scholarly journals On a Result of Levin and Stečkin

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Peng Gao
Keyword(s):  
Best Constant ◽  
Previous Inequality

The following inequality for0<p<1andan≥0originates from a study of Hardy, Littlewood, and Pólya:∑n=1∞((1/n)∑k=n∞ak)p≥cp∑n=1∞anp. Levin and Stečkin proved the previous inequality with the best constantcp=(p/(1-p))pfor0<p≤1/3. In this paper, we extend the result of Levin and Stečkin to0<p≤0.346.

scholarly journals A Note on Generalized Hardy-Sobolev Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
T. V. Anoop
Keyword(s):  
Integral Inequality ◽  
Sobolev Inequality ◽  
Banach Function Space ◽  
Weight Functions ◽  
Sobolev Inequalities ◽  
Best Constant ◽  
One Dimensional ◽  
Muckenhoupt Condition ◽  
Previous Inequality ◽  
The One

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2,   u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

scholarly journals Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Function Method ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Multiple Integral ◽  
Constant Factor ◽  
Best Constant ◽  
Necessary And Sufficient ◽  
Best Constant Factor ◽  
Hilbert Type

AbstractFor the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

scholarly journals A Note on Universal Bilinear Portfolios

2021 ◽  
Vol 9 (1) ◽  
pp. 11
Author(s):  
Alex Garivaltis
Keyword(s):  
Growth Factor ◽  
Trading Strategy ◽  
Weighted Averaging ◽  
Asset Market ◽  
Best Constant ◽  
Capital Growth ◽  
Universal Portfolios ◽  
Universal Portfolio ◽  
Technical Sense ◽  
Constant Rebalanced Portfolios

This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of “universal portfolios”. I generalize Cover’s benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hindsight for the realized sequence of asset prices. A bilinear trading strategy is a mini two-period active strategy whose final capital growth factor is linear separately in each period’s gross return vector for the asset market. I apply Thomas Cover’s ingenious performance-weighted averaging technique to construct a universal bilinear portfolio that is guaranteed (uniformly for all possible market behavior) to compound its money at the same asymptotic rate as the best bilinear trading strategy in hindsight. Thus, the universal bilinear portfolio asymptotically dominates the original (1-linear) universal portfolio in the same technical sense that Cover’s universal portfolios asymptotically dominate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact, like so many Russian dolls, one can get carried away and use these ideas to construct an endless hierarchy of ever more dominant H-linear universal portfolios.

scholarly journals The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

2021 ◽  
Author(s):  
Rupert L. Frank ◽  
David Gontier ◽  
Mathieu Lewin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Nonlinear Schrodinger Equation ◽  
Best Constant ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Bound State Case ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

Ramifications of Landau's inequality

1980 ◽  
Vol 86 (3-4) ◽  
pp. 175-212 ◽  
Author(s):  
Man Kam Kwong ◽  
A. Zettl
Keyword(s):  
Best Constant

SynopsisThe problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications

2021 ◽  
Vol 19 (1) ◽  
pp. 400-411
Author(s):  
Bing He ◽  
Yong Hong ◽  
Qiang Chen
Keyword(s):  
Operator Theory ◽  
Multiple Integral ◽  
Best Constant ◽  
Equivalent Parameter ◽  
Hilbert Type

Abstract In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

On the best constant in a Hardy–Sobolev inequality

2006 ◽  
Vol 85 (1-3) ◽  
pp. 171-180 ◽  
Author(s):  
Angelo Alvino ◽  
Vincenzo Ferone ◽  
Guido Trombetti
Keyword(s):  
Sobolev Inequality ◽  
Best Constant

scholarly journals Optimal and Suboptimal Control of a Standard Brownian Motion

2016 ◽  
Vol 26 (3) ◽  
pp. 383-394
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Cost Function ◽  
Suboptimal Control ◽  
Linear Control ◽  
Standard Brownian Motion ◽  
Final Time ◽  
Best Constant ◽  
Constant Control

Abstract The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.

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