Inequalities related to the Cauchy-Schwarz inequality

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1 Sankhyā : he Indian Journal of Statistics 01, Volume 74-A, Part 1, pp c 01, Indian Statistical Institute Inequalities related to the Cauchy-Schwarz inequality R. Sharma, R. Bhandari and M. Gupta Himachal Pradesh University, Shimla, India Abstract We obtain an inequality complementary to the Cauchy-Schwarz inequality in Hilbert space. he inequalities involving first three powers of a self-adjoint operator are derived. he inequalities include the bounds for the third central moment, as a special case. It is shown that an upper bound for the spectral radius of a matrix is a root of a particular cubic equation, provided all eigenvalues are positive. AMS (000) subject classification. Primary 47A50, 15A4, 60E15; Secondary 47A63, 6D15. Keywords and phrases. Eigenvalues, standard deviation, the Samuelson inequality, skewness, trace, and the Cauchy-Schwarz inequality. 1 Introduction he Cauchy-Schwarz inequality in a Hilbert space X implies that 1 u, u 1 u, u u, u, (1.1) where u is an arbitrary vector in X, 1 and are self-adjoint operators on X to itself. Let 0 <m 1 1 M 1. hen, for every unit vector u in X, 1 u, u 1 1 u, u (m 1 + M 1 ) 4m 1 M 1. (1.) he inequality (1.) is a well-known version of the Kantorovich inequality, see Greub and Rheinboldt (1959). A related inequality asserts that (Krasnoselskii and Krein, 195; Mond and Shisha, 1970) 1 u, u 1 u, u (m 1 + M 1 ). (1.3) 4m 1 M 1

2 10 R. Sharma, R. Bhandari and M. Gupta he inequalities (1.) and (1.3) are special cases of a more general inequality, see Diaz and Metcalf (1965) which says that for 0 <m 1 1 M 1 and 0 <m M, 1 u, u u, u (m 1m + M 1 M ) 1 u, u. (1.4) 4m 1 m M 1 M Various refinements, extensions, and generalizations of such basic inequalities have been considered in the literature; in particular, Bhatia and Davis (000) have obtained the following interesting bound on the variance of 1 : var ( 1 )= 1 u, u 1 u, u ( 1 u, u m 1 )(M 1 1 u, u ). (1.5) We note that the inequality (1.4) is a generalization of the inequalities (1.) and (1.3), and furnishes a complementary inequality to the Cauchy-Schwarz inequality (1.1) when written in the form (see Diaz and Metcalf, 1965), 4m 1 m M 1 M (m 1 m + M 1 M ) 1 u, u u, u 1 u, u 1 u, u u, u. (1.6) Our main result (heorem.1) gives a generalization of the inequality (1.5) for two positive operators and provides yet another complementary lower bound for the Cauchy-Schwarz inequality (1.1). We get an alternative proof of the inequality (1.4) and the bounds for 3 u, u in terms of the ratio of u, u and u,u, 0 <a b (Corollaries.1 and.). his motivates us to investigate the bounds for 3 u, u in terms of the prescribed values of a, b, u,u, and u, u (heorem. and Corollary.3). A special case gives the bounds for the third central moment of a discrete or continuous random variable, taking values in a given finite interval (Corollary.4); see also Sharma et al. (009) and Shohat and amarkin (1963). Wilkins (1944) uses the method of Lagrange multipliers to prove the following upper bound for the measure of skewness: γ 1 n. (1.7) n 1 We show that the bounds for the third central moment together with the Samuelson inequality immediately implies the inequality (1.7), (Corollary.5). A related inequality gives an upper bound for the largest eigenvalue of a positive definite matrix C. his upper bound comes out to be the root of a particular cubic equation whose coefficients are the functions of traces of C, C,andC 3 (heorem 3.1). Our results compare favorably with those obtained by Wolkowicz and Styan (1980).

3 Inequalities related to the Cauchy-Schwarz inequality 103 Main results heorem.1. Let 1 and be self-adjoint commuting operators on a Hilbert space X such that 0 <m 1 1 M 1 and 0 <m M.hen for every unit vector u in X 1 u, u u, u 1 u, u ( 1 u, u m)(m 1 u, u ) (M m), 4 (.1) where m = m 1 M u, u and M = M 1 m u, u. (.) Proof. For 0 <m 1 1 M 1 and 0 <m M,wehave m 1 M 1 M 1 m. herefore, ( 1 m )( ) 1 M1 1 M m is a positive operator. Hence, for every unit vector u in X 1 u, u ( m1 + M ) 1 1 u, u m 1M 1 M m m M u, u. (.3) On combining (.) and (.3) we easily get the left-hand inequality in (.1). he right-hand inequality in (.1) is immediate, apply the arithmetic meangeometric mean inequality, ( ) M m ( 1 u, u m)(m 1 u, u ). An alternative formulation of the inequality (1.4) follows from the lefthand inequality in (.1). Corollary.1. With 1,, m, andm as above 1 u, u u, u (m + M) 4mM 1 u, u. Proof. From (.1), we get 1 u, u u, u 1 u, u f ( 1 u, u ), (.4)

4 104 R. Sharma, R. Bhandari and M. Gupta where he derivative f (x) = f (x) = m + M x 3 (m + M) x mm x. (.5) ( ) mm m + M x, vanishes at x = mm m + M, and f (x) changes its sign from positive to negative while x passes through this value. herefore, f ( 1 u, u ) f ( ) mm. (.6) m + M We have ( ) mm (m + M) f = m + M 4mM. (.7) Combining (.4), (.5), (.6), and (.7) we immediately get the inequality (1.4). Corollary.. Let be a self-adjoint operator on Hilbert space X such that 0 <a b. hen, for every unit vector u in X, u, u u,u 3 u, u ( a + b ) u, u 4a b u,u. (.8) Proof. Substituting 1 = and = 3 with m 1 = a, M 1 = b, m = a 3/ and M = b 3/ in (1.6), we get on simplification the inequalities in (.8). heorem.. Let be a self-adjoint operator on Hilbert space X. hen, for a< <b, a u, u + ( a u,u u, u ) u,u a 3 u, u b u, u ( b u,u u, u ), (.9) b u,u where u is any unit vector in X.

5 Inequalities related to the Cauchy-Schwarz inequality 105 Proof. For x b, wehave (x α) (x b) 0, (.10) where α is any real number. From (.10), we get that f (x) =(α + b) x α (α +b) x + α b x 3 0. herefore, for every unit vector u in X, f ( ) u, u 0. (.11) he inequality (.11) implies that 3 u, u (α + b) u, u α (α +b) u,u + α b. (.1) Let g (α) =(α + b) u, u α (α +b) u,u + α b. he derivative g (α) = ( u, u b u,u + α (b u,u ) ) vanishes at α = α 1, where α 1 = b u,u u, u, (.13) b u,u and g (α) changes its sign from negative to positive while α passes through α 1, the function g (α) therefore assumes its minimum at α = α 1. he inequality (.1) is true for all real values of α and therefore also must hold good when α = α 1. Substituting the value of α = α 1 from (.13) in (.1) we get on simplification the right-hand inequality in (.9). he right-hand inequality applied to operator will immediately yield the left-hand inequality in (.9). Corollary.3. he right-hand inequality in (.9) may be written in several different ways, e.g., 3 u, u ( b u,u b u, u ) ( b u, u ) u,u, (.14) b u,u 3 u, u u, u b ( b u,u u, u ) ( u, u u,u ) + u,u u,u (b u,u ) (.15)

6 106 R. Sharma, R. Bhandari and M. Gupta and ( 3 u, u u,u u, u b u, u ) ( u, u u,u ) +. b u,u (.16) Likewise, we can write the left-hand inequality in (.9) in several different ways. Proof. he inequalities (.14) (.16) follow easily on simplifying and rearranging the terms in the right-hand inequality in (.9). Corollary.4. Let X be a discrete or continuous random variable taking values in [a, b] with a<μ 1 <b.hen μ (μ 1 a) μ μ 1 a μ 3 (b μ 1 ) μ μ b μ 1 (.17) where μ 1 = E [X] and μ r = E [X μ 1 ]r, r =, 3. Proof. It is enough to prove the result for the case when X is a discrete random variable taking finitely many values x 1,x,..., x n with probabilities p 1,p,..., p n, respectively. he argument is similar in all other cases. Start with the Euclidean space R n with standard inner product defined on it. Let u i = p i,i =1,,..., n, where p i are nonnegative real numbers with p i =1. Let : R n R n, defined as u =(x 1 u 1,x u,..., x n u n ). hen is a linear operator on R n, u, u = u,u = u i = x i u i = p i =1, p i x i = μ 1. (.18) Similarly, u, u = μ and 3 u, u = μ 3. (.19) On substituting values of u,u, u, u,and 3 u, u from (.18) and (.19) in (.9), we get aμ + (aμ 1 μ ) μ 1 a μ 3 bμ (bμ 1 μ ) b μ. (.0) 1 Also, on substituting μ 3 = μ 3 +3μ 1 μ + μ 3 1 and μ = μ + μ 1 in (.0), and simplifying a little we get the inequalities in (.17).

7 Inequalities related to the Cauchy-Schwarz inequality 107 Corollary.5. Let x i (i =1,,..., n) denote n real numbers. hen γ 1 n n 1 where γ 1 is the measure of skewness, γ 1 = 1 ( ) xi A 3, (.1) n S and S = 1 n A = 1 n x i (x i A). Proof. On substituting μ 1 = A, μ = S,andμ 3 = m 3 in the right-hand inequality in (.17), we find that m 3 S 3 b A S S b A, (.) where x i b, i =1,,..., n and m 3 = 1 (x i A) 3. n From the inequality (Samuelson, 1968), we have b A n 1. S he function ( ) b A f = b A S S S b A is an increasing function. herefore, ( ) b A f f ( n 1 ). (.3) S his implies that b A S S b A n. (.4) n 1 On combining (.1), (.), (.3), and (.4), we get the Wilkins inequality (1.7). he third central moment is an odd function of the observation. hus the Wilkins inequality applied to x i (i =1,,..., n) shows that it also holds for γ 1.

8 108 R. Sharma, R. Bhandari and M. Gupta 3 A related bound for eigenvalues Let C M n and let all the eigenvalues of C be real, like in the case of a Hermitian matrix. he bounds on eigenvalues using traces have been studied in literature, see Sharma (008), Sharma, Gupta and Kapoor (010) and Wolkowicz and Styan (1980). It is easy to calculate trc and trc.he calculations of trc 3 is relatively costly. However, it is of interest to know if better estimates can be obtained on using the bounds which involve the value of trc 3. A lower bound on the spectral radius using trc 3 is recently reported in Sharma et al. (010). he upper bound on the largest eigenvalue in terms of the traces, on using the Samuelson inequality, is obtained by Wolkowicz and Styan (1980). We show in the following theorem that an upper bound on the largest eigenvalue of C is a root of a cubic equation whose coefficients involve the values of trc, trc,andtrc 3. We give examples, and compare the present estimates with those obtained by Wolkowicz and Styan (1980). heorem 3.1. Let λ i 0(i =1,,..., n), be the eigenvalues of a complex n n matrix C. hen,λ j is less than or equal to the largest real root of the following cubic equation: trcx 3 trc x + trc 3 x + ( trc ) trctrc 3 =0. (3.1) Further, if the roots of (3.1) are all real then for each j =1,,..., n, either λ j is less than or equal to the smallest root of (3.1) or is greater than or equal to the second largest real root of (3.1). Proof. Let m 3 denotes the third order moment of the n eigenvalues λ i, we write he Cauchy-Schwarz inequality gives, m 3 = λ3 j n + n 1 ( ) 1 λ 3 i. (3.) n n 1 1 n 1 λ 3 i 1 n 1 1 n 1 λ i. (3.3) λ i

9 Inequalities related to the Cauchy-Schwarz inequality 109 We also note that the inequality (3.3) is related to the Corollary.. Combining (3.) and (3.3), we get We also have and m 3 λ3 j n + 1 n λ i. (3.4) λ i λ i = trc λ j (3.5) λ i = trc λ j. (3.6) Combining (3.4), (3.5), and (3.6), we arrive at trcλ 3 j trc λ j + trc 3 λ j + ( trc ) trctrc 3 0. (3.7) Let r 1, r,andr 3 denote the roots of the cubic equation (3.7). hen, (λ j r 1 )(λ j r )(λ j r 3 ) 0, (3.8) where j =1,,..., n. Letr 1 be real, r and r 3 are complex. We then have From (3.8) and (3.9), we find that (λ j r )(λ j r 3 ) 0. (3.9) λ j r 1, j =1,,..., n. We now consider the case when the roots of (3.7) are all real. Let r 1 r r 3. hen, it follows from (3.8) that either his proves the theorem. λ j r 1, or r λ j r 3.

10 110 R. Sharma, R. Bhandari and M. Gupta Example 3.1. Let λ 4 be the largest eigenvalue of C 1 = We have, trc 1 =, trc1 = 154, and trc3 1 = 101. he estimates of Wolkowicz and Styan (1980) gives λ It follows from the heorem 3.1 that λ 4 is less than or equal to the largest real root of the following cubic equation: x 3 308x + 101x 706 = 0. (3.10) he cubic equation (3.10) has only one real root, x herefore, λ We note that the value of λ 4 is around Example 3.. Let λ 5 be the largest eigenvalue of C = We have, trc = 30, trc =, and trc3 = 199. he estimates of Wolkowicz and Styan (1980) gives λ It follows from the heorem 3.1 that λ 5 is less than or equal to the largest real root of the following cubic equation: 10x 3 148x + 643x 86 = 0. (3.11) he cubic equation (3.11) has only one real root, x herefore, λ We note that the value of λ 5 is around Example 3.3. Let 1 0 C 3 = We have, trc 3 = 16, trc3 = 114, and trc3 3 = 946. he Gerschogrin theory assures that the eigenvalues of C 3 are positive real numbers, and the largest eigenvalue λ It follows from the heorem 3.1 that λ 3 is less than or equal to the largest real root of the following cubic equation: 8x 3 114x + 473x 1070 = 0. (3.1) he cubic equation (3.1) has only one real root, x herefore, λ We note that the value of λ 3 is around

11 Inequalities related to the Cauchy-Schwarz inequality 111 Acknowledgement. he first two authors wish to express their gratitude to Prof. Rajendra Bhatia for his helpful guidance and suggestions, and also thank Indian Statistical Institute for sponsoring their visit to New Delhi in January 009, when this work had begun. he authors acknowledge the support of the UGC-SAP. References bhatia, r. and davis, c. (000). A better bound on the variance. Amer. Math. Monthly, 107, diaz, j.b. and metcalf, f.t. (1965). Complementary inequalities III: Inequalities complementary to Schwarz s inequality in Hilbert space. Math. Ann., 16, greub, w. and rheinboldt, w. (1959). On a generalisation of an inequality of L.V. Kantorovich. Proc. Amer. Math. Soc., 10, krasnoselskii, m.a. and krein, s.g. (195). An iteration process with minimal residuals. Mat. Sb. N.S., 31, mond, b. and shisha, o. (1970). Difference and Ratio Inequalities In Hilbert Space. In Inequalities-II, (O. Shisha, ed.). Academic Press, New York, samuelson, p.a. (1968). How deviant you can be?. J. Amer. Statist. Assoc., 63, sharma, r. (008). Some more inequalities for arithmetic mean, harmonic mean and variance. J. Math. Inequal.,, sharma, r., devi, s., kapoor, g., ram, s. and barnett, n.s. (009). A brief note on some bounds connecting lower order moments for random variables defined on a finite interval. IJAS, 1, sharma, r., gupta, m. and kapoor, g. (010). Some better bounds on the variance with applications. J. Math. Inequal., 4, shohat, j.a. and tamarkin, j.d. (1963). he problem of moments. Mathematical surveys Number 1, Amer. Math. Soc. wilkins, j.e. (1944). A note on skewness and kurtosis. Ann. Math. Statist., 15, wolkowicz, h., and styan, g.p.h. (1980). Bounds for eigenvalues using traces, Linear Algebra Appl., 9, R. Sharma, R. Bhandari and M. Gupta Department of Mathematics Himachal Pradesh University Shimla, India rajesh hpu math@yahoo.co.in Paper received: 5 January 010; revised: 11 August 010.