Some new quantum derivatives and integrals with their applications in integral error bounds

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

An integral inequality for convex functions is deduced from Jensen's inequality. This gives as a special case a commonly-used inequality in the analysis of call congestion in queueing theory which has previously been derived only by rather longer ad hoc procedures.

In this paper, some new integral inequalities and their applications to special means of real numbers will be given using generalized Hermite-Hadamard’s integral inequalities and Simpson’s type inequalities holding for convex functions. Our results presented here would provide extensions of those given in earlier works.

In the paper, with the help of two known integral identities and by virtue of the classical Hölder integral inequality, the authors establish several new integral inequalities of the Hermite–Hadamard type for convex functions. These newly established inequalities generalize some known results.

The aim of this paper is to establish some generalized integral inequalities for convex functions of 2-variables on the co-ordinat. Then, we will give a generalized identity and with the help of this integral identity, we will investigate some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities involving Riemann integrals and Riemann-Liouville fractional integrals.

In this paper, we introduced an advanced family of numerical composite integration formulas of closed Newton–Cotes-type that uses the function values on uniformly spaced intervals only without any derivative values. To increase the accuracy, we divide the given interval into a number of equal subintervals and integrating on each interval by using integration rules with abscissas outside integration interval. Since there are more unknowns when using including function values outside integration interval in addition to function values of on interval, the order of accuracy of these numerical integration formulas is higher than the standard closed Newton-Cotes formulae. These new formulae are obtained using the method of undetermined coefficients which are based on the concept of the precision of the quadrature formula. The error terms are found using the concept of precision. Also, we compared the errors in an advanced family of numerical composite integration formulas with the errors in composite closed Newton–Cotes-type. Finally, we have presented some examples and then mentioned the related MATLAB codes.

The goal of this article is to establish some fractional proportional integral inequalities for convex functions by employing proportional fractional integral operators. In addition, we establish some classical integral inequalities as the special cases of our main findings.

The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some special cases several new quantum estimations for q-midpoints and q-trapezoidal-type inequalities are derived with an example. The results obtained could be applied in the optimization of several economic geology problems.

New Hermite–Hadamard type integral inequalities for convex functions and their applications

We consider the modified Hermite–Hadamard inequality and related results on integral inequalities, in the context of fractional calculus using the Riemann–Liouville fractional integrals. Our results generalize and modify some existing results. Finally, some applications to special means of real numbers are given. Moreover, some error estimates for the midpoint formula are pointed out.

In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature.

The authors establish some new inequalities for differentiable convex functions, which are similar to the celebrated Hermite-Hadamard's integral inequality for convex functions, and apply these inequalities to construct inequalities for special means of two positive numbers.

In this paper, by using some classical inequalities from the Theory of Inequality and integral identity, we establish two new general inequalities. Our results have some relationships with certain integral inequalities obtained by Sarikaya and Aktan.

The solution to numerical integration problems generally can be achieved using mesh methods. However, mesh methods, commonly known as trapezoidal, rectangular, and midpoint, only apply to Cartesian coordinates. Therefore, this research develops a mesh method that can be used for numerical integration in polar coordinates, specifically using Triangle shapes. This study also analyzes the errors from the results of the Triangle mesh method and provides examples and visualizations of applying the Triangle mesh method to solve numerical integration problems. The steps of this research are as follows: first, determining the form of the integral in the problem of numerical integration in polar coordinates. Then, the area bounded by the curve is divided into several parts, each approximated by a triangle. Next, a numerical integration formula of the triangle mesh method is created by summing the areas of each triangle. After that, the resulting error of the triangle mesh method is analyzed using the Taylor series. Finally, proving that the results of the triangle mesh method approximate the area bounded by the curve in polar coordinates. From this research, the numerical integration formula for the Triangle mesh method is obtained, the error formula with a second-order approximation degree, and based on the proof of the numerical integration formula for the Triangle mesh method, it is concluded that as the number of triangles approaches infinity, the results of the Triangle mesh method will converge to the exact area bounded by the curve.

The construction of numerical integration rules of degree three for product regions