Double Inequality Research Articles

Applications of Differential Inequalities Employing A New ConvolutedOperator Constructed by The Supertrigonometric Function

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea ofsubordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem.

Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

Gender-Based Double Standards and Inequalities in Online Culture: Objectification of Feminine Sexuality, Self-Expression and Appearance, and Cybershaming

Online communities provide a place that facilitates netizens to share their thoughts and ideas freely and swiftly with the speed of a click; this is relatively free and more convenient compared to traditional methods of socialisation. However, the online world can still reflect the traditional community to some extent. Especially with gender-related topics, women are the most often recipients of cyber shaming. This essay will explore the topic of gender-based double standards both online and offline, focusing on cyber shaming received pertaining to women's outfit choices. The essay argues that the cyber shaming of women's outfit choices is not just a matter of individual taste or preference but is rooted in deeply ingrained gender stereotypes and expectations that constrain women's freedom of expression and limit their opportunities for self-determination. This essay will use qualitative and quantitative research methods to investigate the impact of outfits on criticism directed at women and the implications of gender-based double standards. By shedding light on the gender-based double standards that underpin cyber shaming, the essay aims to raise awareness and promote a more equitable and inclusive online culture that values diversity, respect, and empowerment for all. Moreover, this essay demonstrates that online cyberbullying can cause significant consequences in the offline world; some can even cost lives. There is also a need for progressive changes in criminal and privacy law that promote gender equity and prevent cyber-SV against EYW. The implications of these results for academics and practitioners are discussed, and directions for future research are outlined. Throughout the conducted research on three designed inquiry questions, this study concluded with three key findings: Outfit-based content manifests judgments and criticism in the comment to be harsher for women; the impact of cyberbullying on emerging young women is greater than at other ages; and women to some extent have aggravated the effect of the misogyny complex.

Jordan-Type Inequalities and Stratification

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants.

Double asymptotic inequalities for the generalized Wallis ratio

Asymptotic estimates for the generalized Wallis ratio \(W^*(x):=\frac{1 {\sqrt{\pi}}\cdot\frac{\Gamma(x+\frac{1}{2})}{\Gamma(x+1)}\) are presented for \(x\in\mathbb{R}^+\) on the basis of Stirling's approximation formula for the \(\Gamma\) function. For example, for an integer \(p\ge2\) and a real \(x>-\tfrac{1}{2}\) we have the following double asymptotic inequality \[A(p,x)\,<\,W^*(x)\,<\,B(p,x),\] where \begin{align*} A(p,x):=& W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{379(x+p)^3}\right), \\ B(p,x):= & W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{191(x+p)^3}\right),\\ W_p(x):=& \frac{1}{\sqrt{\pi\,(x+p)}}\cdot\frac{(x+1)^{(p)}}{(x+\frac{1}{2})^{(p)}}, \end{align*} with \(y^{(p)}\equiv y(y+1)\cdots(y+p-1)\), the Pochhammer rising (upper) factorial of order \(p\).

Quality and inequality in pre-primary and home environment inputs to early childhood development in Egypt

By the time children start primary school, large socioeconomic disparities are evident in their learning and development. Both pre-primary and home environments can play important roles in influencing school readiness and can contribute to disparities in early childhood development, but there is limited evidence on their relative roles in the Middle East and North Africa. This paper examines how pre-primary quality, stimulation at home, and early childhood development vary by socioeconomic status for pre-primary students in Egypt. The results demonstrate substantial socioeconomic inequality in stimulation at home, more so than in pre-primary quality and inputs, although there is variation in the degree of inequality across different dimensions of pre-primary quality. “Double inequality” is observed, where students with less stimulating home environments experience slightly lower quality pre-primary inputs. There are particularly large pre-primary inequities in structural quality (physical environment) and less inequity in process quality (pedagogy). These results suggest that targeted investments in pre-primary education in Egypt are necessary to reduce inequality in school readiness but are likely insufficient to close the socioeconomic status gap in children's development. Investing in interventions to improve vulnerable children's home learning environments, as well as investing in quality pre-primary, is critical to address disparities in children's development.

A novel nonzero functional method to extended dissipativity analysis for neural networks with Markovian jumps

<abstract><p>This paper explored the topic of extended dissipativity analysis for Markovian jump neural networks (MJNNs) that were influenced by time-varying delays. A distinctive Lyapunov functional, distinguished by a non-zero delay-product types, was presented. This was achieved by combining a Wirtinger-based double integral inequality with a flexible matrix set. This novel methodology addressed the limitations of the slack matrices found in earlier research. As a result, a fresh condition for extended dissipativity in MJNNs was formulated, utilizing an exponential type reciprocally convex inequality in conjunction with the newly introduced nonzero delay-product types. A numerical example was included to demonstrate the effectiveness of the proposed methodology.</p></abstract>

Some New Improvements for Fractional Hermite–Hadamard Inequalities by Jensen–Mercer Inequalities

This article’s objective is to introduce a new double inequality based on the Jensen–Mercer (JM) inequality, known as the Hermite–Hadamard–Mercer inequality. We use the (JM) inequality to build a number of generalized trapezoid‐type inequalities. Moreover, in addition to the (JM) inequality, we also use the Hölder inequality and the power mean inequality. Finally, a few examples are given to highlight the main points of our outcomes.

A new proof of a double inequality of Masjed-Jamei type

<abstract><p>In this paper, we provide a new simple proof of a double inequality of Masjed-Jamei type proved by L. Zhu <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

Composite resilient state tracking and disturbance rejection control for periodic piecewise polynomial delay systems with multiform disturbances

In this study, we scrutinize the problems of resilient state tracking control (RSTC) and disturbance rejection (DR) for periodic piecewise polynomial delay systems in the presence of multiform disturbances and gain fluctuations. Specifically, we address both exosystem-induced disturbances and norm-bounded disturbances. Firstly, to tackle effectively the disturbances that are produced by exosystems, the disturbance observers are tailored, which estimates the disturbance in a precise manner. Secondly, the repercussions of norm-bounded disturbances can be lessened by accounting an extended dissipative performance. Moreover, by amalgamating the outcomes of the disturbance observer with the feedback-based tracking control, a DR-based RSTC is constructed. In addition, some parameter variations in the controller gain are taken into consideration in order to guarantee resiliency against unexpected flaws. Besides, the adequate conditions are derived in the form of linear matrix inequalities by making use of Lyapunov stability theory, new double integral inequality and extended Wirtinger’s inequality. Furthermore, the relation for acquiring the controller and observer gain matrices is formulated based on the established criteria. Finally, a numerical example with simulation depiction is presented to show the favour performance and capability of the designed controller.

Strong unique continuation and global regularity estimates for nanoplates

In this paper, we analyze some properties of a sixth-order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry, we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality by a Carleman estimates approach.

Discrete analytical objects in the body-centered cubic grid

We propose a characterization of discrete analytical spheres, planes and lines in the body-centered cubic (BCC) grid, both in the Cartesian and in the recently proposed alternative compact coordinate system, in which each integer triplet addresses some voxel in the grid. We define spheres and planes through double Diophantine inequalities and investigate their relevant topological features, such as functionality or the interrelation between the thickness of the objects and their connectivity and separation properties. We define lines as the intersection of planes. The number of the planes (up to six) is equal to the number of the pairs of faces of a BCC voxel that are parallel to the line.

On Inequalities for the Ratio of v-Gamma and v-Polygamma Functions

In this paper, the author presents some double inequalities involving a ratio of v-Gamma and v-polygamma functions. The approach makes use of the log-convexity property of v-Gamma function and the monotonicity property of v-polygamma function. Some of the results also give generalizations and extensions of some previous results.

Sharp inequalities related to the volume of the unit ball in mathbb{R}^{n}

Let Omega _{n}=pi ^{n/2}/Gamma (frac{n}{2}+1) (n in mathbb{N}) denote the volume of the unit ball in mathbb{R}^{n}. In this paper, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions is presented, which yields a sharp double inequality for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}). Also, we establish new sharp inequalities for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}).

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed.

Three-dimensional exponential sums under constant perturbation

By generalizing Bombieri and Iwaniec’s double large sieve inequality, we obtain an estimation on three-dimensional exponential sums with constant perturbation. As an application of our estimation on the three-dimensional exponential sums, we get that for any ϵ>0, ∑n⩽xΛ([xn])=x∑d⩾1Λ(d)d(d+1)+Oϵ(x1736+ϵ) as x→∞. This improves a recent result of Liu–Wu–Yang, which requires 9/19 in place of 17/36.

Ensuring Equality in Education Access: The Dynamics and Challenges for Vietnamese Ethnic Minority Girls

Education access is one of the critical human rights that has been decided for the development of everyone and has been codified in the international and national legal systems. Despite important achievements in ensuring and promoting fundamental human rights in general, with the characteristics of being one of the most vulnerable in society, ethnic minority girls experienced discrimination and double inequality in benefiting their rights, including access to education. Regarding the education right of girls in the ethnic minority in Vietnam today, the following article focuses on the following matters: (i) Analysing the Vietnamese legal framework in ensuring the education right of ethnic minority girls; (ii) Presenting the difficulties and challenges in ensuring this right for ethnic minority girls in Vietnam; (iii) Proposing necessary recommendations for the Government to improve the enjoyment of the right for Vietnamese ethnic minority girls in the upcoming time.
 Keywords: Education Access, Equality, Ethnic Minority Girls, Vietnam, Women's rights

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Improved Integral Inequality Based on Free Matrices and Its Application to Stability Analysis of Delayed Neural Networks via Matrix-valued Cubic Polynomial Inequality

This paper deals with the stability analysis of delayed neural networks (DNNs). Firstly, an improved integral inequality based on free matrices (I3BFM) is newly proposed. The I3BFM gives the upper bound of the integral quadratic form including the state-related vector stacked by the state, its derivative, and its integration by introducing some free matrices. The upper bound of the I3BFM fully reflects the information not only on each term in the state-related vector but also on the cross-terms between the three terms. Secondly, the double integral Lyapunov-Krasovskii functional (LKF) of the integral quadratic form including the state-related vector is established to utilize the proposed I3BFM in the stability analysis of DNN. From the derivative of the double integral LKF, a matrix-valued cubic polynomial on a time-varying delay is generated in the process of stability analysis. Therefore, a matrix-valued cubic polynomial inequality is applied to make the matrix-valued cubic polynomial numerically tractable. Finally, the relaxed stability criteria utilizing the I3BFM and the double integral inequality are derived, and the superiority of the derived stability criteria is demonstrated by two well-known numerical examples.

A double inequality for the Apéry constant

A double inequality for the Apéry constant