Inequalities For Functions Research Articles

A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications

Abstract The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper.

Robust finite-time H2/H∞ control for stochastic Markovian jump systems via state feedback

The robust finite-time [Formula: see text] control problems for linear discrete-time stochastic Markovian jump systems (MJSs) with external disturbance are investigated. Firstly, the definitions of robust finite-time boundedness, and robust finite-time [Formula: see text] boundedness are presented. Subsequently, by using the stochastic Lyapunov–Krasovskii functional method and linear matrix inequality technique, sufficient conditions for the existence of a robust finite-time [Formula: see text] controller are provided for stochastic MJSs. A state feedback controller is designed. Furthermore, the finite-time guaranteed cost bounds are given. More specifically, the designed controller not only ensures the finite-time [Formula: see text] bounded but also makes the [Formula: see text] cost function less than the bound given by the performance index of the systems. Finally, two examples are given to illustrate the validity of the proposed approach.

Fractional integral inequalities for Superquadratic functions Via Atangana-Baleanu's operator with applications

Fractional integral inequalities for Superquadratic functions Via Atangana-Baleanu's operator with applications

A New Class of Generalized Starlike Bi-Univalent Functions Subordinated to Legendre Polynomials

A new class of generalized starlike bi-univalent functions is introduced in this paper, which is defined using Legendre Polynomials within the open unit disk D. This paper sheds a light on the properties and behaviors of these starlike bi-univalent functions, providing estimations for the modulus of the initial Taylor series coefficients of the functions falling under this particular class and one of its various subclasses. Additionally, this paper also investigates the classical Fekete-Szegö functional problem for functions f that are part of the aforementioned class. Moreover, we obtain the classical Fekete-Szegö inequalities of functions belonging to this class and to one of its various subclasses.

A New Inclusion on Inequalities of the Hermite–Hadamard–Mercer Type for Three-Times Differentiable Functions

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities.

Sharp Wilker and Huygens type inequalities for the Gudermannian and inverse Gudermannian functions

The aim of the present paper is to present new sharp Wilker and Huygens type inequalities for Gudermannian and inverse Gudermannian functions.

Quantitative profile decomposition and stability for a nonlocal Sobolev inequality

In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given bySHL(∫RN(|x|−μ⁎|u|2μ⁎)|u|2μ⁎dx)12μ⁎≤∫RN|∇u|2dx,∀u∈D1,2(RN), where ⁎ denotes the convolution of functions, 2μ⁎:=2N−μN−2 and SHL are positive constants that depends solely on N and μ. For N≥3 and 0<μ<N, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function W[ξ,λ] that is positive and radially symmetric.Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:‖∇u−∑i=1κ∇W[ξi,λi]‖L2≤C‖Δu+(1|x|μ⁎|u|2μ⁎)|u|2μ⁎−2u‖(D1,2(RN))−1, where the parameter region satisfies κ≥2, 3≤N<6−μ, μ∈(0,N) with 0<μ≤4, or in the case of dimension N≥3 and κ=1, μ∈(0,N) with 0<μ≤4.

Series expansion, higher-order monotonicity properties and inequalities for the modulus of the GrÖtzsch ring

Abstract For $r\in(0,1)$ , let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$ , where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that \begin{equation*} \mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{ 2n}r^{2n}, \end{equation*} with $\theta _{n}\in \left( 0,1\right)$ . Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $ , which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

Functional Inequalities in the Framework of Banach Spaces

This paper aims to illustrate the usefulness of majorization theory and higher-order convexity in generalizing several classical inequalities as functional inequalities. This includes results due to Ball et al. (Invent Math 115:463-482, 1994), Hanner (Ark Mat 3:239-244, 1956), Popoviciu (Analele ştiinţifice Univ “Al. I. Cuza” Iasi, Secţia I-a Mat 11:155-164, 1965) and Zhang (Matrix Theory: Basic Results and Techniques, Springer, New York, 2011). Also, based on a substitute for the parallelogram law due to Clarkson, a quadrilateral inequality established by Schötz (in Quadruple inequalities: Between Cauchy-Schwarz and triangle. ArXiv preprint, arXiv:2307.01361 v.3, 2024) in the context of Hilbert spaces is extended to the general framework of Banach spaces.

Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an \((r-\varepsilon)\) -approximation of this value requires \(\Omega (n^{1-2/t})\) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( \(\mathsf {LDC}\) ) \(\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N\) over large alphabets with recoverability probability at least \(1/r + \varepsilon\) . Using hypercontractivity, we give an exponential lower bound \(N= 2^{\Omega (\varepsilon ^4 n/r^4)}\) for 2-query (possibly non-linear) \(\mathsf {LDC}\) s over \(\mathbb {Z}_r\) and using the non-commutative Khintchine inequality we prove an improved lower bound of \(N= 2^{\Omega (\varepsilon ^2 n/r^2)}\) .

Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums

We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum S n = ∑ j = 1 n X j , π ( j ) S_n=\sum _{j=1}^nX_{j,\pi (j)} of a random n × n n\times n matrix X = ( X j , r ) X=(X_{j,r}) , where X j , r X_{j,r} are independent integer valued random variables, and π \pi denotes a uniformly distributed random permutation on { 1 , … , n } \{1,\dots ,n\} independent of X X . As a measure of smoothness, we consider the total variation distance between the distributions of S n S_n and 1 + S n 1+S_n . Our approach uses new auxiliary inequalities for a generalized normalized matrix hafnian and for inverse moments of non-negative random variables, which could be of independent interest. This approach is also used to prove upper bounds of the Lévy concentration function of S n S_n in the case of independent real valued random variables X j , r X_{j,r} .

Arbitrary Random Variables and Wiman’s Inequality

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

On q-Hermite–Hadamard–Mercer and midpoint-Mercer inequalities for general convex functions with their computational analysis

This paper establishes some new inequalities of Hermite–Hadamard–Mercer type for [Formula: see text]-convex functions in the framework of [Formula: see text]-calculus and classical calculus. Some new [Formula: see text]-midpoint-Mercer type inequalities for the [Formula: see text]-differentiable [Formula: see text]-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and [Formula: see text]-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for [Formula: see text]-convex functions in [Formula: see text]-calculus.

Simpson’s quadrature formula for third differentiable and s-convex functions

This study establishes Newton-type inequalities for third differentiable and s-convex functions that use the Riemann integral. New Newton-type inequalities are also introduced using a summation parameter p≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\geq 1$\\end{document} for various convexity cases.

Analysis of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic function and related fractional integral inequalities with applications

In the present work we establish for the first time a class of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

Revised logarithmic Sobolev inequalities of fractional order

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on Rn with an explicit expression for the constant. Namely, we show that for all 0<s<n2 and a>0 we have the inequality∫Rn|f(x)|2log⁡(|f(x)|2‖f‖L2(Rn)2)dx+ns(1+log⁡a)‖f‖L2(Rn)2≤C(n,s,a)‖(−Δ)s/2f‖L2(Rn)2 with an explicit C(n,s,a) depending on a, the order s, and the dimension n, and investigate the behaviour of C(n,s,a) for large n. Notably, for large n and when s=1, the constant C(n,1,a) is asymptotically the same as the sharp constant of Lieb and Loss that was computed in [13]. Moreover, we prove a similar type inequality for functions f∈Lq(Rn)∩W1,p(Rn) whenever 1<p<n and p<q≤p(n−1)n−p.

On fractional Orlicz-Hardy inequalities

We establish the weighted fractional Orlicz-Hardy inequalities for various Young functions satisfying the △2-condition. Further, we identify the critical cases for such Young function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Young function Φ satisfying the △2-condition and for any Λ>1, the following inequality is establishedΦ(a+b)≤λΦ(a)+C(Φ,Λ)(λ−1)pΦ+−1Φ(b),∀a,b∈[0,∞),∀λ∈(1,Λ], where pΦ+:=sup⁡{tφ(t)/Φ(t):t>0}, φ is the right derivatives of Φ and C(Φ,Λ) is a positive constant that depends only on Φ and Λ.

Effectiveness of gluteal control training in chronic low back pain patients with functional leg length inequality

Chronic low back pain (LBP) is a common musculoskeletal disorder and is often accompanied by functional leg length inequality (FLLI). However, little was known about the effects of gluteal muscle control training in patients with LBP and FLLI. This study was designed to investigate the effects of gluteal control training in patients with LBP and FLLI. This is a double-blinded, randomized controlled study design. Forty-eight LBP patients with FLLI were randomized to the gluteal control training (GT) (47.58 ± 9.42 years) or the regular training (RT) (47.38 ± 11.31 years) group and received allocated training for six weeks. The outcome measures were pelvic inclination (PI), ilium anterior tilt difference (IATD), FLLI, visual analogue scale (VAS), patient specific-functional scale (PSFS), Oswestry disability index (ODI), hip control ability, global rating of change scale (GRoC), and lower extremity strength and flexibility. The intervention effects were compared using two-way repeated measures analysis of variance and chi-square tests with α = 0.05. The results indicated that the GT group showed greater improvement (P < 0.01) in PI (1.03 ± 0.38∘ vs. 1.57 ± 0.51∘), IATD (0.68 ± 0.66∘ vs. 2.31 ± 0.66∘), FLLI (0.3 ± 0.22 vs. 0.59 ± 0.13 cm), VAS (1.41 ± 1.32 vs. 3.38 ± 1.51), hip control ability (2.20 ± 0.45 vs. 0.89 ± 0.74), GRoC at 3rd and 6th week as compared to the RT group. Hip strength and flexibility also improved more in the GT group (P < 0.05). In conclusion, gluteal control training was more effective in improving low back pain and dysfunctions, and should be integrated in the management plan in patients with LBP and FLLI.

Concave function inequalities for sum of matrices

Concave function inequalities for sum of matrices

Functional L1-Lp inequalities in the CAR algebra

In the present paper, we use the semigroup method to investigate various functional inequalities invoking L1 and Lp norms in the framework of canonical anti-commuting relations algebra (CAR algebra for short). As the main results, we obtain the Poincaré inequality for Talagrand type sum, Eldan-Gross inequality for projections, and the Talagrand influence inequality along with its strengthening form in the CAR algebra. All our results strengthen the noncommutative Poincaré inequality of Efraim and Lust-Piquard at several points. We conclude the paper with two applications of our inequalities. In the first application, we apply the noncommutative Eldan-Gross inequality to derive two KKL-type inequalities in the CAR algebra, which are closely related to the quantum KKL conjecture of Montanaro and Osborne. The second application is the CAR algebra counterpart of the superconcentration phenomenon derived from the noncommutative Talagrand influence inequality.