Cauchy inequality

Problem—Show that d2F/da2 > 0 so that Eq. (4-159) actually minimizes F. Hint the Cauchy inequality can be used to show that... 

From the Cauchy inequality, the difference of the first two terms of Eq. (4-228) is non-positive. Writing out the part of the third term in curly brackets, we get... 

Applying the Cauchy inequality, this term is also non-positive, giving... 

Substituting equation (A.6) into (A.5), we obtain (A.4). In a general case of n dimensional Euclidean space, the triangular inequality comes from the Cauchy inequality, which we will discuss below. 

Using the dot product operation, we can prove the Cauchy inequality ... 

We can prove now a very important inequality, which is the generalization of the Cauchy inequality for Euclidean space. 

Relations (6.17), (6.18) give rise to estimates of the mean values of rk for any k > -3 with any desired accuracy, uniformly with respect to large enough R values, when small fixed neighborhoods of the points r = 0 and r = oo are excluded from the region of integration. As for 0 < a < b < oo and r e [a, b, the values for rk are bounded by some constant C2 and one may use the notes of paper [97] to find the following relation due to the Cauchy inequality ... 

The Cauchy inequality [2], also known as the Schwarz inequality, states that for any two vectors a and b... 

Next use the Cauchy inequality, Equation 331, to show the right-hand... 

Relationships between phases and magnitudes that anticipated the later developments were the inequalities of Marker and Kasper. They derived a number of inequalities by application of the Schwarz and Cauchy inequalities to the structure factor equations (15) in the presence of crystallographic symmetry. The Marker-Kasper inequalities have provided valuable... 

After the set of determinantal inequalities (18) had been obtained on the basis of the non-negativity of the electron density distribution in a crystal, it was of interest to investigate their relationship to the inequalities derived by Marker and Kasper from use of the Schwarz and Cauchy inequalities, It was shown that, when the appropriate symmetry was introduced into the third-order determinant inequality by means of certain relationships among the structure factors, the Marker-Kasper inequalities could be derived. Examination of the derivation of the Harker-Kasper inequalities shows, as would be expected, that the non-negativity of the electron density distribution is a requirement for their validity,... 

Lemma 2 For any positive self-adjoint operator A in a real Hilbert space the generalized Cauchy-Bunyakovskii inequality holds ... 

The Cauchy-Bunyakovskii inequality and the e-inequality. In the sequel we shall need the well-known Cauchy-Bunyakovskii inequality... 

Let us estimate the sums on the right-hand side of the preceding relation with the aid of the Cauchy-Bunyakovskii inequality ... 

To majorize the right-hand side, we make use of the Cauchy-Bunyakovskii inequality and the e-inequality a6 < a + i with e = permitting... 

By applying successively the Cauchy-Bunyakovskii inequality and the -inequality we arrive at the chain of the relations... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

Proof The energy identity (13) is involved at the first stage. Estimation of it.s right-hand side 2r(9j, y ) is stipulated by successive use of the generalized Cauchy-Bunyakovskii inequality and the e-inequality... 

The generalized Cauchy-Bunyakovskii inequality and the gr-inequality together yield... 

Thus the average cost per share for John is the arithmetic mean ofpi,p2, , p, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only ifpi = p2 = = Pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors... 

Somewhat weaker conditions are obtained by summing the Cauchy-Schwarz inequalities over r, 5, for example. 

Also known as the Cauchy-Bunyakovskii-Schwarz inequality. 

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