A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

A Bismut–Elworthy inequality for a Wasserstein diffusion on the circle

We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $$L_2$$ L 2 -Wasserstein spaces. For a specific example of a diffusion on the $$L_2$$ L 2 -Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $$\mathcal O(t^{-(2+\varepsilon )})$$ O ( t - ( 2 + ε ) ) .

Partial factorizations of products of binomial coefficients

Let [Formula: see text] the product of the elements of the [Formula: see text]th row of Pascal’s triangle. This paper studies the partial factorizations of [Formula: see text] given by the product [Formula: see text] of all prime factors [Formula: see text] of [Formula: see text] having [Formula: see text], counted with multiplicity. It shows [Formula: see text] as [Formula: see text] for a limit function [Formula: see text] defined for [Formula: see text]. The main results are deduced from study of functions [Formula: see text] that encode statistics of the base [Formula: see text] radix expansions of the integer [Formula: see text] (and smaller integers), where the base [Formula: see text] ranges over primes [Formula: see text]. Asymptotics of [Formula: see text] and [Formula: see text] are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.

Asymptotics with remainder term for moments of the total cycle number of random A-permutation

Dedicated to the memory of Alexander Ivanovich Pavlov. We consider the set of n-permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A-permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

A novel computational method for solving nonlinear Volterra integro-differential equation

In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.

Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space

We study homogenization of a second-order elliptic differential operator Aε = - div a(x/ε)∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)-1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)-1 of the homogenized operator A0 = - div a0 ∇, a0 > 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.