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In this paper, we establish some new Bullen-type inequalities for differentiable convex functions using the generalized fractional integrals. The main advantage of the inequalities and operators used to obtain them is that these inequalities can be turned into some existing inequalities for Riemann integrals and new inequalities for Riemann–Liouville fractional integral inequalities and k -fractional integrals. Finally, we add some applications of special means of real numbers using the newly established inequalities to make these results more interesting. [ABSTRACT FROM AUTHOR]

In this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann–Liouville fractional integral operators and limited second derivative mappings. These established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. Thus, a generalization of many inequalities and new results were obtained. Moreover, some examples of obtained inequalities are given for better understanding by the reader. Furthermore, the theoretical results are supported by graphs in order to illustrate the accuracy of each of the inequalities obtained according to the specific choices of the parameter. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

Abstract In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for ( s + 1 ) $(s+1)$ -convex functions involving the ( k , s ) $(k,s)$ -Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities.

In this paper we establish some Ostrowski and trapezoid type inequalities for the $k$-$g$-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results. [ABSTRACT FROM AUTHOR]

In this paper we establish some Ostrowski and trapezoid type inequalities for the Riemann-Liouville fractional integrals of absolutely continuous functions with bounded derivatives. Applications for mid-point and trapezoid inequalities are provided as well. They generalize the known results holding for the classical Riemann integral. Some examples for convex functions are also given. [ABSTRACT FROM AUTHOR]

In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for (s + 1) -convex functions involving the (k , s) -Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson’s inequalities for differentiable convex functions. Finally, we give some applications of newly established inequalities in the context of quadrature formulas.

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter m∈[0,1]. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable (α,m)-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the qϰ2-quantum integral to show midpoint and trapezoidal inequalities for qϰ2-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via qϰ1-quantum integral is not valid for preinvex functions, and we present its proper form. We use qϰ1-quantum integrals to show midpoint inequalities for qϰ1-differentiable preinvex functions. It is also demonstrated that by considering the limit q→1− and ηϰ2,ϰ1=−ηϰ1,ϰ2=ϰ2−ϰ1 in the newly derived results, the newly proved findings can be turned into certain known results.

Let g be a strictly increasing function on (a; b) ; having a continuous derivative g0 on (a; b) : For the Lebesgue integrable function f : (a, b) C, we define the k-g-left-sided fractional integral of f by ... and the k-g-right-sided fractional integral of f by ... where the kernel k is defined either on (0;1) or on [0, 1) with complex values and integrable on any finite subinterval. In this paper we establish some new inequalities for the k-g-fractional inte- grals of functions of bounded variation. Examples for the generalized left- and right-sided Riemann-Liouville fractional integrals of a function f with respect to another function g and a general exponential fractional integral are also provided. [ABSTRACT FROM AUTHOR]

Let g be a strictly increasing function on (a, b), having a continuous derivative g' on (a; b) : For the Lebesgue integrable function ... In this paper we establish some inequalities for the k-g-fractional integrals of integrable functions satisfying some boundedness conditions. Further bounds for absolutely continuous functions whose derivatives also satisfy some boundedness conditions are given as well. Examples for a general exponential fractional integral are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson's inequalities for differentiable convex functions. Finally, we give some applications of newly established inequalities in the context of quadrature formulas. [ABSTRACT FROM AUTHOR]

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter m ∈ [ 0 , 1 ] . Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable (α , m) -convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field. [ABSTRACT FROM AUTHOR]

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the q ϰ 2 -quantum integral to show midpoint and trapezoidal inequalities for q ϰ 2 -differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via q ϰ 1 -quantum integral is not valid for preinvex functions, and we present its proper form. We use q ϰ 1 -quantum integrals to show midpoint inequalities for q ϰ 1 -differentiable preinvex functions. It is also demonstrated that by considering the limit q → 1 − and η ϰ 2 , ϰ 1 = − η ϰ 1 , ϰ 2 = ϰ 2 − ϰ 1 in the newly derived results, the newly proved findings can be turned into certain known results. [ABSTRACT FROM AUTHOR]