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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]

Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen's difference, a wide range of estimations are formalised by considering Jensen-Grüss differences. [ABSTRACT FROM AUTHOR]

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the L p space for multiple power parameter α ¯ = (α 1 , … , α n + 1) when p > 0 . Based on this L p , α ¯ -curvilinear summation for sets and the concept of compression of sets, the L p , α ¯ -curvilinear-Brunn–Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of L p , α ¯ Borell–Brascamp–Lieb inequality, as well as its normalized version, for functions containing the special case of the L p Borell–Brascamp–Lieb inequality through the L p , α ¯ -curvilinear-Brunn–Minkowski inequality for sets. Moreover, we propose the multiple power L p , α ¯ -supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the L p , α ¯ -curvilinear summation for sets as well as L p , α ¯ -supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for p ≥ 1 , etc. [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]

We describe some chosen ideas and results for more than 100 years prehistory and history of the remarkable development concerning Hardy-type inequalities. In particular, we present a newer convexity approach, which we believe could partly have changed this development if Hardy had discovered it. In order to emphasize the current very active interest in this subject, we finalize by presenting some examples of the recent results, which we believe have potential not only to be of interest for a broad audience from a historical perspective, but also to be useful in various applications. [ABSTRACT FROM AUTHOR]

In this paper, by considering the identity established by Luo et al. in [C. Luo, T.-S. Du, M. Kunt and Y. Zhang, Certain new bounds considering the weighted Simpson-like type inequality and applications, J. Inequal. Appl. 2018 2018, Paper No. 332] and under the assumption of the quasi-convexity of the first derivative, we establish some new error estimates of the Simpson-like type inequalities. We also discuss the case where the first derivative satisfies the Hölder condition. At the end, we provide some applications to special means. The obtained results represent a continuation of the above-mentioned paper. [ABSTRACT FROM AUTHOR]

We obtain global explicit numerical bounds, with best possible constants, for the differences 1 n ∑ k ⩽ n ω (k) - log log n and 1 n ∑ k ⩽ n Ω (k) - log log n , where ω (k) and Ω (k) refer to the number of distinct prime divisors, and the total number of prime divisors of k, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an universal definition (fgh-convex) that results in several types of convexity. Particular cases of the fgh-convex are for instance the harmonically convex, geometrically convex, GA-convex, log-convex, and several others. Also, we obtain some useful inequalities such as Jensen, generalization of Jensen Hermite-Hadamard, Mercer inequalities. Moreover, with the use of these inequalities, we obtained bounds for Shannon's entropy and Kapur's entropy. Finally, we found an application of the obtained inequalities in means. [ABSTRACT FROM AUTHOR]

In this study, the Jensen-Mercer inequality for a uniformly convex function is established. There are also certain application-related inequalities that are presented. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish some new generalized Gamidov type integral inequalities involving a $\psi $-fractional operator. We also give two applications to substantiate the validity of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of the coefficients appearing in the q-series expansion of ∏ n ≥ 1 [ (1 - q n) / (1 - q pn) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p = 3 case. [ABSTRACT FROM AUTHOR]

We introduce a generic inequality of the form f 2 (x) ≤ f (s) f (2 x − s) and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions arcsin ⁡ x and arctan ⁡ x. [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.

In this paper, we discuss new iterated discrete inequalities of Hardy type involving matrix kernels. Under some conditions on matrix entries of the kernels, we establish weight characterizations of these inequalities. [ABSTRACT FROM AUTHOR]

In the paper, the authors present the Schur m-power convexity and concavity for the general geometric Bonferroni mean of multiple parameters and establish comparison inequalities for bounding the general geometric Bonferroni mean in terms of the arithmetic, geometric, and harmonic means. These Schur convexity and concavity provide a unified generalization of the Schur convexity and concavity for the geometric Bonferroni means of two or three parameters. [ABSTRACT FROM AUTHOR]

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]

The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ℝnwith smaller central hyperplane sections necessarily have smaller volumes. The solution has been completed and the answer is affirmative if n ⩽ 4 and negative if n ⩾ 5. In this paper, we investigate the Busemann-Petty problem on entropy of log-concave functions: for even log-concave functions f and g with finite positive integrals in ℝn, if the marginal ∫ ℝ n ∩ H f (x) d x of f is smaller than the marginal ∫ ℝ n ∩ H g (x) d x of g for every hyperplane H passing through the origin, is the entropy Ent(f) of f bigger than the entropy Ent(g) of g? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, and hence its answer is negative when n ⩾ 5. For 2 ⩽ n ⩽ 4, we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions. [ABSTRACT FROM AUTHOR]

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].

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young's and Hölder's inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser's theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice. [ABSTRACT FROM AUTHOR]

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.

In this paper, the concept of co-ordinated (log, (s, m))-preinvex functions is introduced. Some new fractional Hermite–Hadamard type inequalities based on new integral identity are established. [ABSTRACT FROM AUTHOR]

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]

We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space ℓ p with p ∈ (0 , 1 ] , we develop characterizations which enable us to reduce the problem to another one with p = 1 . This, in turn, makes it possible to establish an equivalence of the weighted discrete inequality to an appropriate inequality for iterated Hardy-type operators acting on measurable functions defined on R , for all cases of involved positive exponents. [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]