scholarly journals A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, II

2007 ◽  
Vol 20 (5) ◽  
pp. 532-538 ◽  
Author(s):  
Shanhe Wu ◽  
Lokenath Debnath
Keyword(s):  
Jordan’S Inequality ◽  
Sharp Version

scholarly journals A note on Jordan’s inequality

2020 ◽  
Vol 28 (2) ◽  
pp. 97-102
Author(s):  
Emil C. Popa
Keyword(s):  
Jordan’S Inequality

AbstractIn this paper we obtain some bounds in terms of polynomials for the function {{\sin x} \over x}, x ∈ [0, π].

A SHARP VERSION OF BONSALL’S INEQUALITY

2015 ◽  
Vol 92 (3) ◽  
pp. 397-404
Author(s):  
T. C. PEACHEY
Keyword(s):  
Reverse Inequality ◽  
Classical Inequality ◽  
The Best Possible Constant ◽  
Sharp Version

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.

On Jordan’s inequality

2017 ◽  
Vol 77 (2) ◽  
pp. 191-200 ◽  
Author(s):  
Horst Alzer ◽  
Man Kam Kwong
Keyword(s):  
Jordan’S Inequality

scholarly journals A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

2021 ◽  
Author(s):  
Peter Hintz
Keyword(s):  
Order Term ◽  
Angular Frequency ◽  
Leading Order ◽  
Full Generality ◽  
Asymptotically Flat ◽  
Low Energies ◽  
Flat Spacetimes ◽  
Explicit Constant ◽  
Sharp Version ◽  
Linear Scalar

AbstractWe prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

scholarly journals A new refined Jordan's inequality and its application

2009 ◽  
pp. 255-264 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Young-Ho Kim ◽  
S. K. Sen
Keyword(s):  
Jordan’S Inequality

scholarly journals On the Continuity of the Integrated Density of States in the Disorder

2019 ◽  
Author(s):  
Mira Shamis
Keyword(s):  
Density Of States ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
Quantitative Form ◽  
Alternative Approach ◽  
Ky Fan Inequalities ◽  
Ky Fan ◽  
Sharp Version

Abstract Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\mathbb{R}^d$.

scholarly journals New Refinements and Improvements of Jordan’s Inequality

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 284 ◽  
Author(s):  
Lina Zhang ◽  
Xuesi Ma
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Quadratic Polynomial ◽  
Quartic Polynomial ◽  
Polynomial Bounds ◽  
Jordan’S Inequality

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

scholarly journals Two hyperbolic Schwarz lemmas

2002 ◽  
Vol 66 (1) ◽  
pp. 17-24 ◽  
Author(s):  
L. Bernal-González ◽  
M. C. Calderón-Moreno
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Sharp Version

In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

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