A Discrete Analogue of Opial's Inequality

1967 ◽  
Vol 10 (1) ◽  
pp. 115-118 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Discrete Analogue ◽  
Absolutely Continuous ◽  
Opial’S Inequality ◽  
Image Position ◽  
Opial's Inequality

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,(1)Equality holds only if x(t) = Kt for some constant K.

On a Discrete Analogue of Inequalities of Opial and Yang

1968 ◽  
Vol 11 (1) ◽  
pp. 73-77 ◽  
Author(s):  
Cheng-Ming Lee
Keyword(s):  
Integral Inequality ◽  
Discrete Analogue ◽  
Absolutely Continuous ◽  
Negative Numbers ◽  
Image Position

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we haveYang [3] proved the following integral inequality:If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then

A General and Sharpened form of Opial's Inequality

1974 ◽  
Vol 17 (3) ◽  
pp. 385-389 ◽  
Author(s):  
D. T. Shum
Keyword(s):  
Differentiable Function ◽  
Opial’S Inequality ◽  
Image Position ◽  
Opial's Inequality ◽  
Sharpened Form

Z. Opial [11] proved in 1960 the following theorem:Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then1where the constant b/4 is the best possible.

scholarly journals Weighted Opial inequalities

2002 ◽  
Vol 33 (1) ◽  
pp. 83-92
Author(s):  
J. J. Koliha ◽  
J. Pecaric
Keyword(s):  
Difference Equations ◽  
Fractional Derivatives ◽  
Kluwer Acad ◽  
General Inequality ◽  
Opial’S Inequality ◽  
Opial's Inequality

This paper presents a class of very general weighted Opial type inequalities. The notivation comes from the monograph of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Acad., Dordrecht 1995) and the work of Anastassiou and Pecaric (J. Math. Anal. Appl. 239 (1999), 402-418).  Assuming only a very general inequality, we extend the latter paper in several directions.  A new result generalizing the original Opial's inequality is obtained, and applications to fractional derivatives are given.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 337-342 ◽  
Author(s):  
M. S. Srivastava
Keyword(s):  
Random Variables ◽  
Nonparametric Tests ◽  
Absolutely Continuous ◽  
Bivariate Case ◽  
Image Position ◽  
Tests For Independence ◽  
Mutually Independent

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

On orthogonality of Riesz products

1974 ◽  
Vol 76 (1) ◽  
pp. 173-181 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Haar Measure ◽  
Classical Result ◽  
Weak Limit ◽  
Absolutely Continuous ◽  
Riesz Product ◽  
Image Position ◽  
Riesz Products ◽  
Positive Integers

A typical Riesz product on the circle is the weak* limitwhere – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

A dichotomy for infinite convolutions of discrete measures

1973 ◽  
Vol 73 (2) ◽  
pp. 307-316 ◽  
Author(s):  
G. Brown ◽  
W. Moran
Keyword(s):  
Number Theory ◽  
Basic Property ◽  
Discrete Measure ◽  
Absolutely Continuous ◽  
Image Position ◽  
Discrete Measures ◽  
Infinite Convolution

Measures, μ which can be realized as an infinite convolutionwhere each measure μn is a discrete measure, arise naturally in many parts of analysis and number theory (see (15)). The basic property of these measures is ‘purity’; i.e. such a measure μ 1must be absolutely continuous, continuous and singular, or discrete.

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