Bernstein’s Inequality for the Weyl Derivatives of Trigonometric Polynomials in the Space L0

Mathematical Notes - A logarithmic asymptotics for the behavior with respect to n of the exact constant in Bernstein’s inequality for the Weyl derivative of positive noninteger order of...     

The Möbius Transformation and Smirnov’s Inequality for Polynomials

Differential inequalities for polynomials generalizing the well-known Smirnov, Rahman, Schmeisser, and Bernstein inequalities are obtained.

     

Voronovskaja’s Theorem and Iterations for Complex Bernstein Polynomials in Compact Disks

In this paper, firstly we prove the Voronovskaja’s convergence theorem for complex Bernstein polynomials in compact disks in , centered at origin, with quantitative estimates of this convergence. Secondly, we study the approximation properties of the iterates of complex Bernstein polynomials and we prove that they preserve in the unit disk (beginning with an index) the univalence, starlikeness, convexity and spirallikeness. Received: May 5, 2007 Revised: September 14, 2007 and November 11, 2007 Accepted: November 26, 2007     

Bernstein–Nikol’skii-Type Inequalities for Algebraic Polynomials from the Bergman Space in Domains of the Complex Plane

Ukrainian Mathematical Journal - We study Bernstein-type and Nikol’skii-type estimates for an arbitrary algebraic polynomial in regions of the complex plane.     

Brendle’s Inequality on Static Manifolds

We generalize Brendle’s geometric inequality considered in Brendle (Publ Math Inst Hautes Études Sci 117:247–269, 2013) to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chru?ciel and Simon (J Math Phys 42(4):1779–1817, 2001).     

Putnam’s Inequality for log-Hyponormal Operators

Let T be a bounded linear operator on a complex Hilbert space H. T $/in$ B(H) is called a log-hyponormal operator if T is invertible and log (TT ^*) log (T ^* T). Since a function log : (0,) (-,) is operator monotone, every invertible p-hyponormal operator T, i.e., (TT ^*) ^p (T ^* T ^p is log-hyponormal for 0 < p 1. Putnams inequality for p-hyponormal operator T is the following:$ \| (T^*T)^p-(TT^*)^p \|\leq\frac{p}{\pi}\int\int_{\sigma(T)}r^{2p-1}drd\theta $.In this paper, we prove that if T is log-hyponormal, then$ \| log(T^*T)-log(TT^*) \|\leq\frac{1}{\pi}\int\int_{\sigma(T)}r^{-1}drd\theta $.     

Langton’s Type Theorem on Algebraic Orbifolds

Acta Mathematica Sinica, English Series - In this paper, we prove the Langton’s type theorem on separatedness and properness for the moduli functor of torsion free semistable sheaves on...     

An Effective Łojasiewicz Inequality for Real Polynomials

Periodica Mathematica Hungarica -     

A Brief Account on Lagrange’s Algebraic Identity

In Indiscrete Thoughts [18], G.-C. Rota remarked, ??The mystery, as well as the glory of mathematics, lies not so much in the fact that abstract theories do turn out to be useful in solving problems, but, wonder of wonders, in the fact that a theory meant for one type of problem is often the only way of solving problems of entirely different kinds, problems for which the theory was not intended. These coincidences occur so frequently, that they must belong to the essence of mathematics.?? Indeed, it happens often that abstract mathematics leads to concrete applications, and real-life problems constitute a source of inspiration for sophisticated theories. The strong synergy between pure mathematics and its applications advocates for teaching methods that intertwine physical intuition with mathematical abstraction, and recognize the universality of mathematical laws throughout the sciences.     

Analogs of Korn’s Inequality on Heisenberg Groups

We give two analogs of Korn’s inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.

     

Hardy’s Inequality for the n-Dimensional Hankel Transform

Analysis Mathematica - The purpose of this paper is to prove a Hardy type inequality associated with the n-dimensional Hankel transform (n ≥ 1) for the exponent σ0 = n(2 ? p) +...     

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

On Korn’s Inequality

For β ∈ R, the authors consider the evolution system in the unknown variables u and α { ttu＋ xxxxu＋ xxtα＋（β＋｜｜ xu｜｜L2^2） xxu=f, ttα- xxα- xxtα- xxtu=0} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.     

On Korn’s Inequality

Two models based on the hydrostatic primitive equations are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions are established. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.     

An Extension of Bernstein Polynomials on the Semi–Axis

The paper deals with the approximation of functions f on (0,+), where f can be singular at the origin, by means of Bernstein-type sequences. Error estimates in weighted uniform spaces with some converse results are given.Research was supported in part by grant INDAM-GNIM Progetto Equazioni Integrali e Problemi di Algebra Lineare Connessi.     

Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product where ρ ₀ and ρ ₁ are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc. December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998.