scholarly journals Best Constants between Equivalent Norms in Lorentz Sequence Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
S. Barza ◽  
A. N. Marcoci ◽  
L. E. Persson
Keyword(s):  
Triangle Inequality ◽  
Sequence Spaces ◽  
Best Constants ◽  
Crucial Point ◽  
Dual Norm ◽  
Best Constant ◽  
Equivalent Norms ◽  
Level Sequence

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm∥x∥(p,s):=inf⁡{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representationsx=∑kx(k)in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

2010 ◽  
Vol 88 (1) ◽  
pp. 19-27 ◽  
Author(s):  
SORINA BARZA ◽  
JAVIER SORIA
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Best Constants ◽  
Sharp Constants ◽  
Dual Norm ◽  
Equivalent Norms ◽  
Weighted Lorentz Spaces ◽  
Weighted Lorentz Space

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Best Constants for the Inequalities between Equivalent Norms in Orlicz Spaces

2011 ◽  
Vol 59 (2) ◽  
pp. 165-174
Author(s):  
Ha Huy Bang ◽  
Nguyen Van Hoang ◽  
Vu Nhat Huy
Keyword(s):  
Orlicz Spaces ◽  
Best Constants ◽  
Equivalent Norms

scholarly journals Ekeland Variational Principle in the Variable Exponent Sequence Spaces ℓp(·)

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 375
Author(s):  
Monther R. Alfuraidan ◽  
Mohamed A. Khamsi
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Successful Treatment ◽  
Variable Exponent ◽  
Triangle Inequality ◽  
Sequence Spaces ◽  
Ekeland Variational Principle ◽  
The Core

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces ℓ p ( · ) . The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the establishment of the classical EVP, that has hitherto prevented a successful treatment of the modular case. As an application, we establish a modular version of Caristi’s fixed point theorem in ℓ p ( · ) .

A general version of the Hardy–Littlewood–Polya–Everitt (HELP) inequality

1984 ◽  
Vol 97 ◽  
pp. 9-20 ◽  
Author(s):  
Christer Bennewitz
Keyword(s):  
Differential Equation ◽  
Spectral Properties ◽  
Positive Function ◽  
Best Constants ◽  
The Other ◽  
Valid Inequality ◽  
General Version ◽  
Best Constant ◽  
Sturm Liouville

SynopsisThe inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt.For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.

BEST CONSTANTS IN NORMS OF RETARDED WICK PRODUCTS

2010 ◽  
Vol 13 (03) ◽  
pp. 347-361 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Vector ◽  
Best Constants ◽  
Wick Product ◽  
Best Constant ◽  
Wick Products ◽  
Normally Distributed

We find the best constant c(m, n, r), such that the inequality: [Formula: see text] holds for all polynomials f and g of degree at most m and n, respectively, where X is a normally distributed random vector, ⋄r denotes the r-retarded Wick product and ‖⋅‖ the L2-norm.

Best constants in the L2-Nash inequality

2001 ◽  
Vol 131 (3) ◽  
pp. 621-646 ◽  
Author(s):  
Emmanuel Humbert
Keyword(s):  
Best Constants ◽  
Best Constant ◽  
Second Best ◽  
Nash Inequality

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manif

Best constants in the L2-Nash inequality

2001 ◽  
Vol 131 (3) ◽  
pp. 621-646 ◽  
Author(s):  
Emmanuel Humbert
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Best Constants ◽  
Best Constant ◽  
Second Best ◽  
Nash Inequality

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manifold.

Extremals in Landau's inequality for the difference operator

1987 ◽  
Vol 107 (3-4) ◽  
pp. 299-311
Author(s):  
Man Kam Kwong ◽  
A. Zettl
Keyword(s):  
Difference Operator ◽  
Sequence Spaces ◽  
Best Constants ◽  
The Other ◽  
The Difference

SynopsisThe best constants in Landau's inequality for the difference operator in the classical sequence spaces lp are known explicitly only for p = 1, 2, ∞. This is true in both the infinite N = (0, 1, 2, …) and biinfinite Z= (… − 1, 0, 1, …) cases. It is known that there are no extremals when p = 2 in both the infinite and biinfinite cases. Also, it is known that there are extremals when p = ∞ in the biinfinite case. Here we prove that there are no extremals in the other three cases where the best constants are known explicitly. The proofs for these three cases are quite different from each other.

Discrete and shift Kolmogorov type inequalities

1983 ◽  
Vol 93 (3-4) ◽  
pp. 307-317 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Function Spaces ◽  
Sequence Spaces ◽  
Best Constants ◽  
Kolmogorov Type ◽  
Kolmogorov Inequality

SynopsisThe operators Δhf ≡ f(x) on function spaces and Δxn ≡ xn+1–xn on sequence spaces replace derivatives to yield analogues of the Kolmogorov inequality. Estimates for best constants are given for many spaces and for a few the best constants are actually given.

Similarity, separability, and the triangle inequality.

1982 ◽  
Vol 89 (2) ◽  
pp. 123-154 ◽  
Author(s):  
Amos Tversky ◽  
Itamar Gati
Keyword(s):  
Triangle Inequality
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