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In this paper, we introduce the notion of strongly generalized convex functions which is called as strongly η-convex stochastic processes. We prove the Hermite-Hadamard, Ostrowski type inequality, and obtain some important inequalities for above processes. Some previous results are special cases of the results obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper "Monotonicity properties for a ratio of finite many gamma functions" published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value. [ABSTRACT FROM AUTHOR]

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences. [ABSTRACT FROM AUTHOR]

The main result of this paper is to establish a weighted second-order Adams-type inequality on the whole set of R 4 . As an application of this result, we prove the existence of a solution for a Kirchhoff-type equation involving non-linearity with subcritical or critical exponential growth. In the critical case, the associated energy loses its compactness. To avoid this problem, we add an asymptotic condition to the nonlinearity [ABSTRACT FROM AUTHOR]

Unfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it "earthquake convexity". In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Hölder, Hölder–İşcan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Hölder–İşcan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided. [ABSTRACT FROM AUTHOR]

In this paper, by considering the identity established by Luo et al. in [C. Luo, T.-S. Du, M. Kunt and Y. Zhang, Certain new bounds considering the weighted Simpson-like type inequality and applications, J. Inequal. Appl. 2018 2018, Paper No. 332] and under the assumption of the quasi-convexity of the first derivative, we establish some new error estimates of the Simpson-like type inequalities. We also discuss the case where the first derivative satisfies the Hölder condition. At the end, we provide some applications to special means. The obtained results represent a continuation of the above-mentioned paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder's inequality, chain rule on time scales, and the mean inequality. As particular cases of our results (when T = N and T = R ), we get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying and Cheung (Math. Slovaca 61(1):15–28, 2011). [ABSTRACT FROM AUTHOR]

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

The class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including fundamental inequalities, supermultiplicative type inequalities, Jensen-Mercer inequality, integral inequalities, and refined forms. The obtained results extend some celebrated results from the context of convexity to geometric convexity, with interesting applications to numerical inequalities for the hyperbolic and exponential functions. [ABSTRACT FROM AUTHOR]

In this paper, a new integral identity is provided. Based on this equality, Simpson-type dual integral inequalities for functions whose first derivatives are s-convex via Riemann–Liouville fractional integrals are established. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers' comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

The authors of the paper suggest a novel approach in order to examine an integral equality using conformable fractional operators. By using this identity, some Newton-type inequalities are proved for differentiable convex functions by taking the modulus of the newly established equality. Moreover, we prove some Newton-type inequalities by using the Hölder and power-mean inequality. Furthermore, some new results are presented by using special choices of obtained inequalities. Finally, we give some conformable fractional Newton-type inequalities for functions of bounded variation. [ABSTRACT FROM AUTHOR]

As a first attempt, we obtain the boundedness of the rough fractional Hausdorff operator on variable exponent Herz-type spaces. The method used in this paper enables us to study the operator on some other function spaces with variable exponents. [ABSTRACT FROM AUTHOR]

AbstractThis paper presents new simple lower and upper bounds for the cumulative normal distribution function, Φ(z). The accuracy and closeness of the proposed bounds to the exact Φ(z) are investigated based on the maximum absolute error and the mean absolute error. It is found that the maximum absolute error of the proposed lower bound is 8.55×10−3 and it is 4.1×10−4 for the upper bound. In addition, based on 5001 values between z=0 and z=5 with step 0.001, we found that the mean absolute error is 3.27×10−3 for the lower bound and it is 1.1×10−4 for the upper bound and these two values decrease with increasing the z value. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the boundedness of Volterra integral operators in Lebesgue spaces in the case 1 < q < p < ∞ . These operators belong to the classes O 2 ± satisfying less restrictive conditions than the previously studied classes O 1 ± , which we also discuss in the paper. For operators from the classes O 1 ± we prove alternative boundedness criteria. [ABSTRACT FROM AUTHOR]

In this paper we introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen's integral type inequality in this setting are given. Several examples are also involved. [ABSTRACT FROM AUTHOR]

In this paper we prove a new variant of q-Hermite–Hadamard–Mercer-type inequality for the functions that satisfy the Jensen–Mercer inequality (JMI). Moreover, we establish some new midpoint- and trapezoidal-type inequalities for differentiable functions using the JMI. The newly developed inequalities are also shown to be extensions of preexisting inequalities in the literature. [ABSTRACT FROM AUTHOR]

In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions. [ABSTRACT FROM AUTHOR]

Convexity is a fundamental concept in analysis. Over the past few decades, many significant error bounds have been established for various quadrature rules using different types of convexity. This paper focuses on the Gauss–Radau quadrature formula. Initially, we introduce a novel identity related to 2-point left Radau-type rule. Next, we derive several integral inequalities for functions whose first derivatives are s-convex in the second sense. Finally, we present applications to special means to demonstrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

This paper aims to extend the application of the Ostrowski inequality, a crucial tool for figuring out the error bounds of various numerical quadrature rules, including Simpson's, trapezoidal, and midpoint rules. Specifically, we develop a more comprehensive class of Ostrowski-type inequalities by utilizing the weighted version of Riemann–Liouville (RL) fractional integrals on an increasing function. We apply our findings to estimate the error bounds of Hadamard-type inequalities. Our results are more comprehensive, since we obtain the results of the existing literatures as particular cases for certain parameter values. This research motivates researchers to apply this concept to other fractional operators. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired. [ABSTRACT FROM AUTHOR]

Based on the uncertainty theory, Liu [B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 2009, 1, 3–10] introduced an uncertain integral for applying uncertain differential equation, finance, control, filtering and dynamical systems. Since uncertain integrals are the important content of uncertainty theory, this paper explores an approach of the relationship between uncertain integrals by the well-known Chebyshev-type inequality. Also, we propose the concept of an uncertain fractional integral which is generalized version of an uncertain integral. The definition of a strong comonotonic uncertain process and some new properties of the uncertain integral were presented in [C. You and N. Xiang, Some properties of uncertain integral, Iran. J. Fuzzy Syst. 15 2018, 2, 133–142]. Based on the strong comonotonic uncertain process, as an application, we provide Chebyshev's inequality for a fractional uncertain integral and an uncertain integral. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the concept of multiplicatively p-convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p-convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p-convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas. [ABSTRACT FROM AUTHOR]

In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h (v (x) ln n) and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k λ (v (x) , ln n) . A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications. [ABSTRACT FROM AUTHOR]

In this paper, we establish several new inequalities for q-differentiable coordinated convex functions that are related to the right side of Hermite–Hadamard inequalities for coordinated convex functions. We also show that the inequalities proved in this paper generalize the results given in earlier works. Moreover, we give some examples in order to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new (p , q) -integral inequalities of Simpson's second type for preinvex functions. Many results given in this paper provide generalizations and extensions of the results given in previous research. Moreover, some examples are given to illustrate the investigated results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an equality involving tempered fractional integrals for twice-differentiable functions. By using this equality, we establish several left Hermite–Hadamard-type inequalities for the case of tempered fractional integrals. Moreover, we derive our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish some new generalized Gamidov type integral inequalities involving a $\psi $-fractional operator. We also give two applications to substantiate the validity of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions). [ABSTRACT FROM AUTHOR]

Based on a known sharpened and refined versions of a classical Steffensen inequality, we derive new sharpened and refined versions of a q–Steffensen inequality. Next, we give weaker conditions for some already known generalizations of q–Steffensen's inequality and for new refinements obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new Hermite-Hadamard type inequality by using post-quantum calculus for s-preinvex function and establish a new (p,q)-integral identity to derive some new Ostrowski type inequality for (p,q)-differentiable function. We give some application to special means for our newly proved inequalities. [ABSTRACT FROM AUTHOR]

In this paper, with the help of Green's function and Hermite interpolating polynomial, an extension of Jensen's functional for n-convex functions is deduced from Jensen's inequality involving diamond integrals. Special Hermite conditions, including Taylor two point formula and Lagrange's interpolation, are also deployed to find the further extensions of Jensen's functional. This paper also includes discussion on bounds for Grüss inequality, Ostrowski inequality, and Čebyšev functional associated to the newly defined Jensen's functional. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain an identity for differentiable mappings. Then, we establish some new generalized inequalities for differentiable convex functions involving some parameters and generalized fractional integrals. We show that these results reduce to several new Simpson-, midpoint- and trapezoid-type inequalities. The results given in this study are the generalizations of results proved in several earlier papers. [ABSTRACT FROM AUTHOR]

In this paper, we explore a class of Hermite–Hadamard integral inequalities for convex and m-convex functions. The Hölder inequality is used to create this class, which has a wide range of applications in optimization theory. Some trapezoid-type inequalities and midpoint error estimates are investigated. Inequalities for several q-special functions are highlighted. As particular cases, we have included several previous results. [ABSTRACT FROM AUTHOR]

Let Ω n = π n / 2 / Γ (n 2 + 1) (n ∈ N ) denote the volume of the unit ball in 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 Ω n 2 / (Ω n − 1 Ω n + 1) . Also, we establish new sharp inequalities for the quantity Ω n 2 / (Ω n − 1 Ω n + 1) . [ABSTRACT FROM AUTHOR]

In this paper, we look at the general form of the two-point quadrature formulas, which doesn't have to be symmetric. Under the convexity constraint of the first derivative, we propose a new scheme that gives the best approximation of the two-point quadrature rules. [ABSTRACT FROM AUTHOR]

In this paper, based on a given parameterized identity that generates a quadrature rule family similar to Simpson's second formula, we establish some new Simpson-like type inequalities for functions with bounded as well as Lipchitzian derivatives from which we can deduce the famous 3/8-Simpson's inequality. The study concludes with an application example from management science. [ABSTRACT FROM AUTHOR]

In this paper, new inequalities for the left and right sides of the Hermite–Hadamard inequality are acquired for twice-differentiable mappings. Conformable fractional integrals are used to derive these inequalities. Furthermore, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the symmetry principle, applying the techniques of real analysis and Euler–Maclaurin summation formula, we construct proper weight coefficients and use them to establish a reverse extended Hardy–Hilbert's inequality with multi-parameters. Then, we obtain the equivalent forms and some equivalent statements of the best possible constant factor related to several parameters. Finally, we illustrate how the obtained results can generate some new reverse Hardy–Hilbert-type inequalities. [ABSTRACT FROM AUTHOR]

The widely known hermite-hadamard-Fejer type inequalities are so important in the field of mathematical analysis. Many researchers have studied on these inequalities. In this paper, we have obtained several inequalities related to the Hermite-Hadamard inequality for a special class of the functions called uniformly convex functions. We have also presented applications of these obtained inequalities in some error estimates for higher moments of random variables. [ABSTRACT FROM AUTHOR]

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper deals with generalizations and refinements of Fejér's inequality with some new applications. We introduce a real mapping M f ω (t) and obtain some its functional properties. We present several inequalities in connection with fractional integrals and their refinements by monotone functions. We also give some inequalities related to Euler's gamma and beta functions. Furthermore, we obtain some upper and lower bounds for generalized logarithmic mean and the expected value of random variables by using the arithmetic mean. [ABSTRACT FROM AUTHOR]

In this paper, utilizing an identity given by Yıldız and Sarıkaya in (Yildiz and Sarikaya in Int. J. Anal. Appl. 13(1):64–69, 2017), we establish some weighted Ostrowski type inequalities for co-ordinated convex functions in a rectangle from the plane R 2 . Moreover, as special cases of our main results, we give some weighted Hermite–Hadamard type inequalities. The results given in this paper provide generalizations of some result established in earlier works. [ABSTRACT FROM AUTHOR]

This paper provides improved bounds of inequalities of Mitrinović–Adamović type by using two-parameter functions. It achieves a much better approximation effect than those bounds of prevailing methods. A new method based on the Páde interpolation is used to prove the new bounds, which can also be applied for proving the results of prevailing methods and their much more generalized results. [ABSTRACT FROM AUTHOR]

In this paper we deal with functions related to generalized convexity and refine Jensen type inequalities satisfied by such functions. Specifically, we extend inequalities satisfied by uniformly convex functions, strongly convex functions as well as superquadratic functions. [ABSTRACT FROM AUTHOR]