Type Inequalities Research Articles

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

Some operators of fractional calculus and their applications regarding various complex functions analytic in certain domains

Abstract. In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complex- exponential forms, and also in the light of certain special information in [1], [20], [17] and [38], an extensive result together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out. Mathematics Subject Classification (2010): 26A33, 30A10, 34A40, 35A30, 41A58, 30C45, 30C55, 30C80, 33D15, 26E05, 33E20. Keywords: Complex plane, domains, regular functions, complex exponential, series expansions, fractional calculus, operators of fractional calculus, exponential type inequalities, differential inequalities.

Q-Deformed and delta-parametrized A-generalized logistic function induced Banach space valued multivariate multi layer neural network approximations

Abstract. Here we research the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or RN, N ∈ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. Mathematics Subject Classification (2010): 41A17, 41A25, 41A30, 41A36. Keywords: Multi-layer approximation, q-deformed and λ-parametrized A- generalized logistic function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, iterated approximation.

Spherical Logvinenko–Sereda–Kovrijkine type inequality and null-controllability of the heat equation on the sphere

It is shown that the restriction of a polynomial to a sphere satisfies a Logvinenko–Sereda–Kovrijkine type inequality (a specific type of uncertainty relation). This implies a spectral inequality for the Laplace–Beltrami operator, which, in turn, yields observability and null-controllability with explicit estimates on the control costs for the spherical heat equation that are sharp in the large and in the small time regime.

On Bernstein- and Marcinkiewicz-type inequalities on multivariate [formula omitted]-domains

We prove new Bernstein and Markov type inequalities in Lp spaces associated with the normal and the tangential derivatives on the boundary of a general compact Cα-domain with 1≤α≤2. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of Lp norms of algebraic polynomials on Cα-domains with asymptotically optimal number of function samples used.

Generalizations of the Kantorovich and Wielandt Inequalities with Applications to Statistics

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models.

Some new integral inequalities for F-convex functions via ABK-fractional operator

In this paper, we present Hermite-Hadamard type and midpoint-type integral inequalities for F-convex function via ABK-fractional integral operators. In the context of differentiable mapping Ψ, we provide multiple new identities and prove new results for F-convex functions using ABK-fractional integral operators whose derivatives in the absolute values are F-convex. We establish correlations between our results and a number of significant results in the literature. The results presented in this paper might encourage more study in this field.

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

Do poor psychosocial resources mediate health inequalities in type 2 diabetes mellitus? Findings from the Maastricht study.

Low socioeconomic position (SEP) has been identified as a risk factor for type 2 diabetes mellitus (T2DM), and psychosocial resources might be on the pathway in this association. We examined two poor psychosocial resources, low control beliefs and inferiority beliefs, that might link low SEP with T2DM. 8292 participants aged 40-75 living in Southern Netherlands participated in The Maastricht Study starting from September 2010 to October 2020 and were followed up to 10 years with annual questionnaires. SEP (education, income, occupation), low control beliefs, inferiority beliefs, and (pre)diabetes by oral glucose tolerance test were measured at baseline. Incident T2DM was self-reported per annum. We analysed the mediating roles of poor psychosocial resources by using counterfactual mediation analysis. People with low SEP had more often prevalent and incident T2DM (e.g. low education: HR = 2.13, 95%CI: 1.53-2.97). Low control beliefs and high inferiority beliefs were more common among people with low SEP. Moreover, low control beliefs and high inferiority beliefs were risk factors for T2DM (e.g. low control beliefs: HR = 1.50, 95%CI: 1.08-2.09). The relationship between SEP and T2DM was partially mediated by control beliefs (8.0-13.6%) and inferiority beliefs (2.2-4.5%). We conclude that poor psychosocial resources are important in socioeconomic inequalities in diabetes. Researchers and practitioners should consider the psychosocial profile of people with lower SEP, as such a profile might interfere with the development, treatment, and prevention of T2DM. Further research should explore how poor psychosocial resources interact with chronic stress in relation to socioeconomic health inequalities.

Uniformly strong consistency of the generalized edge frequency polygon density estimator under extended negatively dependent samples

In this paper, we consider a nonparametric density estimator: the generalized edge frequency polygon estimator, which was proposed by Dong and Zheng (Generalized frequency polygon for density estimation. Stat Probab Lett. 2001;55(2):137–145. doi: 10.1016/S0167-7152(01)00077-3). Based on the Bernstein type inequality, we obtain the uniformly strong consistency of the density estimator for extended negatively dependent samples and give the corresponding rate under some mild conditions. The results obtained in the paper generalize and improve the corresponding ones of Xing and Yang (Uniformly strong consistency of frequency polygons for negatively associated samples. Commun Stat Simul Comput. 2015;46(3):2168–2175. doi: 10.1080/03610918.2015.1039129) for frequency polygons in the case of negatively associated samples. Finally, the simulation study are provided to assess the finite sample performance of the theoretical results and a real example is analyzed for illustration.

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.

On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros

Let Πn be the class of algebraic polynomials P of degree n, all of whose zeros lie on the segment [−1,1]. In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: ‖P′‖Lp[−1,1]>c(n)1−1/p+1/q‖P‖Lq[−1,1], P∈Πn, where 0<p≤q≤∞, 1−1/p+1/q≥0 (c>0 is a constant independent of P and n). We show that Zhou’s estimate remains true in the case p=∞, q>1. Some of related Turán type inequalities are also discussed.

Some Hardy and Rellich type inequalities for affine connections

Some Hardy and Rellich type inequalities for affine connections

Proof that a Form of Rubio de Francia’s Conjectured Littlewood-Paley Type Inequality for $$A_{1}\left( {\mathbb {R}}\right) $$-Weighted $$L^{2}\left( {\mathbb {R}}\right) $$ is Valid for Every Even $$A_{1}\left( {\mathbb {R}}\right) $$ Weight

Proof that a Form of Rubio de Francia’s Conjectured Littlewood-Paley Type Inequality for $$A_{1}\left( {\mathbb {R}}\right) $$-Weighted $$L^{2}\left( {\mathbb {R}}\right) $$ is Valid for Every Even $$A_{1}\left( {\mathbb {R}}\right) $$ Weight

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Boundary optimal control problems for parabolic variational inequalities of bilateral obstacle type

This paper is concerned with the optimality system for an optimal control problem governed by parabolic bilateral obstacle problems with Robin boundary conditions. The control-state dynamics is described by variational inequalities and the control on the boundary is considered. Our analysis relies on a regularization method of approximating the variational inequality by a series of semilinear PDEs. The Lipschitz continuity and directionally differentiability in the weak sense for the control-to-state mapping are proved. Then, the necessary optimality conditions for approximate optimal control problems are derived. Finally, the first-order optimality system including complementarity conditions for the original problem is established by a limiting procedure. The result is carefully compared with those obtained in existing literature.

Generalized discrete Grüss and related results with applications

Abstract Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen’s difference, a wide range of estimations are formalised by considering Jensen-Grüss differences.

Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula

Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula

New Quantum Estimates for Midpoint and Trapezoid Type Inequalities Through (α,m)-Convex Functions with Applications

The main goal of current investigation is to present two new q-integral identities for midpoint and trapezoid type inequalities. Then using these identities, we develop several new quantum estimates for midpoint and trapezoid type inequalities via (α, m)-convexity. Some special cases of these new inequalities can be turned into quantum midpoint and quantum trapezoid type inequalities for convex functions, classical midpoint and trapezoid type inequalities for convex functions without having to prove each one separately. Finally, we discuss how the special means can be used to address newly discovered inequalities. 2010 Mathematics Subject Classification. 26D10, 26D15, 26B25.