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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]

Motivated by certain results known from the literature, in this paper we give several new dynamic inequalities for B-monotone functions with respect to time scales nabla calculus. If the time scale represents the set of real numbers, our results reduce to integral inequalities known from the literature. On the other hand, in the setting of positive integers, we obtain new discrete inequalities for B-monotone sequences. [ABSTRACT FROM AUTHOR]

Strongly convex functions as a subclass of convex functions, still equipped with stronger properties, are employed through several generalizations and improvements of the Jensen inequality and the Jensen–Mercer inequality. This paper additionally provides applications of obtained main results in the form of new estimates for so-called strong f-divergences: the concept of the Csiszár f-divergence for strongly convex functions f, together with particular cases (Kullback–Leibler divergence, χ 2 -divergence, Hellinger divergence, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence.) Furthermore, new estimates for the Shannon entropy are obtained, and new Chebyshev-type inequalities are derived. [ABSTRACT FROM AUTHOR]

In the context of Sobolev spaces with variable exponents, Poincaré–Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form ∫ Ω f (x) - ⟨ f ⟩ Ω p (x) d x ⩽ C ∫ Ω | ∇ f (x) | p (x) d x , are known to be false. As a result, all available modular versions of the Poincaré–Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. This obstruction is commonly believed to be an unavoidable anomaly of the variable exponent setting. The main aim of this paper is to show that this is not the case, i.e., that a consistent generalization of the Poincaré–Wirtinger inequality to the variable exponent setting is indeed possible. Our contribution is threefold. First, we establish that a modular Poincaré–Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if Ω ⊂ R n is a bounded Lipschitz domain, and if p ∈ L ∞ (Ω) , p ⩾ 1 , then for every f ∈ C ∞ (Ω ¯) the following generalized Poincaré–Wirtinger inequality holds ∫ Ω f (x) - ⟨ f ⟩ Ω p (x) d x ⩽ C ∫ Ω ∫ Ω | ∇ f (z) | p (x) | z - x | n - 1 d z d x , where ⟨ f ⟩ Ω denotes the mean of f over Ω , and C > 0 is a positive constant depending only on Ω and ‖ p ‖ L ∞ (Ω) . Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré–Wirtinger constant on Lipschitz domains. [ABSTRACT FROM AUTHOR]

Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parameter p. Finally, we present a Milne-type inequality in the fractional context. [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]

This paper investigates a technique that uses Riemann-Liouville fractional integrals to study several Euler-Maclaurin-type inequalities for various function classes. Afterwards, we provide our results by using special cases of obtained theorems and This paper is to derive examples. Moreover, we give some Euler-Maclaurin-type inequalities for bounded functions by fractional integrals. Furthermore, we construct some fractional Euler-Maclaurin-type inequalities for Lipschitzian functions. Finally, we offer some Euler-Maclaurin-type inequalities by fractional integrals of bounded variation. [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]

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]

In this paper, we first introduce four new generalized quantum integrals of two variables functions which are called T q -integrals. We also give some examples of each of these given integrals. Then we establish corresponding T q -Hermite–Hadamard type inequalities for co-ordinated convex functions. In order to show correctness of our results, we apply our main theorems to several co-ordinated convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove Rubio de Francia extrapolation results for pair of non-increasing functions with B p (·) weights. The case of B ∞ weights has also been proved. These results have been applied to obtain the extrapolation results in the framework of variable exponent Lebesgue space L w p (·) with Luxemburg norm. Also, an improved norm estimate for the boundedness of Hardy averaging operator on variable exponent Lebesgue space is given. [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]

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]

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]

In this paper, we discuss new iterated discrete inequalities of Hardy type involving matrix kernels. Under some conditions on matrix entries of the kernels, we establish weight characterizations of these inequalities. [ABSTRACT FROM AUTHOR]

Recently, we have obtained two general theorems dealing with | N ¯ , p n | k summability factors of infinite series and trigonometric Fourier series (Bor in Bull Sci Math 169:102990, 2021) by using almost increasing sequences. In this paper, we have generalized these theorems to | N ¯ , p n ; θ n | k summability methods. Some new and known results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove some operator inequalities associated with an extension of the Kantorovich type inequality for s-convex functions. We also give an application to the order preserving power inequality of three variables and find a better lower bound for the numerical radius of a positive operator under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain several Adams-type inequalities with logarithmic weights on the unit ball B in R n , where n ≥ 3 . Firstly, we show that for any β ∈ 0 , 1 , the following critical Adams' inequality sup u ∈ W 0 , r 2 , n 2 (B , ω) , u ω ≤ 1 ∫ B exp α u n n - 2 1 - β d x < ∞ holds if and only if α ≤ α β = n n - 2 n 2 n V n 2 n - 2 1 - β 1 - β 1 1 - β , where V n = π n 2 Γ n 2 + 1 is the volume of the unit ball B in R n , W 0 , r 2 , n 2 (B , ω) is the radial weighted Sobolev space under the norm u ω = ∫ B Δ u n 2 ω x d x 2 n with ω (x) = log 1 x β (n 2 - 1) or ω (x) = log e x β (n 2 - 1) . Secondly, we prove the following two supercritical Adams' inequalities sup u ∈ W 0 , r 2 , n 2 (B , ω) , u ω ≤ 1 ∫ B exp α β + x m u n n - 2 1 - β d x < ∞ and sup u ∈ W 0 , r 2 , n 2 (B , ω) , u ω ≤ 1 ∫ B exp α β u n n - 2 1 - β + x m d x < ∞ , where m is some positive number. Moreover, we prove the existence of extremals for these Adams-type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we develop a concept of a logarithmically superquadratic function. Such a class of functions is defined via superquadratic functions. We first establish some new properties of superquadratic functions. In particular, we derive the corresponding superadditivity relation and its reverse, as well as the external form of the Jensen inequality and its reverse. Then, as a direct consequence of the established results, we obtain the corresponding properties for logarithmically superquadratic functions. Further, we show that logarithmically superquadratic functions with values greater than or equal to one are convex and logarithmically superadditive. In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Minkowski integral inequality on time scales by Δ Δ -integral and the inverse Minkowski inequality over the set of constant sign functions. We get similar results by replacing Δ Δ -integral with Δ ∇ , ∇ Δ and ∇ ∇ -integrals. For application, we give another proof of the Hardy inequality on time scales according to the Minkowski inequality. [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 give some new power-type refinements and reverses of the weighted arithmetic–geometric mean inequality. Our result is a remarkable generalization of the one due to Furuichi (J Math Inequal 5(1):21–31, 2011), Manasrah and Kittaneh (J Math Anal Appl 361(1):262–269, 2010, Linear Multilinear Algebra 59(9):1031–1037, 2011). When the power is a positive integer number, our result is also a refinement of a very recent generalization established by Ighachane et al. (Math Inequal Appl 23(3):1079–1085, 2020) for this inequality. As an application, we provide some generalized inequalities for determinants of positive definite matrices. [ABSTRACT FROM AUTHOR]

Many of the fastest known algorithms to compute π involve generalized hypergeometric series, such as the Ramanujan–Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic expansions for the error of finite approximation. For example, when using the first n terms of the Chudnovskys' series, we obtain the finite approximation π n ≈ π . It is known that the truncation error satisfies | π n - π | ≈ 53360 - 3 n. In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is π n - π = 53360 - 3 n 106 720 10 005 π 1 672 209 n exp A 1 n + A 2 n 2 + δ n n 3 , with 0.006 907 < δ n < 0.008 429 and the exact rational values of A 1 and A 2 : A 1 = - 1 781 843 197 433 7 456 754 505 816 , A 2 = - 1 080 096 011 925 710 088 395 3 475 199 235 000 451 148 614 116. Thus we demonstrate how to establish precise error bounds for the approximations for π obtained through Ramanujan-like series for 1 / π . [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to studying some sharp bounds for Hardy-type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its conjugate operator, respectively. [ABSTRACT FROM AUTHOR]

Yager has introduced the q-rung orthopair fuzzy set (q-ROFS), which is the most significant generalization of the Pythagorean fuzzy set (PFS). The q-rung orthopair adjusts in the needed boundary range when the q rung value increases. As a result, the q-ROFS input range is more adaptable, resilient, and appropriate than the intuitionistic fuzzy set and PFS. In this paper, we proposed new entropy measure for q-rung orthopair fuzzy set and it is being proved that the proposed entropy measure follows all the requirements of an entropy measure for q-ROFs. Numerical example depicts the efficiency of the proposed entropy measure. Furthermore, in the present paper entropy weighted method is applied to compute weights vector where partial information is used for criteria weights. Then, a new decision-making method using inferior ratio method based on proposed entropy measure is put forward. The working of the proposed multi-criteria decision-making model is explained through a numerical example. At last, a detailed comparison of the proposed model with certain current approaches is given, which demonstrates that the proposed decision model is more effective and beneficial than existing methodologies. [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]

This paper proves an equality for the case of differentiable convex functions including the conformable fractional integrals. By using this equality, we establish several parameterized inequalities with the help of the conformable fractional integrals. Several inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. [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, the improvements in the estimations of some entropies are of our interest. These improvements can also be done sequentially. We find some new and refined bounds for the Tsallis entropy, Tsallis quasilinear entropy, and Tsallis quasilinear divergence. The estimations for the Tsallis entropy are provided with respect to the Shannon entropy and Rényi entropy. We obtain some alternative refined bounds for the Tsallis relative operator entropy and we establish the significant refined relation between the Tsallis relative operator entropy and the generalized relative operator entropy. [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]

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 prove some new characterizations of two weighted functions u and v in norm inequalities of Hardy’s type, in the context of dynamic inequalities on time scales T . These norm inequalities studied the boundedness of the operator of Hardy’s type between the weighted spaces L v p (T) and L u q (T) . The paper covers the different cases when 1 < p ≤ q < ∞ and when 1 < q < p < ∞ . As special cases, when T = R , we obtain the corresponding previously known results from the literature, while for T = N we obtain some discrete results which are essentially new. In seeking applications, we will establish some non-oscillation results for second-order half-linear dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hardy inequalities for the p-Laplace operator associated with Robin boundary conditions in general domains. Moreover, by using the new Hardy inequality, we derive the classical Hardy inequality with the weight of mean distance function. [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]

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 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]

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, 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, 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]

In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a stability result for symmetrized convex stochastic processes are proved. [ABSTRACT FROM AUTHOR]