Inequalities For Functions Research Articles

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.

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.

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.

Geometrical criterions of generalized subadditivity

The motivation of our studies is a theorem of J. Matkowski and T. Świątkowski from 1993, which provides convenient and sufficient conditions for the subadditivity of real functions. In the work was constructed a family of sets P⊂R×R with the property that each function whose graph is contained in P is subadditive. The purpose of this article is to generalize this result in several directions. We will discuss a more general inequality:f(∑i=1Nαixi)⩽∑i=1Nβif(xi). Moreover, our family of sets is constructed in such a way that an essentially larger class of solutions are covered. Geometric method used in presented paper may be applied to other functional inequalities like Hlawka's, submultiplicative and subquadratic inequalities as well as to obtaining irregular functions satisfying certain inequalities.

General Trapezoidal-Type Inequalities in Fuzzy Settings

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of p∈0,1.

Starlikeness and Convexity of Generalized Bessel-Maitland Function

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order δ, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies.

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.

On fractional Orlicz-Hardy inequalities

We establish the weighted fractional Orlicz-Hardy inequalities for various Young functions satisfying the Δ2-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 Δ2-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.

Estimations of global Mittag–Leffler bounds and synchronization for fractional-order permanent magnet synchronous motor systems

This article investigates the global Mittag-Leffler bounds (GM-LBs) and synchronization of fractional-order permanent magnet synchronous motors (FOPMSMs). To begin with, by using some appropriate generalized Lyapunov functions, the extremum principle of function and fractional differential inequalities, a few new bounds for the three-dimensional cylindrical and ellipsoidal domains and a rotatory parabolic body of FOPMSMs are estimated by utilizing the parameters in the systems, which improves the earlier studies and may conclude some new estimates. Moreover, linear feedback control strategies owning one or two inputs and fractional adaptive control owning both single input and single state are accepted to come true the global Mittag-Leffler synchronization (GM-LS) and global asymptotic synchronization (GAS) of two FOPMSMs. Some new criteria for the synchronization are acquired by utilizing some inequality approaches. Finally, the feasibility of the proposed control schemes is verified by numerical simulations.

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.

Inequalities for Basic Special Functions Using Hölder Inequality

Let p,q≥1 be two real numbers such that 1p+1q=1, and let a,b∈R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form |f(ap+bq)|≤|f(a)|1p|f(b)|1q, and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.

An Extension of Left Radau Type Inequalities to Fractal Spaces and Applications

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain previously established findings to fractal sets but also a refinement of these when the fractal dimension μ is equal to one. Finally, to support our findings, we present a practical application to demonstrate the effectiveness of our results.

The incidence of tax rates on workers’ debt stability and demand regimes

This paper explores different tax regimes in a Post Keynesian model where workers get into debt to emulate the consumption of upper-income classes. The government taxes income to fund a social wage that would reduce workers’ need to get into debt. Three tax regimes are analyzed: taxing profits, managers’ wages, or both. In a numerical exercise, we explore the effects of changing the within-wage and functional inequalities. The government’s income tax choices and the wage bill’s distribution matter for the economy’s sustainability and the relationship between distribution and growth. The demand regime of the economy is not independent of policy. Taxing only profits and reducing wage inequality are the best possible outcomes in the model if we were to wind down the unsustainability feature of neoliberalism without sacrificing real performance.

Some inequalities for entire functions

Let $\mathcal{L}_p$ be the subspace of the space $L_p(\bR)$ consisting of the restriction to the real axis of all entire functions of exponential type $\le \pi$. In this paper, for any function $f\in \mathcal{L}_p \,\,(1\le p\le \infty)$, we obtain estimates for the norm of $f$ in terms of the sequence $(f(n/2))_{n\in\bZ},$ namely$$\frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1},$$where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\bZ)} +\|JT_{1/2}f\|_{\ell_p(\bZ)})$. Here $J:\mathcal{L}_p\to\ell_p(\bZ)$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\bZ,$$ and $T_\tau$ is the shift by $\tau\in\bR$ of the function $f$,$$ (T_\tau f)(z):=f(z+\tau), \quad z\in\bC.$$

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.

On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g=(g1,…,gr) defining S. They are polynomial in c and ε−1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.

New weighted Trudinger-Moser inequality for functions not necessarily radially symmetric and applications

In this paper, we prove some new inequality of Trudinger-Moser' type defined in some weighted Sobolev space whose norm is a combination of the norms of the N-Laplacian and the p-Laplacian with 1<p<N in RN,N≥2. The presence and the behavior of the weights prevent us to use of the symmetrization arguments. This inequality holds for functions which are not required to be radially symmetric. The strategy used in order to establish our inequality mainly consists of a proper change of variables which permits to return to the non-weighted case. Various improvements of this inequality are also proved. Finally, one of these improvements is used to study some elliptic equation involving a weighted (N,p)-Laplacian operator.