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Cosine similarity measures are essential in situations that assess the similarities and differences between two potential outcomes. For different extensions of fuzzy sets, soft sets, and hypersoft sets, a wide range of similarity metrics have been examined in the literature. On the other hand, decision-making problems like sustainable supply chain management, in a single-valued neutrosophic cubic hypersoft set (svNCHSS) scenario have not been addressed so far previously using similarity metrics. Improved cosine similarity measures of svNCHSSs based on the cosine function are proposed by combining the cosine similarity measures of simplified neutrosophic sets in vector space that are currently available. It also looks into their properties and discusses their problems. In the svNCHSS environment, we developed a multi-attribute strategy for evaluating sustainable supplier selection (SuSS) in the healthcare sector using cosine and weighted cosine similarity measures. To construct the ranking order of these values and opt for the most appropriate supplier, the method entails calculating the similarity measure values between each assessed supplier and the ideal supplier. Four options are ranked based on thirteen sustainability-related sub-criteria that are connected to each of the three primary criteria like economic, social, and environmental that take into account all aspects of sustainability in a case study of a SuSS procedure for the healthcare sector. The results of the ranking technique are strongly influenced by the weights given to the attributes. For a given circumstance, no ranking method can ensure trustworthy selection. By contrasting their results with current similarity measures, the suggested decision-making problems involving many criteria and their sub-criteria are verified.

The Fermatean fuzzy soft set (FFSS) is a natural extension of the intuitionistic fuzzy soft set (IFSS) and the Pythagorean fuzzy soft set (PyFSS). It retains the corresponding benefits of IFSS, PyFSS, and their extensions, including the ability to represent an entity from both favorable and adverse perspectives. It renders it feasible to mimic real-world situations where uncertainty is prevalent more precisely. It is possible to portray uncertain circumstances more effectively with the incorporated dimension of hesitation degree. The current study presents an innovative form of FFSS, known as complex Fermatean fuzzy soft set (cFRFSS), aiming at addressing the complexities related to data uncertainty and periodicity. The basic idea of cFRFSS, including set-theoretic operations and properties, is demonstrated numerically, and associated results are provided. To mitigate inconsistent information, modified entropy metrics for cFRFSS are put forward in the following section. Then, to assist university managers in assessing the research output of their faculty members within the institution, a robust method employing suggested cFRFSS-based entropy measures is offered. The suggested strategy’s adaptability is assessed by contrasting it with other strategies that have been previously put forth.

Fractional calculus is extremely important and should not be undervalued due to its critical role in the theory of inequalities. In this article, different generalized Hermite-Hadamard type inequalities for functions whose modulus of first derivatives are [Formula: see text]-convex are presented, via Caputo-Fabrizio integrals. Graphical justifications of main results are presented. Graphs enable us to support our conclusions and show the reliability of our findings. Additionally, some applications to probability theory and numerical integration are also established. As special cases, certain established outcomes from different articles are recaptured. This study acts as a stimulant for future studies, inspiring researchers to investigate more thorough results by utilizing generalized fractional operators and broadening the idea of convexity.

Abstract The process of motorcycles evaluation is intricate and influenced by several factors that are uncertain, indeterminate and vague. It is difficult to estimate to what extent the choice will satisfy all requirements and change with the times. The theoretical frameworks such as hypersoft sets, interval and single-valued neutrosophic sets are useful in managing such vagueness and uncertainty. This study introduces a novel theoretical framework; the single-valued neutrosophic cubic hypersoft set (svNCHSS), to mitigate the loss of useful data. In comparison to the frameworks mentioned above, the svNCHSS is more adaptable in handling ambiguous and unpredictable circumstances because it incorporates all of their limitations as a single model. The set-theoretic operations like P-union $$(\sqcup _{P})$$ ( ⊔ P ) , P-intersection $$( \sqcap _{P})$$ ( ⊓ P ) , R-union $$( \sqcup _{R})$$ ( ⊔ R ) , R-intersection $$(\sqcap _{R})$$ ( ⊓ R ) , AND- and OR-operations are investigated for svNCHSS environment. Two potent algorithms are proposed in the context of a decision-assisted mechanism that employs AND- and OR-operations for the evaluation of motorbikes taking into account multi-argument-based factors. To evaluate the suggested algorithms’ effectiveness, a prototype case study is used for validation. The computed results are then compared. The structural comparison reveals the flexibility of svNCHSS, making its aggregation operations a useful tool for a variety of industrial applications, including risk assessment, process improvement, and decision-making.

Abstract The Hermite–Hadamard–Fejér-type inequality is an effective utensil for examining upper and lower estimations of the integrals of convex functions. In this study, the power mean inequality and Hölder inequality are employed. The outcomes are new Hermite–Hadamard–Fejér-type inequalities via Raina fractional integrals for m-convex functions. The findings are extensions of many previous investigations.

Selecting the best power source that is legal, affordable, environmentally friendly, and able to ensure long-term viability is a difficult but vital task. Existing frameworks based on traditional fuzzy and soft sets are unable to adequately capture the complexity of the optimal energy system selection (ESS). These decision models may also be complex, especially when rough data and integrity need to be taken into account. In this study, the imperative concepts of rough set and hypersoft set are integrated into a novel theoretical framework called hypersoft rough set (HSRS). The former provides a broad theoretical framework to address information-based ambiguities and uncertainties, while the latter can be thought of as a trustworthy aid for incomplete data analysis using approximate methods. Elementary notions of HSRS, its relevant approximation space, lower and upper approximations, and operations are characterized along with essential properties and results. A rigorous algorithmic strategy for assessing the feasibility of ESS based on the operations of HSRS is suggested to assist decision-makers in identifying appropriate strategies to address the electric power deficit. Potential benefits of the newly suggested approach include improved versatility in modeling complex decision-making scenarios, better discriminating ability, suitability for handling abnormalities in data, and parametrization. The algorithm's adaptability is evaluated through a practical application to a real-world problem about the identification of the best ESS in Pakistan. The outcomes show that the suggested approach effectively ascertains the ideal ESS. Compared to the methods currently in use, the analytical framework that has been suggested seems to be more robust.

In this work, Korkine and Sonin's identities are defined on arbitrary time scales. These identities are utilized to establish the Chebyshev and Grüss-type inequalities on time scales. By applying these inequalities, we determine the bounds of the remainders in Montgomery identities that incorporate Taylor's formula on time scales. Moreover, we derive discrete and quantum inequalities based on these results.

Decision-making approaches are often used in uncertain environments by people who must make difficult judgments in daily life, including elements of varied qualities and costs. These methods assist decision-makers in managing ambiguity and uncertainty, allowing for more informed and risk-reduced decisions. This research introduces an advanced framework called a complex single-valued neutrosophic soft set (csvNSS) to address uncertainties inherent in decision-making. The csvNSS framework is capable of managing information periodicity by introducing two components: amplitude and phase. The first deals with fuzzy membership, while the second manages periodicity within a complex plane. Some rudiments of csvNSS like properties, set operations and aggregations, are investigated. To make these ideas practically applicable in choosing an appropriate location for the hospital, an algorithm for handling csvNSS is proposed. An enhanced strategy is validated through the use of a specific example that takes site selection for hospital into account. The outcome demonstrates the efficacy of the suggested strategy. The method can be used in other domains where selection issues arise.

In this study, midpoint-type integral inequalities for [Formula: see text]-convex function in the third sense, involving Caputo fractional derivatives and Caputo–Fabrizio integral operators, are demonstrated. Inequalities, containing Caputo–Fabrizio integral operators for functions, whose derivatives are [Formula: see text]-convex function in the third sense, are also established. This article utilized [Formula: see text]-convex function in the third sense to generalized midpoint-type integral inequalities which give more sharp inequalities.

In this paper, an identity involving Riemann–Liouville fractional integrals is newly produced. Some new extensions of integral inequalities are proved with the help of Hölder, power-mean integral inequalities and newly established identity. Whereas, the absolute values of first derivatives of real valued functions, which are appeared in these inequalities, are exponentially ([Formula: see text]-convex. Some special cases are discussed and graphical evidences are presented for validity of proved results.

To address the shortcomings of the cubic intuitionistic fuzzy sets (CIFSs) for the entitlement of multi-argument approximate function, the cubic intuitionistic fuzzy hypersoft set (Ω-set) is an emerging study area. This type of setting associates the sub-parametric tuples with the collection of CIFSs. Categorizing the evaluation of parameters into their corresponding sub-parametric values based on non-overlapping sets has significance in decision making and optimization related situations. Some operations of Ω-set are proposed in this study, along with certain practical features. We provide the complement, P-order, and R-order subsets, P-union (∪P), R-union (∪R), P-intersection (∩P) and R-intersection (∩R) of Ω-sets. The internal cubic intuitionistic fuzzy hypersoft set (ΩI-set) and the external cubic intuitionistic fuzzy hypersoft set (ΩE-set) are also proposed in this paper, which will aid researchers in applying this new theory to other areas of study. We show a few examples in this context and look into some more aspects of ∪P, ∪R, ∩P and ∩R of ΩI-sets and ΩE-sets. Arguments for a few significant theorems about ΩI-sets and ΩE-sets are also presented. Lastly, an algorithm is presented that assists decision-makers in evaluating appropriate solar panels to establish solar plants. The proposed algorithm uses the idea of ∪P and ∪R for two Ω-sets constructed based on expert opinions of decision makers.

Abstract In this study, Levinson-type inequalities for the class of n-convex ( n ≥ 3 $n \geq 3$ ) functions are generalized using new Green functions and the Hermite interpolating polynomial involving two types of data points. Some estimations for novel functionals are derived using f-divergence. Furthermore, different inequalities involving Shannon entropies are presented.

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

AbstractIn the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen’s inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen’s inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results.

The generalization of strongly convex and strongly m-convex functions is presented in this paper. We began by proving the properties of a strongly modified (h,m)-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class.

Abstract In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and the Zipf–Mandelbrot law are also used to apply the main results to information theory.

In this paper, we present an identity for differentiable functions that has played an important role in proving Hermite–Hadamard type inequalities for functions whose absolute values of first derivatives are s-convex functions. Meanwhile, some Hermite–Hadamard type inequalities for the functions whose absolute values of second derivatives are s-convex are also established with the help of an existing identity in literature. Many limiting results are deduced from the main results which are stated in remarks. Some applications of proved results are also discussed in the present study.

Generalizations of some Steffensen's inequalities for the class of higher order convex functions are obtained in present study. To prove the results, some preliminary lemmas are established by using extended Montgomery identities via Taylor's formula on time scales. Furthermore, Ostrowski type inequalities are also established, along with prove of optimization of the constant. Moreover, newly obtained results are discussed in discrete and quantum calculus as well.

A new area of research called intuitionistic fuzzy soft expert set is expected to overcome the drawbacks of an intuitionistic fuzzy soft set in terms of eligibility for soft expert-argument approximate function. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert-argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, [Formula: see text]-convexity(concave) sets, strongly [Formula: see text]-convexity (concave) sets, strictly [Formula: see text]-convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and results on the basis of these specified notions. Additionally, by using a theoretic cum analytical approach, various elements of computational geometry, such as convex hull and convex cone, are theorized and some pertinent results are generalized.

Abstract In this paper, Csiszár f-divergence via diamond integral is introduced and some inequalities related to Csiszár f-divergence involving diamond integrals are presented. Some examples are presented for different divergence measures by fixing time scales. Some divergence measures are estimated in terms of logarithmic, identric, geometric, and arithmetic means. The obtained results generalize some known results in the literature and provide new bounds for some divergence measures in q-calculus.

Interval-valued fuzzy hypersoft set ($ \mathbb{IVFHSS} $) is considered a pertinent fuzzy set-like model that is the combination of an interval-valued fuzzy set and a hypersoft set. It is more flexible and trustworthy for dealing with information-based uncertainties due to the consideration of interval-based hypersoft settings. This kind of setting enables the decision makers to approximate the alternatives in terms of interval-type opinions by considering multiple arguments concurrently. These features make it a fitting model for dealing with uncertain decision-making scenarios like the recruitment process. The vagueness arises in the recruitment process when the data obtained is hesitant. The analogous educational norms among the candidates may increase its complexity. Evaluation techniques focus on leveling hypersoft sets for grading several alternatives based on multi-arguments. When several alternatives have an identical status, such grading systems frequently encounter problems, making it challenging for decision-makers to select the preeminent alternative. This settlement of such an issue is the basis of this article. Thus, in this study, first the axiomatic notions of similarity measures between $ \mathbb{IVFHSS}s $ are characterized, and then their relevant theorem is proved. In order to provide a consistent decision-support framework for the recruitment process, a robust algorithm is proposed. Finally, the effectiveness, feasibility, and efficiency of the proposed model are demonstrated through the depiction of recruitment-based pattern recognition.

Abstract The paper generalizes Shannon-type inequalities for diamond integrals. It includes two-dimensional Hölder’s inequality and Cauchy–Schwartz’s inequality, which help to prove weighted Grüss’s inequality for diamond integrals. Jensen’s inequality and Grüss’s inequality are used to provide Shannon-type inequalities for diamond integrals. Shannon-type inequalities for multiple integrals are also part of the discussion. Moreover, many distributions are deduced from the main results, and new bounds are obtained.

In this work, Hölder-Isçan inequality is used for the class of n-times differentiable (s,m)-convex functions. The outcomes are new Hermite-Hadamard type inequalities and modified integrals are estimated by better bounds. Special cases are deduced as the existing results from literature. Furthermore, some applications to arithmetic, geometric and logarithmic means are also presented.

In this paper, the Montgomery identity is generalized for double integrals on time scales by employing a novel analytical approach to develop the generalized Ostrowski type integral inequalities involving double integrals. Some inimitable cases are discussed for different parameters and parametric functions. Moreover, applications to some particular time scales are also presented.

Abstract In the current paper, a generalized Montgomery identity is obtained with the help of Taylor’s formula on time scales. The obtained identity is used to establish Ostrowski inequality, mid-point inequality, and trapezoid inequality. Moreover, the weighted versions of generalized Montgomery identity and respective Ostrowski inequality are also discussed. Special cases are obtained for different time scales to obtain new and existing results.

Abstract Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher order convex functions containing Csiszár divergence on time scales. New entropic bounds in q-calculus and h-discrete calculus are also deduced. Some estimates for Zipf–Mandelbrot entropy are also given.

Abstract Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

Abstract The aim of the present paper is to obtain new generalizations of an inequality for n-convex functions involving Csiszár divergence on time scales using the Green function along with Fink’s identity. Some new results in h-discrete calculus and quantum calculus are also presented. Moreover, inequalities for some divergence measures are also deduced.

Abstract The main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

Abstract In this paper, we present some inequalities for Csiszár f-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.

Abstract In this paper we give generalized results of a majorization inequality by using extension of the Montgomery identity and newly defined Green’s functions (Mehmood et al. in J. Inequal. Appl. 2017(1):108, 2017). We obtain a generalized majorization theorem for the class of n-convex functions. We use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

In the paper we extend some Hardy and Littlewood type inequalities on time scales for the function of n variables. Special cases of obtained results include generalized Wirtinger, Hardy and Littlewood type inequalities.

Abstract The aim of this paper is to establish new generalized fractional versions of the Hadamard and the Fejér–Hadamard integral inequalities for harmonically convex functions. Fractional integral operators involving an extended generalized Mittag-Leffler function which are further generalized via a monotone increasing function are utilized to get these generalized fractional versions. The results of this paper give several consequent fractional inequalities for harmonically convex functions for known fractional integral operators deducible from utilized generalized fractional integral operators.

Abstract This paper is devoted to obtain generalized results related to majorization-type inequalities by using well-known Fink’s identity and new types of Green functions, introduced by Mehmood et al. (J. Inequal. Appl. 2017:108, 2017). We give a generalized majorization theorem for the class of n-convex functions. We utilize the Csiszár f-divergence and generalized majorization-type inequalities in providing the corresponding generalizations. As an application, we present the obtained results in terms of Shannon entropy and Kullback–Leibler distance.

Abstract Levinson type inequalities are generalized by using Hermite interpolating polynomial for the class of n $\mathfrak{n}$ -convex functions. In seek of application to information theory, some estimates for new functional are obtained based on f divergence. Inequalities involving Shannon entropies are also discussed.

Abstract The aim of present paper is to obtain Shannon type inequalities using the extended version of Jensen’s inequality in time scales settings. The concept of differential entropy of a continuous random variable on time scales is introduced, and its bounds for some particular distributions are also estimated.

Interval-valued fuzzy hypersoft set is an emerging field of study which is projected to address the limitations of interval-valued fuzzy soft set for the entitlement of multiargument approximate function. This kind of function maps the subparametric tuples to power set of universe. It emphasizes on the partitioning of attributes into their respective subattribute values in the form of disjoint sets. These features make it a completely new mathematical tool for solving problems dealing with uncertainties. In this study, after characterization of essential properties, operations, and set-inclusions (L-inclusion and J-inclusion) of interval-valued fuzzy hypersoft set, some of its modular inequalities are discussed via set-inclusions. It is proved that all set-inclusion-based properties and inequalities are preserved when ordinary approximate function of interval-valued fuzzy soft set is replaced with multiargument approximate function of interval-valued fuzzy hypersoft set.

In this paper, the well-known Hölder’s inequality is proved via Hahn differential and integral operators, which is a helping tool to establish some Opial-type inequalities via Hahn’s calculus. The weight functions involved in these Opial-type inequalities are positive and monotone. In search of applications, some new as well as some existing inequalities in the literature are obtained by applying suitable limits.

The main motivation of this study is to introduce a novel auxiliary result of Simpson’s formula by employing the Mercer scheme for twice differentiable functions involving the Atangana-Baleanu (AB) fractional integral operator concerned with the Mittag-Leffler as a nonsingular or nonlocal kernel. Thus, by employing Mercer’s convexity on twice differentiable mappings along with Hölder’s and power-mean inequalities, one can develop a variety of new Simpson’s error estimates. Lastly, some applications to q-digamma function and modified Bessel functions are presented. Furthermore, the graphical illustrations described the efficiency and applicability of the proposed technique with success. We make links between our findings and a number of well-known discoveries in the literature. It is hoped that the proposed methodology will provide a new venue in the numerical techniques for calculating the quadrature formulae.

Based on the Riemann–Liouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as concavity are established. We expect that present outcomes are the generalization of already obtained results. Applications to beta, q-digamma, and Bessel functions are also provided.

The new outcomes of the present paper are q-analogues (q stands for quantum calculus) of Hermite-Hadamard type inequality, Montgomery identity, and Ostrowski type inequalities for s-convex mappings. Some new bounds of Ostrowski type functionals are obtained by using Hölder, Minkowski, and power mean inequalities via quantum calculus. Special cases of new results include existing results from the literature.

Hypersoft set is a novel area of study which is established as an extension of soft set to handle its limitations. It employs a new approximate mapping called multi-argument approximate function which considers the Cartesian product of attribute-valued disjoint sets as its domain and the power set of universe as its co-domain. The domain of this function is broader as compared to the domain of soft approximate function. It is capable of handling the scenario where sub-attribute-valued sets are considered more significant than taking merely single set of attributes. In this study, notions of set inclusion, convex (concave) sets, strongly convex (concave) sets, strictly convex (concave) sets, convex hull, and convex cone are conceptualized for the multi-argument approximate function. Based on these characterized notions, some set-theoretic inequalities are established with generalized properties and results. The set-theoretic version of classical Jensen’s type inequalities is also discussed with the help of proposed notions.

In the present paper, q-fractional integral operators are used to construct quantum analogue of Ostrowski type inequalities for the class of s-convex functions. The limiting cases include the nonfractional existing cases from literature. Specially, Ostrowski type inequalities for q-integrals and Ostrowski type inequalities for convex functions are deduced.

This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are (s,m)-convex. Inequalities involving the digamma function and special means are deduced as applications.

Abstract By using new Green’s functions and Lidstone interpolating polynomial, some new generalizations of Levinson type inequalities for (2p+1) $(2p+1)$-convex functions are obtained. In seek of applications of our results to information theory, new generalizations based on f-divergence estimates are also proven. Moreover, some inequalities for Shannon entropies are deduced as well.

Abstract In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel–Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

In this work, we estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Taylor one point and Taylor two points interpolations are used to generalize the new inequalities for m-convex function.

Abstract In the presented paper, Levinson’s inequality for 3-convex function is generalized by using two Green’s functions. Čebyšev, Grüss, and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, the main results are applied to information theory via f-divergence, Rényi divergence, Rényi entropy, Shannon entropy, and Zipf–Mandelbrot law.

Abstract In the presented paper, Levinson’s inequality for the 3-convex function is generalized by using two Green functions. Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, the main results are applied to information theory via the f-divergence, the Rényi divergence, the Rényi entropy, the Shannon entropy and the Zipf–Mandelbrot law.