Search Results (103)

A refined version of the q-Hermite–Hadamard inequalities for strongly convex functions is introduced in this paper, utilizing both left and right q-integrals. Tighter bounds and more accurate estimates are derived by incorporating strong convexity. New q-trapezoidal and q-midpoint estimates are also presented to enhance the precision of the results. The improvements in the results compared to previous work are demonstrated through numerical examples in terms of precision and tighter bounds, and the advantages of using strongly convex functions are showcased. Full article

Starting from ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ${\psi }_k$-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), $\psi$-Riemann–Liouville ($\psi$-R-L), ${\psi }_k$-Riemann–Liouville (${\psi }_k$-R-L), Raina’s, k-Raina’s, and $\psi$-Raina’s fractional integrals ($\psi$-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator ${\mathbf{S}}_{\mu ,\delta ,q}^{n,m}$ and subordination are used in this study to define and construct new classes of $\alpha$-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes ${\mathcal{R}}_q^{\alpha }(a,c,m,L,P),$ of $\alpha$-convex functions in the open unit disc ${\mathcal{U}}_s$, such as initial coefficient bounds, coefficient inequalities, Fekete–Szegö problems, the second Hankel determinants, and logarithmic coefficients. The results presented in this paper are simple to comprehend and demonstrate how current research relates to earlier research. We found all of the estimates, and they are sharp. Full article

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when $\beta =0$ and to classical Lebesgue spaces when $\mathtt{q}=\infty ,\beta =0$. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis. 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

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings. Full article

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

The rapidly exploring random tree star (RRT*) algorithm is widely used to solve path planning problems. However, the RRT* algorithm and its variants fall short of achieving a balanced consideration of path quality and safety. To address this issue, an improved discretized artificial potential field-QRRT* (IDAPF-QRRT*) path planning strategy is introduced. Initially, the APF method is integrated into the Quick-RRT* (Q-RRT*) algorithm, utilizing the attraction of goal points and the repulsion of obstacles to optimize the tree expansion process, swiftly achieving superior initial solutions. Subsequently, a triangle inequality-based path reconnection mechanism is introduced to create and reconnect path points, optimize the path length, and accelerate the generation of sub-optimal paths. Finally, by refining the traditional APF method, a repulsive orthogonal vector field is obtained, achieving the orthogonalization between repulsive and attractive vector fields. This places key path points within the optimized vector field and adjusts their positions, thereby enhancing path safety. Compared to the Q-RRT* algorithm, the DPF-QRRT* algorithm achieves a 37.66% reduction in the time taken to achieve 1.05 times the optimal solution, and the IDAPF-QRRT* strategy nearly doubles generated path safety. Full article

In this article, we consider a unified generalized version of extended Euler’s Beta function’s integral form a involving Macdonald function in the kernel. Moreover, we establish functional upper and lower bounds for this extended Beta function. Here, we consider the most general case of the four-parameter Macdonald function $K_{\nu +{\displaystyle \frac{1}{2}}}\left(p{t}^{-\lambda }+q{(1-t)}^{-\mu }\right)$ when $\lambda \ne \mu$ in the argument of the kernel. We prove related bounding inequalities, simultaneously complementing the recent results by Parmar and Pogány in which the extended Beta function case $\lambda =\mu$ is resolved. The main mathematical tools are integral representations and fixed-point iterations that are used for obtaining the stationary points of the argument of the Macdonald kernel function $K_{\nu +{\displaystyle \frac{1}{2}}}$. Full article

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (${\mathcal{l}}^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$). Moreover, it was developed using classical Lebesgue space (${\mathtt{L}}_{\mathtt{p}}$) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not. Full article

The principal aim of this paper is to introduce the extended Voigt-type function ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and its counterpart extension ${\mathbb{W}}_{\mu ,\nu }(x,y)$, involving the Neumann function $Y_{\nu }$ in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions $K(x,y)$ and $L(x,y)$, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions ${\mathbb{V}}_{\mu ,\nu }^{\left(r\right)}(x,y),{\mathbb{W}}_{\mu ,\nu }^{\left(r\right)}(x,y)$ and the r positive integer of the initial extensions ${\mathbb{V}}_{\mu ,\nu }(x,y),{\mathbb{W}}_{\mu ,\nu }(x,y)$. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions ${\Psi }_2^{\left(r\right)}$. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind $Q_{\eta }^{-\nu }$ in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and ${\mathbb{W}}_{\mu ,\nu }(x,y)$. Particularly interesting results are presented for the Neumann function $Y_{\nu }$ and for the Struve ${\mathbf{H}}_{\nu }$ function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ${\ell }^p\to {\ell }^q$ bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. Full article

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval $[{\mathrm{a}}_0,{\mathrm{a}}_1]\times [{\mathrm{c}}_0,{\mathrm{c}}_1]\subseteq {\Re }^2$, we introduce the notion of partial ${\mathrm{q}}_{\theta }$-, ${\mathrm{q}}_{\varphi }$-, and ${\mathrm{q}}_{\theta }{\mathrm{q}}_{\varphi }$-symmetric derivatives and a ${\mathrm{q}}_{\theta }{\mathrm{q}}_{\varphi }$-symmetric integral. Moreover, we will construct the ${\mathrm{q}}_{\theta }{\mathrm{q}}_{\varphi }$-symmetric Hölder’s inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications. Full article

This paper aims to unify q-derivative/q-integrals and h-derivative/h-integrals into a single definition, called $q-h$-derivative/$q-h$-integral. These notions are further extended on the finite interval $[a,b]$ in the form of left and right $q-h$-derivatives and $q-h$-integrals. Some inequalities for $q-h$-integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified using the $q-h$-derivative/$q-h$-integral concept. Full article

This study uses Raina’s function to obtain a new coordinated $pq$-integral identity. Using this identity, we construct several new $pq$-Simpson’s type inequalities for generalized convex functions on coordinates. Setting $p_1=p_2=1$ in these inequalities yields well-known quantum Simpson’s type inequalities for coordinated generalized convex functions. Our results have important implications for the creation of post quantum mathematical frameworks. Full article