Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been ...actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field.
The current reprint presents studies of uncertain variational problems. It aims to develop the research in this field by formulating and demonstrating some characterization results of well-posedness ...and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations.
The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p-Laplacian. The generalized hyperbolic functions are defined similarly. Some ...classical inequalities for trigonometric and hyperbolic functions, such as Mitrinović–Adamović’s inequality, Lazarević’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.
In this paper, we study on ℝ 2 some new types of the sharp subcritical and critical Trudinger-Moser inequality that have close connections to the study of the optimizers for the classical ...Trudinger-Moser inequalities. For instance, one of our results can be read as follows: Let 0 ≤ β < 2, p ≥ 0, α ≥ 0. Then sup ∇ u 2 2 + u 2 2 ≤ 1 u 2 p ∫ ℝ 2 exp α 1 − β 2 u 2 u 2 d x x β < ∞ if and only if α < 4π or α = 4π, p ≥ 2. The attainability and inattainability of these sharp inequalties will be also investigated using a new approach, namely the relations between the supremums of the sharp subcritical and critical ones. This new method will enable us to compute explicitly the supremums of the subcritical Trudinger-Moser inequalities in some special cases. Also, a version of Concentration-compactness principle in the spirit of Lions ( Lions, I. Rev. Mat. Iberoam. 1(1) 145–01 1985) will also be studied.
Tackling inequalities in health and healthcare is more important than ever. The COVID-19 pandemic starkly illustrated the disproportional impact of the virus on those who already faced disadvantage ...and discrimination. Moreover, there is evidence that the public health measures taken to contain the virus are likely to have longstanding differential impacts across populations. Numerous studies have documented avoidable differences in health, within and between populations. Similarly, studies have consistently shown inequalities in access, use, experience and outcomes from healthcare and public health programmes. The focus has often been on individual determinants, such as gender, age and ethnicity. Less attention has been paid to structural or contextual determinants, except for area-level socioeconomic conditions. In addition, to tackle inequalities, there is a need to move beyond measuring; to understand why inequalities arise and how they can be addressed. This Special Issue sought to extend the parameters of inequalities research in health and healthcare beyond measuring and documenting inequalities. Reviews, observational studies, and quasi-experimental and other evaluation designs (using quantitative, qualitative or mixed methods), focusing on the following were welcomed: • understanding inequalities across health and care systems; • methodological developments to understand drivers of inequalities; • efforts to reduce inequalities, particularly in evidence-based healthcare or public health policy and practice; • understanding and mitigating the adverse impact of the COVID-19 pandemic on inequalities.
We introduce the log-h-convex concept for the interval-valued functions (IVFs) and establish some of the new Jensen's, Hadamard's, and Hadamard-Fejér's inequalities for this kind of functions, which ...generalize some known results. Meanwhile, some interesting examples are given.
We prove a subset of inequalities of Caffarelli–Kohn–Nirenberg type in the hyperbolic space H N , N ≥ 2 , based on invariance with respect to a certain nonlinear scaling group, and study existence of ...corresponding minimizers. Earlier results concerning the Moser–Trudinger inequality are now interpreted in terms of CKN inequalities on the Poincaré disk.
In this paper, we prove a Poincaré inequality in Musielak–Orlicz spaces under the assumption that the Musielak function depends only on N-1 coordinates of the spatial variable x. We will use this ...inequality to show the existence of solutions for some elliptic inequalities with lower order terms and L1 data in Musielak–Orlicz spaces.
The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the ...Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.
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