Sharp variational inequalities for average operators over finite type curves in the plane

Generalized existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature

A sharp volume inequality for mixed LYZ ellipsoids

Sharp radial Sobolev inequalities with inverse square potentials and applications to nonlinear Schrödinger equations

A sharp Sobolev trace inequality of order four on three-balls

Purpose − Social inequality refers to conditions of inequality of access, opportunity and utilization of resources between community groups. This phenomenon can be seen in differences in income, quality of education, infrastructure development, and legal treatment. That issue is also a structural issue in India, a country that still faces sharp inequalities between social classes, rural and urban areas, and powerful groups and marginalized communities. Film media has a strategic role as a means of representing social reality, because it is able to convey critical messages visually and narratively. The Indian biographical film “12th Fail” by Vidhu Vinod Chopra has become an object of study because it tells the story of the struggle of the character Manoj from a poor family in facing an unequal social system. Method − Descriptive qualitative research methods are applied through Roland Barthes' semiotics approach, including denotation, connotation and myth analysis. Primary data are sourced from scenes and film dialogue, while secondary data are from relevant scientific literature. Findings − The results showed four main forms of social inequality, including inequalities in quality and educational facilities, socioeconomic status inequalities, rural and municipal infrastructure inequalities and legal justice inequalities. The representation shows how social systems tend to favour the powerful and limit the social mobility of the poor.

In this paper, we establish, among other results, that if f and g are absolutely continuous functions on [a, b] with values in the complex numbers, satisfying f(a) = 0 and g(b) = 0, then the following inequality holds: integral from a to b of |f(t) g(t)| dt <= (1/4)(b - a)^2 integral from a to b (|f'(t)|^2 + |g'(t)|^2) dt - (1/4) integral from a to b ((t - a)^2 |f'(t)|^2 + (b - t)^2 |g'(t)|^2) dt, provided that the integrals on the right-hand side are finite. In particular, if f(a) = f(b) = 0, then the following sharp inequality holds: integral from a to b |f(t)|^2 dt <= integral from a to b (b - t)(t - a) |f'(t)|^2 dt. We also derive several trapezoid-type and Gruss-type inequalities.

Let 4 be a bounded domain with smooth boundary .In this paper, we establish the following sharp form of the trace Adams inequality in W 2,2 ( ) with zero mean value and zero Neumann boundary condition:Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space 4+ .With this classification result, we can show that S(12 2 ) is attained by using the blow-up analysis and capacitary estimate.As an application, we prove a sharp trace Adams-Onofri-type inequality in general four-dimensional bounded domains with smooth boundary.

This study investigates continuing vocational education and training (CVET) in three institutional contexts: Germany, Sweden, and the United Kingdom. Drawing on PIAAC Cycle 1’s unique questions and transparency, we analyse institutional regimes and socio-economic gradients in CVET participation. Barriers to and engagement in CVET is embedded in educational, labour market, and welfare state institutions. Results show that Sweden’s universal-collectivist regime fosters the highest participation with minimal disparities, while Germany’s corporatist regime shows lower participation and higher stratification. The UK’s liberal-individualist model exhibits moderate participation but sharp socio-economic inequalities, emphasising financial and employer-related barriers. Resource constraints such as time and money are prominent barriers across institutional contexts. Our exploratory measurement of CVET interest opens pathways for future research that was not thought possible with existing PIAAC data. These findings contribute to long-term institutional analysis and theory, and inform policy interventions to reduce barriers, increase interest, and support disadvantaged groups.

This work presents a pair of sharp geometric inequalities that connect the normalized scalar curvature with the generalized normalized δ-Casorati curvature for θ-slant submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). Alongside establishing these estimates, we rigorously describe the geometric conditions under which equality is achieved. The results not only generalize prior findings related to Casorati curvature but also offer new insights into the extrinsic geometry of submanifolds under non-standard connections. To conclude, we propose several open problems that invite further exploration in this direction.

Sharp Inequalities for Rational Functions with Fixed Poles and Constrained Zeros

The concept of a trigonometrically convex function consists in replacing the linear weights inherent in the classical concept of convexity with trigonometric ones. It emerged in 2018, combined with a framework of integral inequalities. In this article, we innovate by establishing valuable bounds for the integral of the product of two functions, including at least one such trigonometrically convex function. To derive these bounds, we use a number of analytical tools, including sharp trigonometric inequalities and the Hermite–Hadamard, Chebyshev and Hölder integral inequalities. Our results are flexible; they can accommodate different convexity conditions. Detailed proofs are given for completeness.

In this paper, we solve the longstanding Gaussian curvature conjecture of a minimal graph S over the unit disk. The conjecture asserts that for any minimal graph above the unit disk, the Gaussian curvature at the point directly above the origin satisfies the sharp inequality |K|<π22. We first reduce the conjecture to the problem of estimating the Gaussian curvature of certain Scherk-type minimal surfaces defined over bicentric quadrilaterals inscribed in the unit disk, containing the origin. We then provide a sharp estimate for the Gaussian curvature of these minimal surfaces at the point above the origin. Our proof employs complex-analytic methods, as the minimal surfaces in question allow a conformal harmonic parameterization.

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion bounds for the mappings and their complex derivatives. The exact region where the log harmonic mappings of the form f(z)=zh(z)h′(z)¯, with h being starlike analytic, give the starlike image is determined. Subordination relations for logarithmic derivatives are established, connecting the mappings with the Schwarz lemma and the Carathéodory class. Furthermore, we obtain the growth estimates for the underlying starlike functions h and their derivatives, as well as accurate inequalities governing the arclength of the circle image under log-harmonic mappings. These findings contribute to the geometric function theory of log-harmonic mappings.

In this article, novel results on the maximality of discrete fractional Green's functions are established and corresponding explicit Lyapunov inequalities for delta fractional systems, with applications to chaos analysis and robust control design, are derived. For the proposed Riemann-Liouville fractional difference system with the delta boundary conditions, explicit expressions for the maximum values of the associated Green's function over its domain are obtained. These results lead to a refined Lyapunov delta-type inequality establishing a necessary condition for the existence of nontrivial solutions, where the lower bound explicitly depends on the maximum values of the fractional order and the Green's function. Furthermore, it is demonstrated that violation of this inequality implies the existence of nontrivial solutions and can induce chaotic behavior in fractional difference systems. For control applications, robust stability conditions for uncertain fractional systems are established and stabilizing state feedback controllers is designed. Finally, the numerical examples validate the emergence of chaos under inequality violation and confirm the control design's efficacy for robust stability.

In many countries around the world, globalization has intensified hyper-consumerism not only among successful people but also among young people, encouraging individuals to use luxury goods as tools for self-expression and distinction. Nigeria’s socio-economic situation – characterized by sharp inequalities, a young population, a wealthy diaspora, and a growing urban elite – further reinforces the symbolic power of luxury. This article analyzes how the consumption of luxury goods functions as a visible indicator of social status in contemporary Nigeria. The article is based on the literature on the subject, ethnographic observations in the digital environment conducted between 2015 and 2025, and an exploratory survey. The research draws on Veblen’s concept of conspicuous consumption, Bourdieu’s trans-substantialization, Lipovetsky’s shift towards experience, and Social Identity Theory. The results of the study indicate that Nigerians increasingly rely on luxury products to signal their membership in high-status groups and gain access to the privileges associated with elevated social position. Social media platforms serve as key arenas for showcasing a luxurious lifestyle, reinforcing status competition and shaping popular notions of success. Respondents associate luxury with quality, prestige, recognition and social advancement, while also using it to reward themselves and satisfy their emotional needs. In Nigeria, buying luxury goods has become an important way of expressing who you are and where you are going. It shows the changes taking place in society and the economy but also reveals the tension between people’s dreams and their everyday lives.

A geometrical approach to the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces

Abstract The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in the convex geometry which forged new connections between projection bodies, centroid bodies, and mixed volume inequalities. We establish several different empirical forms of the Petty projection inequality by re‐examining these key relationships from a stochastic perspective. In particular, we derive sharp extremal inequalities for several multiple‐entry functionals of random convex sets, including mixed projection bodies and mixed volumes.

This ethnographic study analyzes the economic dynamics and cultural resilience of the fishing community in Anyer Beach, a historic coastal area that has undergone profound transformation due to the tourism industry. The purpose of this study is to examine the pattern of livelihood transition from the traditional maritime sector to the service sector, measure the implications of emerging social disparities, and analyze how local wisdom practices are maintained amid the tide of commercialization. Ethnographic qualitative methods were used through participatory observation and in-depth interviews with various parties at the study sites, such as Kamasan Village and Sambolo 2 Beach. The results of the study show a radical transition in livelihoods driven by economic rationality, resulting in a significant surge in per capita monthly income (a wealth shock) of 400% to 900%. However, this rapid economic growth, supported by simple management, simultaneously triggered sharp social inequality and negative lifestyle changes. Culturally, local maritime wisdom, such as the Ruwat Laut ritual, demonstrated resilience, functional adaptation, and commodification as cultural tourism assets. It was concluded that Anyer's economic success was not yet supported by social and ecological sustainability. Therefore, an immediate transition to a structural co-management model, supported by a local regulatory framework, is recommended to more evenly and sustainably distribute tourism benefits.

In this paper, we investigate contact skew CR-warped product submanifolds of locally conformal almost cosymplectic manifolds, a framework that simultaneously generalizes warped product pseudo-slant, semi-slant, and contact CR-submanifolds. We first establish a necessary and sufficient characterization theorem showing that a proper contact skew CR-submanifold with integrable slant distribution admits a local warped product structure if and only if certain shape operator conditions involving the slant angle and the warping function are satisfied. Subsequently, we derive sharp geometric inequalities for the squared norm of the second fundamental form in terms of the warping function, the slant angle, and the conformal factor of the ambient manifold. The equality cases are completely characterized and lead to strong rigidity results, namely that the base manifold is totally geodesic while the slant fiber is totally umbilical in the ambient space. Several applications are presented, showing that our results recover and extend a number of known inequalities and classification theorems for warped product submanifolds in cosymplectic, Kenmotsu, and Sasakian geometries as special cases.