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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 this study, some mappings related to the Fejér-type inequalities for GAGA-convex functions are defined over the interval [0,1]{[}0,1]. Some Fejér-type inequalities for GAGA-convex functions are proved using these mappings. Properties of these mappings are considered and consequently we obtain refinements of some known results.

We introduce a new unified extension of the integral form of Euler's beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer's confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function. [ABSTRACT FROM AUTHOR]

In this study, Levinson-type inequalities for the class of n-convex (n ≥ 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. [ABSTRACT FROM AUTHOR]

In the article, we establish some monotonicity and convexity (concavity) properties for certain combinations of polynomials and the Ramanujan R-function by use of the monotone form of L’Hôpital’s rule and present serval new asymptotically sharp bounds for the Ramanujan R-function that improve some previously known results.

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results. [ABSTRACT FROM AUTHOR]

Keywords: Fractional differential equations (primary); Caputo's derivative; Sharp bound; Polynomial coefficients; 35R11 (primary); 26A33; 26D20; 41A10 EN Fractional differential equations (primary) Caputo's derivative Sharp bound Polynomial coefficients 35R11 (primary) 26A33 26D20 41A10 1785 1800 16 10/11/22 20221001 NES 221001 Preliminaries The fractional differential equations (FDEs) are generalizations of the conventional differential equations to an arbitrary order. Fractional differential equations (primary), Sharp bound, 35R11 (primary), 26D20, 41A10, 41A10, Caputo's derivative, Polynomial coefficients, 26A33 Anal.2019223807824398788710.1515/fca-2019-00441426.39024 14 PodlubnyIFractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications1998New YorkAcademic Press0924.34008 15 Postavaru, O., Dragoi, F., Toma, A.: Enhancing the sccuracy of solving Riccati fractional differential equations. [Extracted from the article]

In this paper, we establish some estimates of the left side of the generalized Gauss-Jacobi quadrature formula for harmonic h-preinvex functions involving Euler’s beta and hypergeometric functions. The obtained results are mainly based on the identity given by M. A. Noor, K. I. Noor and S. Iftikhar in [17].

In this paper, motivated by the Hölder–Łojasiewicz inequality, we present some inequalities to uniformly estimate the volumes of the images or pre-images of definable families of sets through maps definable in o-minimal structures, via the volumes of the families involved. [ABSTRACT FROM AUTHOR]

In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on f{\mathfrak{f}} divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.

A new proof of Euler’s formular for calculating ζ(2k) is given. Some new inequalities and identities for ζ(2k + 1) have also been given. The Riemann’s functional equation together with trigonometric identities were used to establish the results.

This paper is concerned to present and apply a new generalized fractional derivative, that is the Generalized Hilfer-type (GH) fractional derivative. This derivative unifies various previously defined fractional derivatives of the types Hilfer-Katugampola, Hilfer-Hadamard, Caputo-Hadamard, Hadamard, Hilfer, Riemann-Liouville, Caputo etc into a single form. The mellin transform of this new GH fractional derivative is obtained. As an application, a generalized space-time fractional diffusion equation is constructed using the newly obtained GH fractional derivative in the time-variable and the already existing Riesz fractional derivative in the space-variable. For the solution of this new generalized fractional diffusion equation, the similarity transformations and the mellin transform methods are used where an explicit solution in terms of the Fox’s H-function is derived. The diffusive behavior and the role of various parameters involved in the newly obtained GH fractional derivative are described with the help of 3D plots.

In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain. [ABSTRACT FROM AUTHOR]

In this paper, we prove some properties of weighted Cesàro and Copson sequences spaces by establishing some factorization theorems. The results lead to two-sided norm discrete inequalities with best possible constants and also give conditions for the boundedness of the generalized discrete weighted Hardy and Copson operators. [ABSTRACT FROM AUTHOR]

Using an extension of Montgomery's identity and the Green's function, we obtain new identities and related inequalities for weighted averages of n-convex functions, i.e. the sum ∑ i = 1 m ρ i h (λ i) and the integral ∫ a b ρ (λ) h (γ (λ)) d λ where h is an n-convex function. [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]

We present new generalizations of the weighted Montgomery identity constructed by using the Hermite interpolating polynomial. The obtained identities are used to establish new generalizations of weighted Ostrowski type inequalities for differentiable functions of class C n . Also, we consider new bounds for the remainder of the obtained identities by using the Chebyshev functional and certain Grüss type inequalities for this functional. By applying those results we derive inequalities for the class of n-convex functions. [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]

By the utilization of diamond-α derivatives, certain new generalizations of Ou-Iang type of dynamic integral inequalities of one independent variable on time scales are examined. The resulting inequalities are significant in the study of various fields of dynamic equations. A few mathematical applications are also presented. [ABSTRACT FROM AUTHOR]

In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series. [ABSTRACT FROM AUTHOR]

In this paper we prove some essential complements of the paper (J. Inequal. Appl. 2019:171, 2019) on the same theme. We prove some new Fourier inequalities in the case of the Lorentz–Zygmund function spaces L q , r (log L) α involved and in the case with an unbounded orthonormal system. More exactly, in this paper we prove and discuss some new Fourier inequalities of this type for the limit case L 2 , r (log L) α , which could not be proved with the techniques used in the paper (J. Inequal. Appl. 2019:171, 2019). [ABSTRACT FROM AUTHOR]

In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.

In the article, we present several Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions and give an application to the product of the moment of two continuous and independent random variables. Our results are generalizations of some earlier results. Additionally, an illustrative example on the probability distribution is given to support our results. [ABSTRACT FROM AUTHOR]

We establish a Simpson-type identity of multiparameter and certain Simpson-type inequalities via k-fractional integrals. Worth mentioning, the obtained inequalities in this article generalize some results presented by Set et al. [Simpson type integral inequalities for convex functions via Riemann-Liouville integrals. Filomat. 2017;31(14):4415–4420] and Sarikaya et al. [On new inequalities of Simpson's type for s-convex functions. Comput Math Appl. 2010;60:2191–2199]. As applications, we also provide several inequalities for f-divergence measures and probability density functions. We expect that this study will be result in the new k-fractional integration explorations for Simpson-type inequalities. [ABSTRACT FROM AUTHOR]

Using Euler-type identities some new generalizations of Steffensen’s inequality for n–convex functions are obtained. Moreover, the Ostrowski-type inequalities related to obtained generalizations are given. Furthermore, using inequalities for the Čebyšev functional in terms of the first derivative some new bounds for the remainder in identities related to generalizations of Steffensen’s inequality are proven.

New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results. [ABSTRACT FROM AUTHOR]

Any function f from (0,∞) onto (0,∞1) which is decreasing and convex has an inverse g which is positive, decreasing and convex. When f has some form of generalized convexity we determine additional convexity properties inherited by g. When f is positive, decreasing and p; q-convex, its inverse g is q; p-convex. Related properties which pertain when f is a Stieltjes function are developed. The results are illustrated with the Stieltjes function f (x) = arctan(1/sqrt(x))/sqrt(x) via a transcendental equation. [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]

The purpose of the present paper is to establish some new retarded weakly singular integral inequalities of Gronwall-Bellman type for discontinuous functions, which generalize some known weakly singular and impulsive integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of singular differential equations and singular impulsive equations. [ABSTRACT FROM AUTHOR]

We provide the monotonicity and convexity properties and sharp bounds for the generalized elliptic integrals $\mathscr{K}_{a}(r)$ and $\mathscr {E}_{a}(r)$ depending on a parameter $a\in(0,1)$ , which contains an earlier result in the particular case $a=1/2$ . [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]

By some new analysis techniques, we generalize the results presented by Pachpatte in [Integral and Finite Difference Inequalities and Applications, volume 205, Elsevier, 2006] and and by S. D. Kendre in [Some nonlinear integral inequalities for Volterra–Fredholm integral equations, Adv. Inequal. App, 2014:Article 21, 2014.] to nonlinear retarded inequalities, and investigate also some new forms. Some applications are also presented in order to illustrate the usefulness of some of our results. [ABSTRACT FROM AUTHOR]

The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green's functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green's functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and show that the weighted multivariate operator geometric mean possess several attractive properties and means inequalities. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a new concept "extended ((s1, m1)-(s2, m2))-convex function on co-ordinates" and establish some new inequalities of the Hermite-Hadamard type for extended ((s1, m1)-(s2, m2))-convex functions of 2 variables on co-ordinates. [ABSTRACT FROM AUTHOR]

The authors first introduce the concepts of generalized $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results. [ABSTRACT FROM AUTHOR]