sharpen several known inequalities for convex functions. As applications, we establish new integral inequalities which improve some recently published estimates. In addition, we obtain inequalities involving generalized logarithmic means. Finally, we derive corresponding probabilistic versions of the main results, which lead to refinements of several known results from the literature. Full article

Classical information theory (CIT) characterizes the transmission limit for communication systems under syntactic accuracy, whereas semantic information theory (SIT) studies communication from the perspective of semantic fidelity induced by synonymous mapping. In this paper, we investigate the semantic channel capacity of Rayleigh fading channels under synonymous mapping of the channel gain and additive noise. We first derive the semantic capacity formula when synonymous mapping is applied to the channel fading coefficient and establish corresponding upper and lower bounds using Jensen’s inequality. To determine an optimized synonymous partition, the partition design is formulated as a constrained optimization problem and solved numerically using a neural network-based approach with the Adam optimizer. Furthermore, we extend the framework by applying synonymous mapping to both the channel fading coefficient and the additive noise and derive the corresponding semantic capacity formula together with its theoretical bounds. The numerical results illustrate the theoretical semantic channel capacity under synonymous mapping and validate the compatibility of the proposed framework with both CIT and SIT. At a 20-dB SNR with $K=8$ channel gain intervals and $J=4$ noise intervals, the semantic capacity reached 9.86 sebits/s/Hz. Full article

Some equivalent statements and basic properties for the generalized convexity assumption $p\left(x\right)f\left(x\right)+q\left(x\right)f^{\prime }\left(x\right)+f^{″}\left(x\right)\ge 0$ are proved. Then based on these conclusions, various Jensen type inequalities under the generalized convexity are established. The idea is to transform such $p\left(x\right),q\left(x\right)$-convex functions to some simpler $p\left(x\right)$, 0-convex or 0, $q\left(x\right)$-convex functions, or even convex functions. Ky Fan and Wang-Wang type inequalities are also generalized as applications. 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

Spacecraft pursuit–evasion in contested environments is complicated by strategic incompleteness: the evader can switch maneuvering modes and deploy multi-domain countermeasures that degrade the pursuer’s perception, leading to non-stationary information and distributionally ambiguous interference statistics. A dynamic time-window Nash equilibrium framework is developed for linearized Local Vertical Local Horizontal (LVLH) relative motion under interference-induced uncertainty. Perceptual degradation is modeled via an evidence–theoretic belief representation, and the Jensen–Shannon (JS) divergence is introduced to quantify discrepancies between nominal and interference-corrupted beliefs. The divergence metric drives an adaptive time-window partitioning policy and an uncertainty-aware running cost that balances nominal performance objectives with robustness regularization during high-degradation intervals. In each time window, sufficient conditions are provided for the existence of a local Nash equilibrium, and equilibrium strategies are characterized by the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. A global consistency result is established: assuming state continuity, additive cost decomposition, and dynamic-programming compatibility at window boundaries, concatenating the window-wise equilibria yields a Nash equilibrium over the entire horizon. Unlike conventional receding-horizon differential games with a fixed replanning grid, the proposed policy partitions the horizon online in response to perceptual-degradation events and stitches adjacent windows through a continuation value. This boundary stitching enables the global consistency guarantee under additive costs and state continuity. To hedge against ambiguity in interference intensity, a variational distributionally robust optimization (DRO) problem with moment-constrained ambiguity sets is formulated, and the dual worst-case distribution is derived. The resulting Karush–Kuhn–Tucker (KKT) system is reformulated as a finite-dimensional variational inequality, for which an accelerated Alternating Direction Method of Multipliers (ADMM) operator-splitting solver is proposed for efficient real-time computation. Numerical simulations validate the framework and demonstrate improved robustness and computational scalability under time-varying interference compared with fixed-window baselines. Full article

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. 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

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