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In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

In this paper, we study the generalized Riemann–Liouville fractional integral for the functions with fractal support. The aim of this article is to investigate reverse Minkowski’s inequalities and certain other related inequalities by employing the generalized Riemann–Liouville fractional integral for the functions with fractal support.

In this article, we develop a new class of trapezium-type inequalities up to twice differentiable $ h $-convex mappings for fractional integrals of Riemann-type. We conclude numerous existing results in literature from our general inequalities. Based on our consequences, we will obtain some quadrature formulas as applications.

Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.

In this paper, we at first develop a generalized integral identity by associating Riemann-Liouville (RL) fractional integral of a function concerning another function. By using this identity estimates for various convexities are accomplish which are fractional integral inequalities. From our results, we obtained bounds of known fractional results which are discussed in detail. As applications of the derived results, we obtain the mid-point-type inequalities. These outcomes might be helpful in the investigation of the uniqueness of partial differential equations and fractional boundary value problems.

In this paper, we investigate some properties of the Pochhammer p , s , k -symbol ξ n , k , s p and gamma p , s , k -function Γ s , k p ξ . We then prove several identities for newly defined symbol ξ n , k , s p and the function Γ s , k p ξ . The integral representations for the gamma p , s , k -function and beta p , s , k -function are presented. Also, we define a new Mittag-Leffler p , s , k -function and study its analytic properties and its transforms.

The devotion of this paper is to study the Bessel function of two variables in k-calculus. we discuss the generating function of k-Bessel function in two variables and develop its relations. After this we introduce the generalized (s, k)-Bessel function of two variables which help to develop its generating function. The s-analogy of k-Bessel function in two variables is also discussed. Some recurrence relations of the generalized (s, k)-Bessel function in two variables are also derived. © 2022 All rights reserved.

In this article, we introduce a class of functions U(p) with integral representation defined over measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.AMS Subject Classification: 26D15; 26D10; 26A33; 34B27

Aim: To examine the prevalence of complications related with transulnar approach in patients undergoing elective percutaneous coronary interventions. Study Design: Cross-sectional/observational study. Place & Duration: The study was conducted at cardiology department of Cat A Hospital Batkhela and Fauji Foundation Hospital Peshawar for six months duration from January 2020 to December 2020. Methods: One hundred and eighteen patients of both genders with ages 20 to 75 years who underwent percutaneous coronary interventions were included. Patients’ detailed demographics including age, sex, BMI and com-morbidities were recorded after taking informed written consent from all the patients. All the patients had percutaneous coronary procedure through transulnar approach and periprocedural complications were examined. Data was analyzed using SPSS 24.0. Results: Out of 118 patients 85 (72.03%) were males and 33 (27.97%) were females with mean age of 55.74±11.71 years. Mean BMI was 28.09±7.33 kg/m2. Hypertension was the most common morbidity found in 63 (53.4%) patients followed by diabetes mellitus and smoking. Minor bleeding was the commonest complication found in 28 (23.7%) patients followed by ulnar artery occlusion, excessive bleeding, ulnar nerve injury and hematoma in 10 (8.5%), 8 (6.8%), 6 (5.08%) and 2 (2.5%) patients respectively. Conclusion: It is concluded that transulnar approach for coronary interventions is safe and effective with fewer rate of complications. Keywords: Coronary Intervention, Angiography, Transulnar Approach, Complications.

The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on $\omega (\theta )$ ω ( θ ) and $\Psi (\theta )$ Ψ ( θ ) . Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results with certain choices of $\omega (\theta )$ ω ( θ ) and $\Psi (\theta )$ Ψ ( θ ) . Additionally, examples of constructing bounded functions are also presented in the paper.

In this paper, we establish some new integral inequalities involving general kernels. We obtain the related broad range of fractional integral inequalities. Also, we apply the Young inequality to find new forms of inequalities for generalized kernels. These new and motivated results generalize the results for fractional integrals such that fractional integral of a function with respect to an increasing function, Riemann–Lioville fractional integrals, Erdélyi–Kober fractional integrals, Hadamard fractional integrals, generalized factional integral integrals in addition to the corresponding k-fractional integrals.

In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, Laplace transform, Mellin transforms, and some relations of extension of extended generalized multi-index Bessel function (E1GMBF) with the Laguerre polynomial and Whittaker functions. Further, we also discuss the composition of the generalized fractional integral operator having Appell function as a kernel with the extension of extended generalized multi-index Bessel function and establish these results in terms of Wright functions.

The main objective of this paper is to introduce a further extension of the extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Additionally, we investigate the extended fractional derivative of some familiar special functions, the Mellin transform of the newly defined Caputo fractional derivative operator and generating relations for the extensions of extended hypergeometric functions.

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

In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Holder type inequalities. Furthermore, novel consequences connected with this inequality, together with statements and confirmation of various variants for the advocated generalized conformable fractional integral operator, are elaborated. Moreover, our derived results are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions. The numerical approximations of our consequence have several utilities in applied sciences and fractional integro-differential equations., Prince Sultan University [RG-DES-2017-01-17], The author T. Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

We study a generalized left sided tempered fractional (GTF)-integral concerning another function Ψ in the kernel. Then we investigate several kinds of inequalities such as Grüss-type and certain other related inequalities by utilizing the GTF-integral. Additionally, we present various special cases of the main result. By utilizing the connection between GTF-integral and Riemann–Liouville integral concerning another function Ψ in the kernel, certain distinct particular cases of the main result are also presented. Furthermore, certain other inequalities can be formed by applying various kinds of conditions on the function Ψ.

The main aim of this present paper is to establish fractional conformable inequalities for the weighted and extended Chebyshev functionals. We present some special cases of our main result in terms of the Riemann–Liouville fractional integral operator and classical inequalities.

The principal aim of this paper is to introduce $\mathtt{k}$-fractional derivative operator by using the definition of $\mathtt{k}$-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to $\mathtt{k}$-hypergeometric and $\mathtt{k}$-Appell's functions. Also, we investigate the $\mathtt{k}$-fractional derivative of $\mathtt{k}$-Mittag-Leffler and the Wright hypergeometric functions.

In modern times, the organizational managements greatly depend on decision-making (DM). DM is considered the management’s fundamental function that helps the businesses and organizations to accomplish their targets. Several techniques and processes are proposed for the efficient DM. Sometimes, the situations are unclear and several factors make the process of DM uncertain. Fuzzy set theory has numerous tools to tackle such tentative and uncertain events. The complex picture fuzzy set (CPFS) is a super powerful fuzzy-based structure to cope with the various types of uncertainties. In this article, an innovative DM algorithm is designed which runs for several types of fuzzy information. In addition, a number of new notions are defined which act as the building blocks for the proposed algorithm, such as information energy of a CPFS, correlation between CPFSs, correlation coefficient of CPFSs, matrix of correlation coefficients, and composition of these matrices. Furthermore, some useful results and properties of the novel definitions have been presented. As an illustration, the proposed algorithm is applied to a clustering problem where a company intends to classify its products on the basis of features. Moreover, some experiments are performed for the purpose of comparison. Finally, a comprehensive analysis of the experimental results has been carried out, and the proposed technique is validated.

The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the proposed modified weighted-type fractional integral incorporating another function in the kernel F(θ). For the weighted and extended Chebyshev’s functionals, we also propose weighted fractional integral inequalities. With specific choices of ϖ(θ) and F(θ) as stated in the literature, one may easily study certain new inequalities involving all other types of weighted fractional integrals related to Chebyshev’s functionals. Furthermore, the inequalities for all other type of fractional integrals associated with Chebyshev’s functionals with certain choices of ϖ(θ) and F(θ) are covered from the obtained generalized weighted-type fractional integral inequalities.

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and [Formula: see text]-Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature.

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

In this paper, we adopt conformable fractional integral to develop integral inequalities such as Minkowski and Hermite–Hadamard inequalities. Our results are the generalization of the inequalities obtained by Dahmani and Bougoffa cited in the literature.

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function$\mathcal{G}$Gin the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of$\omega (\theta )$ω(θ)and$\mathcal{G}(\theta )$G(θ)as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of$\omega (\theta )$ω(θ)and$\mathcal{G}(\theta )$G(θ).

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

In this paper we introduce the (k, s)-Hilfer-Prabhakar fractional derivative and discuss its properties. We find the generalized Laplace transform of this newly proposed operator. As an application, we develop the generalized fractional model of the free-electron laser equation, the generalized time-fractional heat equation, and the generalized fractional kinetic equation using the (k, s)-Hilfer-Prabhakar derivative.

In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of n ( n &isin, N ) positive functions by utilizing Hadamard proportional fractional integral operator. The inequalities presented in this paper are more general than the inequalities existing in the literature.

The goal of this article is to establish some fractional proportional integral inequalities for convex functions by employing proportional fractional integral operators. In addition, we establish some classical integral inequalities as the special cases of our main findings.

In this present article, we establish certain new Pólya–Szegö-type tempered fractional integral inequalities by considering the generalized tempered fractional integral concerning another function Ψ in the kernel. We then prove certain new Chebyshev-type tempered fractional integral inequalities for the said operator with the help of newly established Pólya–Szegö-type tempered fractional integral inequalities. Also, some new particular cases in the sense of classical tempered fractional integrals are discussed. Additionally, examples of constructing bounded functions are considered. Furthermore, one can easily form new inequalities for Katugampola fractional integrals, generalized Riemann–Liouville fractional integral concerning another function Ψ in the kernel, and generalized fractional conformable integral by applying different conditions.

In this paper, we define a new extension of Srivastava's triple hypergeometric functions by using a new extension of Pochhammer's symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601-2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.

Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) function εa,b;p,vκz1 in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function εa,b;p,vκz1. Some particular cases and outcomes of the main results are also established.

In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.

In the paper, the authors establish some new inequalities of the Gruss type for conformable fractional integrals. These inequalities generalize some known results.

In the paper, the authors establish several new inequalities of the Gruss type for conformable k-fractional integral operators. These inequalities generalize some known results.

Since Gruss presented an interesting integral inequality in [9], its various generalizations and variants, which are called Gruss type inequalities, have been investigated. Very recently, certain Gruss type inequalities involving ? ? s , k-fractional integral operator have been established. Motivated by the above mentioned works, we aim to present a Young's type inequality and a weighted AM-GM type inequality involving the ? ? ? , k-fractional integral operator. Some special cases of the main results here are also considered.

In this present paper, we investigate generalized integration formulas containing the generalized $k$-Bessel function $W_{v,c}^{k}(z)$ based on the well known Oberhettinger formula [12] and obtain the results in term of Wright-type function. Also, we establish certain special cases of our main result.

In this paper, we present certain composition formulae of the pathway fractional integral operators associated with two extended Mittag-Leffler functions. Here, we find out the relevant connections of some particular cases of the main results with those earlier ones.

This paper is devoted to the study of fractional calculus with an integral and differential operators containing the following family of extended Mittag–Leffler function: $$\begin{aligned} E_{\alpha ,\beta }^{\gamma ;c}(z; p)=\sum \limits _{n=0}^{\infty }\frac{B_p(\gamma +n, c-\gamma )(c)_{n}}{B(\gamma , c-\gamma )\Gamma (\alpha n+\beta )}\frac{z^n}{n!}, (z,\beta , \gamma \in \mathbb {C}), \end{aligned}$$ in its kernel. Also, we further introduce a certain number of consequences of fractional integral and differential operators containing the said function in their kernels.

In this paper, we introduce the $(k, s)$ -fractional integral and differential operators involving k-Mittag-Leffler function $E_{k,\rho,\beta}^{\delta}(z)$ as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results.

In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

Our main goal in this present paper is to define first a new extension of the Pochhammer symbol and the gamma functions which involving the Mittag-Leffler function in their kernels. By using this extended Pochhammer symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric functions and of some of its special cases. Also, we establish the basic properties and results for the extended $$\tau $$-Gauss hypergeometric function, which includes integral and derivative formulas involving the Mellin transform and fractional calculus approaches. Some new and known results as consequences of our proposed extension of the $$\tau $$-Gauss hypergeometric function are also established.