Minkowski’s inequality and sums of squares

Positive polynomials arising from Muirhead’s inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski’s inequality can be rewritten as sums of squares.     

Refinements of Fejér’s inequality for convex functions

In this paper, we establish some new refinements for the celebrated Fejér??s and Hermite-Hadamard??s integral inequalities for convex functions.     

Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ? Y ? L ₁[0, 1] ⁿ and {e _k } _k be an expanding sequence of finite sets in ? ⁿ with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? ⁿ . The properties of the sequence of norms ${ left| {S_{e_k } (f)} right|x} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L _p [0, 1], the Lorentz spaces L _p,q [0, 1], L _p,q [0, 1] ⁿ , and the anisotropic Lorentz spaces L _p,q*[0, 1] ⁿ are considered. In the one-dimensional case, the sequence {e _k } _k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? ⁿ . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L _p,q [0, 1] ⁿ and L _p,q*[0, 1] ⁿ are obtained.     

A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements

We obtain a new generalization of Chebyshev’s inequality for random vectors. Then we extend this result to random elements taking values in a separable Hilbert space.     

Estimates of Weyl’s exponential sums for very small values of the variable

We consider Weyls exponential sums for very small values of the variable. In this case we can give an asymptotic transformation formula. Weyls exponential sums will be usually estimated by means of Weyls or Vinogradovs method. Here we use van der Corputs method and obtain sufficiently good results in the present case.Received: 14 January 2002     

A spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that

Generalization of Lundberg’s inequality for the case of stock insurance company

The ruin probability of an insurance company paying dividends according to a barrier strategy with a step barrier function is considered. Upper bounds for the probability of ruin are obtained within the framework of Sparre Andersen and Cramer–Lundberg risk models.     

A new sharp approximation for the Gamma function related to Burnside’s formula

In this paper, based on Burnside’s formula, a similar continued fraction approximation of the factorial function and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside’s formula and the classical Stirling’s series, some numerical computations are given.     

Necessary and sufficient conditions for the validity of Jensen’s inequality

We consider the d-dimensional Jensen inequality $$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$ T [ φ ( f 1 , … , f d ) ] ≥ φ ( T [ f 1 ] , … , T [ f d ] ) ( * ) as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set ${K\subset \mathbb{R}^d}$ K ? R d , and f ₁, . . . , f _d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen’s inequality holds for a sublinear functional T if and only if T is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen’s inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results.     

A generalization of the Chung-Erdös inequality for the probability of a union of events

A generalization of the Chung-Erdös inequality for the probability of a union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least m events from a set of events A₁,..., A_n, where 1 ≤ m ≤ n. Bibliography: 6 titles.     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

A sharp lower bound of Burkholder’s functional for K-quasiconformal mappings and its applications

In this paper, for \(K\) -quasiconformal mappings of a bounded domain into the complex plane, we build a sharp lower bound of Burkholder’s functional. As an application, we give two explicit and sharp lower bounds of Burkholder’s integrals for two subclasses of \(K\) -quasiconformal mappings, respectively. As the second application, we obtain a sharp upper bound of the \(L^p\) -integral of \(\sqrt{J_f}\) for certain \(K\) -quasiconformal mappings.     

Gårding’s inequality for elliptic operators with degeneracy

In this paper, we prove Gårding’s weighted inequality for degenerate elliptic operators in an arbitrary (bounded or unbounded) domain of n-dimensional Euclidean space ? ⁿ and use this inequality to study the unique solvability of a specific variational problem. It is assumed that the lower coefficients of the operators under consideration belong to some weighted L _p -spaces.     

On the sharp Jackson-Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant C _n,m in the Jackson-Nikol’skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n ≥ 0 on the unit sphere \(\mathbb{S}^{m - 1} \) of the Euclidean space ? ^m is studied. Two-sided estimates for the constant C _n,m are obtained, which, in particular, give the order n ^m?1 of its behavior with respect to n as n → +∞ for a fixed m.     

On the sharp Jackson-Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant C _n,m in the Jackson-Nikol’skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n ≥ 0 on the unit sphere $ \mathbb{S}^{m - 1} $ \mathbb{S}^{m - 1} of the Euclidean space ℝ ^m is studied. Two-sided estimates for the constant C _n,m are obtained, which, in particular, give the order n ^m−1 of its behavior with respect to n as n → +∞ for a fixed m.