Search

In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section. [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 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 introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen's integral type inequality in this setting are given. Several examples are also involved. [ABSTRACT FROM AUTHOR]

In this paper, we determine the monotonicity of three classes of functions involving modified Bessel functions of the second kind. We also presented some inequalities and bounds for modified Bessel functions of the second kind. In particular, our results derive bounds for the Airy function and its derivative. [ABSTRACT FROM AUTHOR]

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

In this paper, 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, we establish an identity involving Sarikaya fractional integrals for twice differentiable functions. We obtain some new generalized fractional inequalities for the functions whose second derivatives in absolute value are convex by utilizing obtained equality. Utilizing the new inequalities obtained, some new inequalities for Riemann–Liouville fractional integrals and k-Riemann–Liouville fractional integrals are obtained. In addition, some of these results generalize ones obtained in earlier works. [ABSTRACT FROM AUTHOR]

In this paper, we present several sharp bounds for the reciprocals of the Neuman-Sándor mean in terms of the harmonic, arithmetic, quadratic, and contraharmonic means by using classical analysis and inequalities theory. Finally, we give a remark based on the generalized trigonometric and generalized hyperbolic functions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an asymptotic expansion for the determinant of the matrix [ Ω n − i + j ] 1 ⩽ i , j ⩽ r , as n → ∞ , where Ω n denotes the volume of the unit ball in R n . [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, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained. [ABSTRACT FROM AUTHOR]

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class. [ABSTRACT FROM AUTHOR]

Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions Γ q (x + 1) / Γ q (x + s) for all real number s and 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all 0 < s < 1 when letting q → 1 and a new inequality for all s > 1 . [ABSTRACT FROM AUTHOR]

A new identity, with two parameters for differentiable function with respect to another functions via generalized integral operators, is first obtained. By applying the established identity, the trapezium, midpoint and Simpson type integral inequalities pertaining to generalized (m, h1, h2) -preinvex functions are deduced. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of the our results to special means and new error estimates for the trapezium and midpoint quadrature formula have been presented. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels ( a − 1 A B R ∇ δ , γ y) (η) of order 0 < δ < 0.5 , β = 1 , 0 < γ ≤ 1 starting at a − 1 . If ( a − 1 A B R ∇ δ , γ y) (η) ≥ 0 , then we deduce that y (η) is δ 2 γ -increasing. That is, y (η + 1) ≥ δ 2 γ y (η) for each η ∈ N a : = { a , a + 1 , ... } . Conversely, if y (η) is increasing with y (a) ≥ 0 , then we deduce that ( a − 1 A B R ∇ δ , γ y) (η) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we establish some new results on trapezium type inequalities of coordinated distance-disturbed (ℓ 1 , h 1) – (ℓ 2 , h 2) -convex functions of higher orders (σ 1 , σ 2) by using the Katugampola (k 1 , k 2) -fractional integrals. As special cases of our general results, we recapture some earlier proved results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hermite–Hadamard–Fejér type inclusions for the product of two co-ordinated convex interval-valued functions. These inclusions are generalizations of some results given in earlier works. [ABSTRACT FROM AUTHOR]

In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for (s + 1) -convex functions involving the (k , s) -Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-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. [ABSTRACT FROM AUTHOR]

This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Hölder 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. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, for x > 0 , 1 − exp (− x 2 x 2 + 1 ) < tanh x < 1 − exp (− x 3 x 3 + 1 ) 3. This solves an open problem proposed by Ivády. [ABSTRACT FROM AUTHOR]

In this paper, the author establishes a new identity for differentiable functions and obtain some new inequalities for differentiable functions based on s–convexity via Riemann-Liouville fractional integrals involving Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a generalized integral identity for twice local fractional differentiable mappings on fractal sets R α (0 < α ≤ 1) of real line numbers. Then, using twice local fractional differentiable mappings that are in absolute value at certain powers generalized strongly m-convex, we obtain some new estimates on generalization of trapezium-like inequalities. We also discuss some new special cases which can be deduced from our main results. [ABSTRACT FROM AUTHOR]

In this paper, we present a new approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation over other classical ones. [ABSTRACT FROM AUTHOR]

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013-2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow-Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance. [ABSTRACT FROM AUTHOR]

In this paper, a new general identity for differentiable mappings via k-fractional integrals is derived. By using the concept of $(h,m)$ -convexity, $(\alpha,m)$ -convexity and the obtained equation, some new trapezium-like integral inequalities are established. The results presented provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the 'two-point right focal' problem. We use Fink's identity and a new Abel-Gontscharoff-type Green's function for a 'two-point right focal' to generalize the refinement of Jensen's inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen's inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we show several Turán type inequalities for a generalized Mittag-Leffler function with four parameters via the $(p,k)$ -gamma function. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the fractional Hermite-Hadamard inequality on , where is a set of positive real numbers such that for . Using this inequality, we present a new characterization of convexity of a function defined on . [ABSTRACT FROM AUTHOR]

Based on the Padé approximation method, in this paper we determine the coefficients $a_{j}$ and $b_{j}$ such that where $k\geq0$ is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results. [ABSTRACT FROM AUTHOR]

Generalized Jacobian elliptic functions play an important role in study of some bifurcation problems involving one-dimensional p-Laplacian. This paper deals with concavity and monotonicity of those functions. Also, inequalities involving functions discussion are established. In particular, Wilker-type and Van der Corput inequalities are derived. [ABSTRACT FROM PUBLISHER]

In this paper, we present Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions. [ABSTRACT FROM AUTHOR]

In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function. [ABSTRACT FROM AUTHOR]

Generalized Jacobian elliptic functions play an important role in some bifurcation problems involving one-dimensionalp-Laplacian. One of the goals of this paper is to investigate various kinds of concavity of these functions. Also, inequalities involving functions under discussion, and in particular Slater inequality is established. Transformations changing the variable and modulus are obtained. [ABSTRACT FROM AUTHOR]

This paper deals with a class of degenerate quasilinear elliptic equations of the form $-\operatorname{div}(a(x,u,\nabla u))+F(x,u,\nabla u)=f $, where $a(x,u,\nabla u)$ is allowed to degenerate with the unknown u. Under some hypothesis on a, F, and f, we obtain the existence of bounded solutions $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$. For the case $f\in L^{1}(\Omega)$, we also prove that there exists at least one renormalized solution. [ABSTRACT FROM AUTHOR]

In this paper, we present some sharp inequalities for the ratio of gamma functions. The main tool is the multiple-correction method formulated in (Cao et al. in J. Number Theory 149:327-350, 2015; Cao in J. Math. Anal. Appl. 424:1425-1446, 2015). Some conjectures are proposed. [ABSTRACT FROM AUTHOR]

A new one-parameter bivariate mean is introduced and investigated. The mean under discussion is defined in terms of the incomplete symmetric elliptic integral of the second kind. Several inequalities involving the mean discussed in this paper are obtained. Computable bounds and Wilker's type inequalities are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we present some sharp inequalities for the ratio of gamma functions. The main tool is the multiple-correction method formulated in (Cao et al. in J. Number Theory 149:327-350, 2015; Cao in J. Math. Anal. Appl. 424:1425-1446, 2015). Some conjectures are proposed. [ABSTRACT FROM AUTHOR]

In this paper, we present new asymptotic expansions of the gamma function. Based on our expansions, we establish some symmetric double inequalities for the gamma function. [ABSTRACT FROM AUTHOR]

In this paper, new integral inequalities of Hadamard type involving several differentiable φ –r– convex functions are given. [ABSTRACT FROM PUBLISHER]

In the paper, the authors establish some generalized fractional integral inequalities of the Hermite–Hadamard type for (α , m) -convex functions, show that one can find some Riemann–Liouville fractional integral inequalities and classical integral inequalities of the Hermite–Hadamard type, and generalize and extend some known results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Hermite-Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann-Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite-Hadamard inequality via the Riemann-Liouville fractional integral. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by Wν,ck(x):=∑r=0∞(−c)rΓk(rk+ν+k)r!(x2)2r+νk. We also indicate some representation formulae for the function introduced. Further, we show that the function Wν,ck is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function Wν,ck, particularly, in the case c=−1. We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of Wν,ck. [ABSTRACT FROM AUTHOR]