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This article presents some new inequalities of Simpson’s type for differentiable functions by using $(\alpha ,m) $-convexity. Some results for concavity are also obtained. These new estimates improve on the previously known ones. Some applications for special means of real numbers are also provided.

In this research article, the authors have presented the modelling of quintessence compact stars, which satisfies the Karmarkar conditions. For this purpose, we have formulated the set of Einstein field equations with the static metric, anisotropic perfect fluid, and quintessence field. The equation of state [p.sub.r] = [alpha][rho] and Karmarkar condition have been used to solve the set of field equations. The unknown constant in the metric functions (appearing due to the Karmarkar conditions) have been found by matching the interior metric with the Schwarzschild exterior metric. The observed value of mass and radius of some well-known classes of stars has been used. The fluid variables density, radial and transverse pressures, and anisotropic parameter have been plotted graphically. The first and second derivatives of density and radial pressure have been evaluated to discuss the regularity of the model. The speed of sound for the radial and transverse directions determines the stability of the proposed model. Moreover, the redshift for the proposed model of the star has been discussed. Key words: general relativity, compact star, embedding class, anisotropy. Nous presentons une modelisation d'etoiles compactes quintessences qui satisfont la condition de Karmarkar. A cette fin, nous formulons les equations de champ d'Einstein avec une metrique statique, un fluide parfait anisotrope et un champ de quintessence. Nous utilisons l'equation d'etat [p.sub.r] = [alpha][rho] et la condition de Karmarkar pour resoudre les equations de champ. Les constantes inconnues dans les fonctions de la metrique (apparaissant a la suite de la condition de Karmarkar) sont determinees en ajustant la metrique interne avec la metrique exterieure de Schwarzschild. Nous utilisons les valeurs de masse et de rayon d'une certaine classe bien connue d'etoiles. Nous presentons de facon graphique les variables de densite, de pressions radiale et transverse et du parametre d'anisotropie. Nous calculons les premiere et deuxieme derivees de la densite et de la pression radiale dans une discussion de la stabilite du modele. Les valeurs de la vitesse du son, dans les directions radiale et transverse, determinent la stabilite du present modele. Nous analysons aussi le deplacement vers le rouge pour le modele d'etoile presente ici. [Traduit par la Redaction] Mots-cles: relativite generale, etoiles compactes, champ de quintessence, condition de Karmarkar, sources anisotropes., 1. Introduction A century ago, the first exact solution of the Einstein field equations was gained by Karl Schwarzchild [1]. After this, a lot of exact solutions have been represented [...]

The advancement in coloring schemes of graphs is expanding over time to solve emerging problems. Recently, a new form of coloring, namely P 3 -coloring, was introduced. A simple graph is called a P 3 -colorable graph if its vertices can be colored so that all the vertices in each P 3 path of the graph have different colors; this is called the P 3 -coloring of the graph. The minimum number of colors required to form a P 3 -coloring of a graph is called the P 3 -chromatic number of the graph. The aim of this article is to determine the P 3 -chromatic number of different well-known classes of bipartite graphs such as complete bipartite graphs, tree graphs, grid graphs, and some special types of bipartite graphs. Moreover, we have also presented some algorithms to produce a P 3 -coloring of these classes with a minimum number of colors required. [ABSTRACT FROM AUTHOR]

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly -convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means. [ABSTRACT FROM AUTHOR]

Using Atangana-Baleanu (AB) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown. [ABSTRACT FROM AUTHOR]

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring P 2 coloring, where P 2 is a path graph. However, there are situations in which this type of coloring cannot give us the solution to the problem at hand. To answer such questions, in this article, we introduce a novel graph coloring called P 3 coloring. A graph is called P 3 -colorable if we can assign colors to the vertices of the graph such that the vertices of every P 3 path are distinct. The minimum number of colors required for a graph to have P 3 coloring is called the P 3 chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the P 3 chromatic number for different symmetric families of graphs. [ABSTRACT FROM AUTHOR]

This article is aimed at studying novel generalizations of Hermite-Mercer-type inequalities within the Riemann-Liouville k -fractional integral operators by employing s -convex functions. Two new auxiliary results are derived to govern the novel fractional variants of Hadamard-Mercer-type inequalities for differentiable mapping Ψ whose derivatives in the absolute values are convex. Moreover, the results also indicate new lemmas for Ψ ′ , Ψ ′ ′ , and Ψ ′ ′ ′ and new bounds for the Hadamard-Mercer-type inequalities via the well-known Hölder's inequality. As an application viewpoint, certain estimates in respect of special functions and special means of real numbers are also illustrated to demonstrate the applicability and effectiveness of the suggested scheme. [ABSTRACT FROM AUTHOR]

In the present article, we establish an integral identity for Riemann-Liouville fractional integrals. Some Simpson type integral inequalities utilizing this integral identity are obtained. It is worth mentioning that the presented results have close connection with those in [M. Z Sarikaya, E. Set, M. E Ozdemir, On new inequalities of Simpson's type for s-convex functions, Computers and Mathematics with Applications, 60 (2010), 2191{2199)]. [ABSTRACT FROM AUTHOR]

In this article, we establish some new Hermite-Hadamard type integral inequalities for (s;m)-convex functions, utilizing the fractional integrals. We observe and prove that these results have some relationship with the obtained results of [10, 11]. [ABSTRACT FROM AUTHOR]

In this short note, a Riemann-Liouville fractional integral identity including first order derivative of a given function is established. With the help of this fractional-type integral identity, some new Hermite-Hadamard-type inequality involving Riemann-Liouville fractional integrals for(m; h1; h2)-convex function are considered. Our method considered here may be a stimulant for further investigations concerning Hermite-Hadamard-type inequalities involving fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper,we have established some Hermite-Hadamard in-equalities for preinvex functions via fractional integrals and these results have some relationship with the obtained results. Application of the obtained results are given as well. [ABSTRACT FROM AUTHOR]