Sharpening Jordan’s inequality and the Yang Le inequality

In this work, the following inequality: $\frac{sinx}{x}\le \frac{2}{\pi }+\frac{\pi -2}{{\pi }^3}({\pi }^2-4x^2)\text{,}x\in (0,\pi /2]$ is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]: where $A_i>0(i=1,2,\dots ,n),{\sum }_{i=1}^n{A}_i\le \pi ,0\le \lambda \le 1$, and $n\ge 2$ is a natural number.     

On the ?ojasiewicz-Simon gradient inequality

We prove a general version of the ?ojasiewicz-Simon inequality, and we show how to apply the abstract result to study energy functionals E of the form

     

Lebesgue points via the Poincar inequality

We show that in a Q doubling space(X, d, μ), Q 1, which satisfies a chain condition, if we have a Q Poincar′e inequality for a pair of functions(u, g) where g ∈ LQ(X), then u has Lebesgue points Hh a.e. for h(t) = log1-Q-ε(1/t). We also discuss how the existence of Lebesgue points follows for u ∈ W 1,Q(X) where(X, d, μ) is a complete Q doubling space supporting a Q Poincar′e inequality for Q 1.     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.     

A remark on the Alexandrov–Fenchel inequality

In this article, we give a complex-geometric proof of the Alexandrov–Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp–Lieb proof of the Prékopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge–Riemann bilinear relation and a mixed norm version of Hörmander's $L^2$-estimate, which also implies a non-compact version of the Khovanski?–Teissier inequality.     

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”. Both authors were partially supported by the University of Bologna, funds for selected research topics.     

An inequality for the function π(n)

We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π ²(m+ n) ≤ π ²(m) + π ²(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n.     

Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.     

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.     

On extension and refinement of the Poincaré inequality

The aim of this paper is to analyze the heat semigroup ${(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}$ generated by the usual Laplacian operator Δ on ${\mathbb{R}^{d}}$ equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup.     

The hyperbolic meaning of the Milnor–Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.     

On the Dirichlet problem generated by the Maz’ya–Sobolev inequality

We discuss the attainability of sharp constants for the Maz’ya–Sobolev inequalities in wedges, “perturbed” wedges and bounded domains. This gives also the solvability of boundary value problems to semilinear equations with critical growth and “fat” singularity at the boundary.     

Hardy–Littlewood–Sobolev inequality on the parabolic biangle

The Ramanujan Journal - We define the heat semigroup associated with a system of bivariate Jacobi polynomials which are orthogonal with respect to a probability measure on the parabolic biangle...