Differentiable Convex Functions Research Articles

Fractal-fractional estimations of Bullen-type inequalities with applications

The study of inequalities inside fractal domains has been stimulated by the growing interest in fractional calculus for the applied and mathematical sciences. This work uses extended fractional integrals in fractal domains to prove new Bullen-type inequalities for differentiable convex functions. By showing how these operators and inequalities can convert classical inequalities into fractal sets, it fills a vacuum in the literature on fractal-fractional integral inequalities. Using extended fractional integral operators, we derive new fractal-fractional estimates and Bullen-type inequalities, accompanied by thorough mathematical derivations and graphical validations. We show optimal results derived from new Bullen-type inequalities for fractal domains. We confirm our results with mathematical examples and graphs, improving models for complicated problems in fractal contexts. In particular, our findings have important ramifications for special means on fractal sets, probability density functions, and quadrature formulas, as well as for future study and applications.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

New cyclic Jensen type bounds via weighted HH inequalities and Montgomery identity with application to information inequalities

In this article a new method is introduced to give weighted estimations of the differences of cyclic refinements of Jensen functionals. We utilize weighted form of Hermite–Hadamard (HH) inequality along with the approximations of Montgomery two- and one-point formulae with Peano type kernel along with the consequences of the n-times differentiable convex functions. As a result, we present new upper and lower bounds that are also verified with concrete examples to show the correctness of the bounds obtained for special cases. As a result, we provide estimations in terms of cyclic power means. We also improvise our results to give new uniform estimations for useful distance functions in information theory. Finally, some estimations of Zipf and Hybrid Zipf law are provided as well.

Accelerated optimization through time-scale analysis of inertial dynamics with asymptotic vanishing and Hessian-driven dampings

ABSTRACT Gradient optimization algorithms can be effectively studied from the perspective of ordinary differential equations (ODEs), where these algorithms are obtained through the temporal discretization of the continuous dynamics. This approach provides a powerful tool to comprehend acceleration phenomena in optimization, thanks to time-scale techniques and Lyapunov analysis. In the context of a Hilbert setting, our study focuses on the rapid optimization properties of inertial dynamics, which combine asymptotically vanishing viscous damping with Hessian-driven damping. These dynamics are derived as high-resolution ODEs from both the Nesterov accelerated gradient method and the Ravine method. For a general differentiable convex function f, choosing the viscous damping coefficient in the form of α / t with 3 $ ]]> α > 3 guarantees an inverse quadratic convergence rate of the values, i.e. o ( 1 / t 2 ) , the weak convergence of trajectories towards optimal solutions, and the fast convergence of gradients towards zero. By carefully scaling the dynamics temporally and based on the tuning of the damping coefficient in front of the Hessian term, we identify the limit dynamic as α becomes large. In particular, we examine the case where the limit dynamic corresponds to the Levenberg-Marquardt regularization of Newton's continuous method. This explanation accounts for the aforementioned fast convergence properties and provides fresh insights into the complexity of these methods, which play a central role in optimization for high-dimensional problems. The interplay between numerical algorithms and continuous-time ODEs plays a fundamental role in our analysis for the design and understanding of accelerated optimization algorithms. Numerical experiments are presented to illustrate and support the theoretical results.

Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution

In a Hilbertian framework, for the minimization of a general convex differentiable function f, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t, is associated with the strongly convex function obtained by adding to f a Tikhonov regularization term with vanishing coefficient ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. By adjusting the speed of convergence of ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document} towards zero, we will obtain both rapid convergence towards the infimal value of f, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f, we study the proximal algorithms in detail, and show that they benefit from similar properties.

New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature.

A study on error bounds for Newton-type inequalities in conformable fractional integrals

Abstract The authors of the paper suggest a novel approach in order to examine an integral equality using conformable fractional operators. By using this identity, some Newton-type inequalities are proved for differentiable convex functions by taking the modulus of the newly established equality. Moreover, we prove some Newton-type inequalities by using the Hölder and power-mean inequality. Furthermore, some new results are presented by using special choices of obtained inequalities. Finally, we give some conformable fractional Newton-type inequalities for functions of bounded variation.

Strong geometric derivatives

For a real valued function f, defined on an open interval I, and an arbitrary real number h, we consider the lower and upper limits of2n(f(y+h2n)−f(y)) whenever n tends to infinity and y tends to a fixed element x of I. We consider two families of functions determined by the properties of these limits. The first interesting property is when these lower and upper limits are finite and equal to each other for every real number h and every x in I. The second notable family is determined by the property that both limits are finite and increasing (in a specific sense) with respect to x for every positive number h. These properties are motivated by the families of continuously differentiable functions and convex functions, respectively. However, restrictions of additive mappings belong to these classes as well. Our decomposition theorems establish that these motivating examples generate the whole classes. Namely, every function belonging to the first family can be represented as the sum of a continuously differentiable function and an additive one, while every function taken from the second family turns out to be the sum of a convex function and an additive one. We apply our results in order to give a local and approximate characterization of affine functions and Wright-convex functions, respectively.

A comprehensive study on Milne-type inequalities with tempered fractional integrals

AbstractIn the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.

Error Bounds for Fractional Integral Inequalities with Applications

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.

Distributed Stochastic Proximal Algorithm With Random Reshuffling for Nonsmooth Finite-Sum Optimization.

The nonsmooth finite-sum minimization is a fundamental problem in machine learning. This article develops a distributed stochastic proximal-gradient algorithm with random reshuffling to solve the finite-sum minimization over time-varying multiagent networks. The objective function is a sum of differentiable convex functions and nonsmooth regularization. Each agent in the network updates local variables by local information exchange and cooperates to seek an optimal solution. We prove that local variable estimates generated by the proposed algorithm achieve consensus and are attracted to a neighborhood of the optimal solution with an O((1/T)+(1/√T)) convergence rate, where T is the total number of iterations. Finally, some comparative simulations are provided to verify the convergence performance of the proposed algorithm.

On Conformable Fractional Milne-Type Inequalities

Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities.

On the Optimal Point of the Weighted Simpson Index

In this short communication, following a brief introduction, we undertake a comprehensive analytical study of the weighted Simpson index. Our primary emphasis concerns the precise determination of the optimal point (minimizer) coordinates and of the minimum value of the index, a differentiable convex function, which is related to the harmonic mean concept. Furthermore, we address and solve the inversion problem and show the tight connection between both approaches. Last, we give some insights and final remarks on this subject.

Novel results of Milne-type inequalities involving tempered fractional integrals

In this current research, we focus on the domain of tempered fractional integrals, establishing a novel identity that serves as the cornerstone of our study. This identity paves the way for the Milne-type inequalities, which are explored through the framework of differentiable convex mappings inclusive of tempered fractional integrals. The significance of these mappings in the realm of fractional calculus is underscored by their ability to extend classical concepts into more complex, fractional dimensions. In addition, by using the Hölder inequality and power-mean inequality, we acquire some new Milne-type inequalities. Moreover, the practicality and theoretical relevance of our findings are further demonstrated through the application of specific cases derived from the theorems.

Some new Inequalities for Differentiable Arithmetic-Harmonically Convex Functions

In this study, by using an integral identity together with both the Hölder and the power-mean inequalities for integrals we establish several new inequalities for differentiable arithmetic-harmonically-convex function. Also, we give some applications for special means.

Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

<abstract><p>The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.</p></abstract>

Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes

<abstract><p>This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.</p></abstract>

Generalization of Hermite-Hadamard Inequality for Twice Differentiable Convex and Quasi-Convex Functions

Generalization of Hermite-Hadamard Inequality for Twice Differentiable Convex and Quasi-Convex Functions

Variational principles for monotone variational inequalities: The single-valued case

We consider a parameterized variational inequality \((A,Y)\) in a Banach space \(E\) defined on a closed, convex and bounded subset \(Y\) of \(E\) by a monotone operator \(A\) depending on a parameter. We prove that under suitable conditions, there exists an arbitrarily small monotone perturbation of \(A\) such that the perturbed variational inequality has a solution which is a continuous function of the parameter, and is near to a given approximate solution. In the nonparametric case this can be considered as a variational principle for variational inequalities, an analogue of the Borwein-Preiss smooth variational principle.
 Some applications are given: an analogue of the Nash equilibrium problem, defined by a partially monotone operator, and a variant of the parametric Borwein-Preiss variational principle for Gâteaux differentiable convex functions under relaxed assumtions.
 The tool for proving the main result is a useful lemma about existence of continuous \(\varepsilon\)-solutions of a variational inequality depending on a parameter. It has an independent interest and allows a direct proof of an analogue of Ky Fan's inequality for monotone operators, introduced here, which leads to a new proof of the Schauder fixed point theorem in Gâteaux smooth Banach spaces.

New cyclic Jensen type estimations VIA weighted Hadamard inequalities under the effect of Fink's identity

We take into account differences resulting from the discrete and integral cyclic Jensen's inequalities and provide upper and lower bounds using weighted Hermite-Hadamard inequalities with the support of Fink's identity. The theory of n-times differentiable convex functions is used to demonstrate this case. Our findings are applicable to any n ​≥ ​2, and we present specific examples to demonstrate the accuracy of the bounds found for unique situations. We also formulate results for power means and quasi-arithmetic means for higher order convexity. Last but not least, we provide applications in information theory by offering new uniform estimations of the generalized Csiszár, Rényi, Shannon, Kullback-Leibler, χ2 ​− ​divergence, Bhattacharyya, Zipf, and Hybrid Zipf-Mandelbrot entropies.