In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η-convex function. Moreover, our results include several new and known results as special cases.

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η- convex function. Moreover, our results include several new and known results as special cases.

We introduce and analyse a simple formal thought experiment designed to reflect a qualitative decision dilemma humanity might currently face in view of climate change. In it, each generation can choose between just two options, either setting humanity on a pathway to certain high wellbeing after one generation of suffering, or leaving the next generation in the same state as this one with the same options, but facing a continuous risk of permanent collapse. We analyse this abstract setup regarding the question of what the right choice would be both in a rationality-based framework including optimal control, welfare economics and game theory, and by means of other approaches based on the notions of responsibility, safe operating spaces, and sustainability paradigms. Despite the simplicity of the setup, we find a large diversity and disagreement of assessments both between and within these different approaches.

This article sets out to deconstruct and challenge the psychologised and pathologising approach that has come to dominate child protection practice in Aotearoa-New Zealand and comparable societies in neoliberal times. Within a risk and protection focused paradigm circumstances and behaviours associated with material deprivation are construed as indicators of heightened danger and harm as opposed to a means of better understanding family life. In this way, although poverty may be classified as an issue that is worthy of attention in the realm of broader economic and social policy, structural inequality is rendered largely irrelevant to the practice of statutory child protection. This article sets out to trouble this construction. It will be argued that understandings of how the effects of material inequality are played out in the lives of children and their families are critical to the development of more effective child protection social work. This ‘life-world’ is generally populated by young women parenting in poverty. Poverty exacerbates the everyday struggle of parenting - it shames and dis-empowers; reducing confidence and perceptions of competence (Gupta, 2015). A paradigm shift is needed. Child protection policy and practice needs to re-engage with the every-day struggles that accompany the lives of socially marginalised families in increasingly stratified late capitalist society. The future of social work in child protection depends on it.

Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality which unifies both the Brascamp-Lieb inequality and Barthe's inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.

This paper models the factors explaining households members economic well-being in Burkina Faso, with a focus on the relative influence of gender inequality in literacy status. It does so, using data from the 2014 survey on household living conditions and a semi-parametric bi-variate sample selection modeling approach. This approach compared to the classic Heckman two-step estimator is methodologically innovative because it deals simultaneously with non-random sample selection using conventional systems of two equations, non-linear covariates' effects using spline approach, and the non-normal bivariate distribution using copula functions. The graphical results from the Lorenz curves combined with the numerical Atkinson and Gini coefficients suggest that inequality in overall per-capita consumption spending among households headed by literates is higher than that of their illiterate counterparts in 2009 and 2014. However, independently of the head of household’s literacy status, the level of inequality in total economic well-being decreased from 2009 to 2014. Using the poverty indices of Watts, Sen, Foster ( alpha= 1)) we found that poverty among households headed by literates is lower than that of their illiterates counterparts for both years, although overall poverty decreased nationally between 2009 and 2014. The results also show that although the gender inequality in literacy status does not translate into inequalities in non-food wellness, it does however for food-wellness as female headed households have 38.9% less per-capita food consumption spending than their men counterparts. Combining both food and non-food consumption spending, total economic well-being also seems to exhibit gender inequality as female headed households now have relatively 26.7% less combined per-capita consumption spending.

In this paper we survey some recent results due to the author concerning various inequalities and approximations for the finite Hilbert transform of a function belonging to several classes of functions, such as: Lipschitzian, monotonic, convex or with the derivative of bounded variation or absolutely continuous. More accurate estimates in the case that the higher order derivatives are absolutely continuous, are also provided. Some quadrature rules with error bounds are derived. They can be used in the numerical integration of the finite Hilbert transform and, due to the explicit form of the error bounds, enable the user to predict a priory the accuracy.

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, let M : [0, 1] → [a, b] be a Mean function given by M (t) = M (t; a, b); where by M (t; a, b) we mean one of the following functions: A_t (a, b) := (1 − t) a + tb, G_t (a, b) = a^1−tb^t and H_t (a, b) := $\frac{ab}{ta+\left(1-t\right)b}=\frac{1}{A_t\left(\frac{1}{a},\frac{1}{b}\right)}$ with the property that M (0; a, b) = a and M (1; a, b) = b. Let I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [a, b] ⊆ I. Consider a non-negative function h : J → (0, ∞), a function f : I → (0, ∞) is said to be h-MN-convex (concave) if the inequality f (M (t; x, y)) ≤ (≥) N (h(t); f (x), f (y)), holds for all x, y ∈ I and t ∈ [0, 1]. In this way, nine classes of h-MN-convex functions are established, and therefore some analytic properties for each class of functions are explored and investigated. Characterizations of each type are given. Various Jensen’s type inequalities and their converses are proved.

In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M and N are denote to Arithmetic, Geometric and Harmonic means and h is a non-negative superadditive or subadditive function.

In China, environmental pollution generated via industrialization as well as the profound changes in the social structure and gradual maturation of the social hierarchy have jointly contributed to the Chinese people's increased environmental consciousness and appeals for environmental justice (EJ). Because of the absence of an EJ theory and a lack of empirical research focused on China, this paper proposes a ‘Pyramid Model’ for China’s EJ research that involves the following three factors: basic demographic and socioeconomic factors, U.S.-based EJ principles, and Chinese characteristics. A nationwide pilot analysis of environmental inequity at the prefecture level is conducted by empirically studying the association of demographics and socioeconomic status with sources of industrial pollution in China. The prefecture-based results are shown to be robust and indicate that areas inhabited by ethnic minorities and western regions in China carry disproportionate environmental burdens. However, a different picture for migrants is presented, revealing that Chinese migrants are not currently exposed to greater levels of industrial pollution, and relevant interpretations of these findings are provided. The results also show that environmental inequality associated with income level, which is observed in the U.S., does not occur in China.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.

In this paper, some convexity properties and some inequalities for the (p,k)-analogue of the Gamma function, Г_p,k(x) are given. In particular, a (p,k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p,k)-analogue of the Riemann zeta function, ζ_p,k(x) is introduced and some associated inequalities are derived. The established results provide the (p,k)-generalizations of some known results concerning the classical Gamma function.

In this study, based on a new quantum integral identity, the authors establish some quantum estimates of Hermite-Hadamard type inequalities for convex functions. These results generalize and improve some known results given in literatures.

In this paper, we establish some inequalities bounding the ratio Г_p,k(x)/Г_p,k(y), where Г_p,k(.) is the (p,k)-analogue of the Gamma function. Consequently, some previous results are recovered from the obtained results.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In a recent paper, Sarikaya et al. introduced a new analogue of the classical Gamma function which they called, the conformable Gamma function. Motivated by their results, this paper seeks to establish some inequalities for the conformable Gamma and Polygamma functions. Among other analytical tools, the procedure relies on the generalized forms of some classical inequalities.

“De-[Constructing] Growth” is offered as a deeper and more useful conceptualization that avoids the negative connotations of, and resistance to, “degrowth” by decoupling profit from unsustainable consumption, production, and inequality.

This study analyses regional development in one of the poorest provinces in China, Guizhou Province, between 2000 and 2012 using a multiscale and multi-mechanism framework. In general, regional inequality has been declining since 2000. In addition, economic development in Guizhou Province presented spatial agglomeration and club convergence, which shows how the development pattern of one core area, two-wing areas and a contiguous area at the edge of the province have been developed between 2006 and 2012. Multilevel regression analysis revealed that industrialization and investment level were the primary driving forces of regional economic disparity in Guizhou Province. The influences of marketization and decentralization on regional economic disparity were relatively weak. Investment level reinforced regional economic disparity and the development of core-periphery structure in the province. However, investment level actually weakened the regional economic disparity in Guizhou Province when the variable of time was considered. In addition, both the topography and urban–rural differentiation were the two main reasons for forming a core-periphery structure in Guizhou Province.

In this paper, we have established Hermite-Hadamard inequalities for r-preinvex functions via fractional integrals.