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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, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014). [ABSTRACT FROM AUTHOR]

This paper proposes a mechanism to produce equivalent Lipschitz surrogates for zero-norm and rank optimization problems by means of the global exact penalty for their equivalent mathematical programs with an equilibrium constraint (MPECs). Specifically, we reformulate these combinatorial problems as equivalent MPECs by the variational characterization of the zero-norm and rank function, show that their penalized problems, yielded by moving the equilibrium constraint into the objective, are the global exact penalization, and obtain the equivalent Lipschitz surrogates by eliminating the dual variable in the global exact penalty. These surrogates, including the popular SCAD function in statistics, are also difference of two convex functions (D.C.) if the function and constraint set involved in zero-norm and rank optimization problems are convex. We illustrate an application by designing a multi-stage convex relaxation approach to the rank plus zero-norm regularized minimization problem. [ABSTRACT FROM AUTHOR]

In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs. [ABSTRACT FROM AUTHOR]

In this paper a(x)-convex functions are defined and Jensen, Hermite-Hadamard, Lah-Ribarić and other inequalities for them are derived. Related analogous of the Lagrange and the Cauchy mean value theorems are also obtained and means of the Cauchy type are generated. [ABSTRACT FROM AUTHOR]

Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J , ( t). In this paper, We can compute the constant J , (1) under the absolute normalized norms on ℝ by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented. 2000 Mathematics Subject Classification: 46B20. [ABSTRACT FROM AUTHOR]

The article provides an overview of the study which investigates the Jensen inequality for convex-concave antisymmetric functions and applications. It pointed out that a class of functions which is between of convex functions and the class of Jensen-convex functions is the class of Wright-convex functions. It presented several theorems which were the main motivations of the paper. It noted that it has proven that without loss of generality, it can suppose that (a,b) = (-1,1), thus, f is an odd function.

By using an integral arithmetic mean, a generalization of Levinson's inequality given in (Pečarić et al. in Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering, vol. 187, 1992) and results from (Vukelić in Appl. Anal. Discrete Math. 14:670–684, 2020), we give extension of Wulbert's result from (Wulbert in Math. Comput. Model. 37:1383–1391, 2003). Also, we obtain inequalities with divided differences for the functions of higher order. [ABSTRACT FROM AUTHOR]

An approach to the sensitivity analysis for discrete optimization problems with perturbed objective function is presented. The problem is stated in the following general form: min{ΣeϵYc(e):Y ϵ F}, where c = (c(e), e ϵ E) is a vector of weights for some finite set E, and F ⊂ 2e is a given family of feasible subsets. It is assumed that the set of feasible solutions F is fixed, but the coefficients of the vector c may vary. The main problem considered in this paper concerns the following question: How does the maximum relative error of a given feasible solution depend on inaccuracies of the problem data? Two particular perturbations of the vector c are considered: (i) it is assumed that c is in a closed ball with radius δ ≥ 0 and center c0 ≥ 0 or (ii) the relative deviation of c(e) from the value c0e) is not greater than δ for any e ϵ E, i.e., ∣c(e) - c0(e)∣ ≤ δc0(e), e ϵ E. For a given feasible solution X ϵ F, two functions of the parameter δ are introduced: the sensitivity function s(X, δ) and the accuracy function a(X, δ). The values of s(X, δ) and a(X, δ) are equal to the maximum relative error of the solution X when the perturbations of problem data are of the type (i) or (ii), respectively. Some general, but computationally inefficient formulae for calculating these functions are derived and their properties are studied. It is shown that s(X, δ) and a(X, δ) are nondecreasing, convex functions with a limited number of breakpoints. A practically efficient method of calculating upper and lower envelopes for the accuracy function is presented. This method is based on the notion of k-best solutions of the problem. It gives an interval to which the maximum relative error of the solution X must belong when the coefficients of vector c are given with accuracy δ. The approach is illustrated with an example of the symmetric traveling salesman problem. [ABSTRACT FROM AUTHOR]

This research investigates bounds of the sum of left-sided and right-sided fractional integrals in a compact form. These bounds are established by using (α , m) -convex functions. The results of this paper also have connection with some known and already published results (Dragomir and Agarwal in Appl. Math. Lett. 11(5):91–95, 1998; Farid in J. Anal. 2018, https://doi.org/10.1007/s41478-0079-4; Farid et al. in Mathematics 2018(6):248, 2018). Moreover, the investigated results are applied to the end points as well as the mid point of the domain and some interesting consequences have been obtained. [ABSTRACT FROM AUTHOR]

Motivated by Pečarić et al. in (J. Math. Inequal. 11(2):543–5502017), we established here weighted discrete dynamic inequalities for the difference of second divided difference of 4-convex functions. Further we extend and unify the two inequalities, by establishing the theory of n-convexity on time scales having constant graininess function. [ABSTRACT FROM AUTHOR]

We introduce a new class of functions called strongly (η , ω) -convex functions. This class of functions generalizes some recently introduced notions of convexity, namely, the η-convex functions and strongly η-convex functions. We also establish inequalities of the Hermite–Hadamard–Fejér's type, which generalize results of Delavar and Dragomir (Math. Inequal. Appl. 20(1):203–216, 2017) and Awan et al. (Filomat 31(18):5783–5790, 2017). In addition, we obtain some new results for this class of functions. Finally, we apply our results to the k-Riemann–Liouville fractional integral operators to obtain more results in this direction. [ABSTRACT FROM AUTHOR]

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations. [ABSTRACT FROM AUTHOR]

We use a concept of abstract convexity to define the almost S-KKMc property, al-S-KKMc(X, Y) family, and almost Φ-spaces. We get some new approximate fixed point theorems and fixed point theorems in almost Φ-spaces. Our results extend some results of other authors. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem on an infinite time interval. The system is linear in the control, the functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the functional at infinity, which is weaker than the previously known conditions, and prove the existence theorem for the solution under this condition. We consider several special cases and propose a general abstract scheme. [ABSTRACT FROM AUTHOR]

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular. [ABSTRACT FROM AUTHOR]

In the present article, the authors have established some Hermite–Hadamard type integral inequalities via Riemann–Liouville fractional integrals that generalize Hermite–Hadamard type inequalities and a few other results (Dragomir and Agarwal in Appl. Math. Lett. 11(5):91–95, 1998; Dragomir, Chob and Kimc in J. Math. Anal. Appl. 245(2):489–501, 2000; Yang, Hwang and Tseng in Comput. Math. Appl. 47(2–3):207–216, 2004). [ABSTRACT FROM AUTHOR]