DISCRETE INEQUALITIES OF WIRTINGER’S

Recent Progress in Inequalities A Volume Dedicated to Prof D S Mitrinovi´c c 1997 Kluwer Academic Publishers DISCRETE INEQUALITIES OF WIRTINGER’S TYPE ´ and IGOR Z ˇ MILOVANOVIC ´ GRADIMIR V MILOVANOVIC Faculty of Electronic Engineering, Department of Mathematics, P.O Box 73, 18000 Niˇs, Yugoslavia Abstract Various discrete versions of Wirtinger’s type inequalities are considered A short account on the first results in this field given by Fan, Taussky and Todd [10] as well as some generalisations of these discrete inequalities are done Also, a general method for finding the best possible constants An and Bn in inequalities of the form An n X k=1 pk x2k ≤ n X k=0 rk (xk − xk+1 )2 ≤ Bn n X pk x2k , k=1 where p = (pk ) and r = (rk ) are given weight sequences and x = (xk ) is an arbitrary sequence of the real numbers, is presented Two types of problems are investigated and several corollaries of the basic results are obtained Further generalisations of discrete inequalities of Wirtinger’s type for higher differences are also treated Introduction and Preliminaries In the well-known monograph written by Hardy, Littlewood and P´ olya [13, pp 184–187] the following result was mentioned as the Wirtinger’s inequality: Theorem 1.1 Let f be a periodic function with period (2π) and such that f ′ ∈ 2π L2 (0, 2π) If f (x) dx = then 2π 2π f ′ (x)2 dx, (1.1) f (x) dx ≤ with equality in (1.1) if and only if f (x) = A cos x + B sin x, where A and B are constants Also, this inequality can be found in the monograph of Beckenbach and Bellman [4, pp 177–180] and, especially, in one written by Mitrinovi´c in cooperation with 1991 Mathematics Subject Classification Primary 26D15; Secondary 41A44, 33C45 Key words and phrases Discrete inequalities; Difference; Eigenvalues and eigenvectors; Best constants; Orthogonal polynomials This work was supported in part by the Serbian Scientific Foundation, grant number 04M03 Typeset by AMS-TEX 289 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 290 Vasi´c [25, pp 141–154], including many other inequalities of the same type The proof of W Wirtinger was first published in 1916 in the book [5] by Blaschke However, inequality (1.1) was known before this, though with other conditions on the function f The French and Italian mathematical literature not mention the name of Wirtinger in connection with this inequality A historical review on the priority in this subject was given by Mitrinovi´c and Vasi´c [24] (see also [25–26]) They have mentioned various generalisations and variations of inequality (1.1), as well as possibility of applications of such kind of inequalities in many branches in mathematics as Calculus of Variations, Differential and Integral Equations, Spectral Operator Theory, Numerical Analysis, Approximation Theory, Mathematical Physics, etc Under some condition of f , there are also many generalisations of (1.1) which give certain estimates of quotients of the form b w(x, y)f (x, y)2 dxdy w(x)f (x)2 dx a D , b f ′ (x)2 dx D a ∂f ∂x + ∂f ∂y , dxdy where w is a weight function (in one or two variables) and D is a simply connected plane domain There are various discrete versions of Wirtinger type inequalities In this survey we will deal only with such kind of inequalities The paper is organised as follows In Section we give a summary on the first results in this field given by Fan, Taussky and Todd [10] as well as some generalisations of these discrete inequalities In Section we present a general method for finding the best possible constants An and Bn in inequalities of the form n An k=1 n pk x2k ≤ k=0 n rk (xk − xk+1 )2 ≤ Bn pk x2k , k=1 where p = (pk ) and r = (rk ) are given weight sequences and x = (xk ) is an arbitrary sequence of the real numbers This method was introduced by authors [19] and later used by other mathematicians (see e.g., [1] and [36]) In the same section we give several corollaries of the basic results Finally, generalisations of discrete inequalities of Wirtinger’s type for higher differences are treated in Section Discrete Fan-Taussky-Todd Inequalities and Some Generalisations The basic discrete analogues of inequalities of Wirtinger were given by Fan, Taussky and Todd [10] Their paper has been inspiration for many investigations in this subject We will mention now three basic results from [10]: DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 291 Theorem 2.1 If x1 , x2 , , xn are n real numbers and x1 = 0, then n−1 (2.1) k=1 (xk − xk+1 )2 ≥ sin2 π 2(2n − 1) n x2k , k=2 with equality in (2.1) if and only if xk = A sin (k − 1)π , 2n − k = 1, 2, , n, where A is an arbitrary constant Theorem 2.2 If x0 (= 0), x1 , x2 , , xn , xn+1 (= 0) are given real numbers, then n (2.2) k=0 (xk − xk+1 )2 ≥ sin2 π 2(n + 1) with equality in (2.2) if and only if xk = A sin an arbitrary constant n x2k , k=1 kπ , k = 1, 2, , n, where A is n+1 Theorem 2.3 If x1 , x2 , , xn , xn+1 are given real numbers such that x1 = xn+1 and n (2.3) xk = 0, k=1 then n (2.4) k=1 (xk − xk+1 )2 ≥ sin2 π n n x2k k=1 The equality in (2.4) is attained if and only if xk = A cos 2kπ 2kπ + B sin , n n k = 1, 2, , n, where A and B are arbitrary constants Let A be a real symmetric matrix of the order n, and R be a diagonal matrix of the order n with positive diagonal elements For the generalised matrix eigenvalue problem (2.5) Ax = λRx, x = [ x1 T xn ] , the following results are well known (cf Agarwal [1, Ch 11]): ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 292 1◦ There exist exactly n real eigenvalues λ = λν , ν = 1, , n, which need not be distinct 2◦ Corresponding to each eigenvalue λν there exists an eigenvector xν which can be so chosen that n vectors x1 , , xn are mutually orthogonal with respect to the matrix R = diag (r11 , , rnn ), i.e., n i T rkk xik xjk = j (x ) Rx = (i = j), k=1 In particular, these vectors are linearly independent 3◦ If A is a tridiagonal real symmetric matrix of the form a1  b1   Hn (a, b) =     (2.6) b1 a2 b2 O O b2 a3 bn−1 bn−1 an     ,   where a = (a1 , , an ), b = (b1 , , bn−1 ) and b2k > for k = 1, , n − 1, then the eigenvalues λν of the matrix A are real and distinct 4◦ If R = I and the eigenvalues λν of A are arranged in an increasing order, i.e., λ1 ≤ · · · ≤ λn , then for any vector x ∈ Rn , we have that (2.7) λ1 (x, x) ≤ (Ax, x) ≤ λn (x, x), n xk yk is the scalar product of the vectors where (x, y) = k=1 x = [ x1 ··· T xn ] and y = [ y1 ··· T yn ] In the case λ1 < λ2 the equality λ1 (x, x) = (Ax, x) holds if and only if x is a scalar multiple of x1 Similarly, if λn > λn−1 the equality (Ax, x) = λn (x, x) holds if and only if x is a scalar multiple of xn Further, for any vector x orthogonal to x1 ((x, x1 ) = 0), we have (2.8) λ2 (x, x) ≤ (Ax, x) If λ1 < λ2 = λ3 < λ4 , then a vector x orthogonal to x1 satisfies the equality λ2 (x, x) = (Ax, x) if and only if x is a linear combination of x2 and x3 5◦ If the real symmetric matrix A is positive definite, i.e., for every nonzero x ∈ Rn , (Ax, x) > 0, then the eigenvalues λν (ν = 1, , n) are positive In a particular case when R = I and A = Hn (a, b) is positive definite, then the DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 293 eigenvalues λν (ν = 1, , n) can be arranged in a strictly increasing order, < λ1 < · · · < λn Note that inequalities (2.1), (2.2) and (2.4) are based on the left inequality in (2.7) (i.e., (2.8)) The right inequality in (2.7) has not been used, so that in [10] we cannot find some opposite inequalities of (2.1), (2.2) and (2.4) As special cases of certain general inequalities, the opposite inequalities of (2.1), (2.2) and (2.4) were first proved in [19] (see also [2]) Using a method similar to one from [10], Block [6] obtained several inequalities related to (2.1), (2.2) and (2.4), as well as some generalisations of such inequalities For example, Block has proved the following result: Theorem 2.4 For real numbers x1 , x2 , , xn (= 0), xn+1 = x1 , the inequality n (2.9) k=1 (xk − xk+1 )2 ≥ sin π 2n n x2k k=1 holds, with equality in (2.9) if and only if xk = A sin(kπ/n), k = 1, 2, , n, where A is an arbitrary constant A number of generalisations of (2.1), (2.2) and (2.4) were given by Novotna ([27] and [29]) We mention here three of them Theorem 2.5 For real numbers x1 , x2 , , xn satisfying (2.3), the inequality n−1 (2.10) k=1 (xk − xk+1 )2 ≥ sin2 π 2n n x2k k=1 holds, with equality in (2.10) if and only if xk = A sin((2k − 1)π/(2n)), k = 1, 2, , n, where A is an arbitrary constant Theorem 2.6 Let n = 2m and let x1 , x2 , , xn , xn+1 = x1 be real numbers such that (2.3) holds Then n k=1 (xk − xk+1 )2 ≥ sin2 π n n x2k + n sin k=1 2π π π sin (xm + x2m )2 , − sin n n n with equality if and only if xk = A cos(2kπ/n) + B sin(2kπ/n), where A and B are arbitrary constants k = 1, 2, , n, ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 294 Theorem 2.7 For real numbers x1 , x2 , , xn satisfying (2.3), the inequality n−1 k=1 (xk − xk+1 )2 ≥ sin2 π 2n n x2k + 2n sin k=1 π π π sin − sin (x1 + xn )2 2n n 2n holds, with equality if and only if xk = A sin((2k − 1)π/(2n)), k = 1, 2, , n, where A is an arbitrary constant Using some appropriate changes, Novotna [27] showed that inequalities (2.1), (2.2) and (2.10) can be obtained from (2.4) She proved the basic Theorem 2.3 using the real trigonometric polynomials Namely, she used the fact that for every number xi there exist the Fourier coefficients Ck and Cj∗ (k = 0, 1, , m; j = 1, , m − 1) such that m−1 Ck cos xi = C0 + k=1 2πki 2πki + (−1)i Cm , + Ck∗ sin n n ≤ i ≤ n For details on this method see for example [1] New proofs of inequalities (2.1), (2.2) and (2.4) were given by Cheng [8] His method is based on a connection with discrete boundary problems of the SturmLiouville type (2.11) ∆ p(k − 1)∆u(k − 1) + q(k)u(k) + λr(k)u(k) = 0, k = 1, , n, u(0) = λu(1), u(n + 1) = βu(n) For some details of this method see Agarwal [1, Ch 11] Another method of proving these inequalities was based on geometric facts in Euclidean space (cf Shisha [32]) A Spectral Method and Using Orthogonal Polynomials In this section we consider our method (see [19]) for determining the best constants An and Bn in the inequalities n (3.1) n An k=1 pk x2k ≤ k=0 n rk (xk − xk+1 )2 ≤ Bn pk x2k , k=1 under some conditions for a sequence of real numbers x = (xk ), where p = (pk ) and r = (rk ) are given weight sequences The method is based on the minimal and maximal zeros of certain class of orthogonal polynomials, which satisfy a three-term recurrence relation For two N -dimensional real vectors z = [ z1 zN ] T and w = [ w1 wN ]T DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 295 N we define the usual inner product by (z, w) = zk wk and consider the sums k=1 n F = k=0 If we put form n rk (xk − xk+1 )2 and pk x2k G= k=1 √ pk xk = yk (k = 1, , n), then F and G can be transformed in the n F = k=0 √ rk √ pk+1 yk − pk yk+1 pk pk+1 and = (HN (a, b)y, y) n yk2 = (y, y), G= k=1 N where y ∈ R and HN (a, b) is a three-diagonal matrix like (2.6), with N = n or N = n − 1, depending on the conditions for the sequence x = (xk ) Especially, we will consider the following two cases: 1◦ x0 = xn+1 = and x1 , , xn are arbitrary real numbers (N = n); 2◦ x1 = and x2 , , xn are arbitrary real numbers (N = n − 1) For such three-diagonal matrices we can prove the following auxiliary result ([19]): Lemma 3.1 Let p = (pk ) and r = (rk ) be positive sequences and the matrix Hn (a, b) be given by (2.6) 1◦ If the sequences a = (a1 , , an ) and b = (b1 , , bn−1 ) are defined by r0 + r1 rn−1 + rn , , , p1 pn r1 rn−1 b = −√ , , , −√ p1 p2 pn−1 pn a= (3.3) then the matrix Hn (a, b) is positive definite 2◦ If the sequences a = (a1 , , an−1 ) and b = (b1 , , bn−2 ) are defined by rn−2 + rn−1 rn−1 r1 + r2 , , , , p2 pn−1 pn r2 rn−1 , b = −√ , , −√ p2 p3 pn−1 pn a= (3.2) then the matrix Hn−1 (a, b) is positive definite We will formulate our results in terms of the monic orthogonal polynomials (πk ) instead of orthonormal polynomials as we made in [19] Such an approach gives a simpler and nicer formulation than the previous one 296 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC The monic polynomials orthogonal on the real line with respect to the inner product (f, g) = R f (t)g(t)dµ(t) (with a given measure dµ(t) on R) satisfy a fundamental three-term recurrence relation of the form (3.5) πk+1 (t) = (t − αk )πk (t) − βk πk−1 (t), with π0 (t) = and π−1 (t) = (by definition) The coefficients βk are positive The coefficient β0 , which multiplies π−1 (t) = in three-term recurrence relation may be arbitrary Sometimes, it is convenient to define it by β0 = R dµ(t) Then the norm of πk can be express in the form (3.6) (π, πk ) = πk = β0 β · · · β k An interesting and very important property of polynomials πk (t), k ≥ 1, is the distribution of zeros Namely, all zeros of πn (t) are real and distinct and are (n) located in the interior of the interval of orthogonality Let τν , ν = 1, , n, denote the zeros of πn (t) in an increasing order (3.7) (n) τ1 (n) < τ2 (n) It is easy to prove that the zeros τν the following tridiagonal matrix α √0  β1   Jn = Jn (dµ) =      O < · · · < τn(n) of πn (t) are the same as the eigenvalues of √ β1 α1 √ β2 O √ β2 α2 βn−1     ,   βn−1  αn−1 which is known as the Jacobi matrix Also, the monic polynomial πn (t) can be expressed in the following determinant form πn (t) = det(tIn − Jn ), where In is the identity matrix of the order n For some details on orthogonal polynomials see [17] and [23] Regarding to the conditions on the sequence x = (xk ), we consider now two important cases: √ Case 1◦ (x0 = xn+1 = 0) If we take αk−1 = −ak and βk = −bk (i.e., βk = b2k > 0), k ≥ 1, then we can consider the matrix Hn (−a, −b) = −Hn (a, b), defined by (2.6), as a Jacobi matrix for certain class of orthogonal polynomials (πk ) Thus, for every y ∈ Rn we have (Hn (a, b)y, y) = (−Hn (−a, −b)y, y) = (−Jn y, y) DISCRETE INEQUALITIES OF WIRTINGER’S TYPE and 297 (n) −τn(n) (y, y) ≤ (−Jn y, y) ≤ −τ1 (y, y), (n) where the zeros τν , ν = 1, , n, of πn (t) are given in an increasing order (3.7) On the other hand, putting π ∗ (t) = [ π0∗ (t) π1∗ (t) T ∗ πn−1 (t) ] and en = [ 0 T 1] , where πk∗ (t) = πk (t)/ πk , we have (cf Milovanovi´c [18, p 178]) tπ ∗ (t) = Jn π ∗ (t) + βn πn∗ (t)en (n) This means that for the eigenvalue t = τν of Jn , the corresponding eigenvector (n) is given by π ∗ (τν ) Notice also that the same eigenvector corresponds to the (n) eigenvalue −τν of the matrix −Jn Therefore, the following theorem holds Theorem 3.2 Let p = (pk )k∈N0 and r = (rk )k∈N0 be two positive sequences, αk−1 = − rk−1 + rk , pk βk = rk2 pk pk+1 (k ≥ 1), and let (πk ) be a sequence of polynomials satisfying (3.5) Then for any sequence of real numbers x0 (= 0), x1 , , xn , xn+1 (= 0), inequalities n (3.8) An k=1 n pk x2k ≤ k=0 n rk (xk − xk+1 )2 ≤ Bn (n) (n) pk x2k , k=1 (n) hold, with An = −τn and Bn = −τ1 , where τν , ν = 1, , n, are zeros of πn (t) in an increasing order (3.7) Equality in the left (right ) inequality (3.8) holds if and only if C πk−1 (t) xk = √ · , pk πk−1 (n) where t = τn k = 1, , n, (n) (t = τ1 ), πk is given by (3.6) and C is an arbitrary constant Some corollaries of this theorem are the following results: Corollary 3.3 For each sequence of the real numbers x0 (= 0), x1 , , xn , xn+1 (= 0), the following inequalities hold : (3.9) sin2 π 2(n + 1) n n k=1 x2k ≤ k=0 (xk − xk+1 )2 ≤ cos2 π 2(n + 1) n x2k k=1 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 298 Equality in the left inequality (3.9) holds if and only if xk = C sin kπ , n+1 k = 1, , n, where C is an arbitrary constant Equality in the right inequality (3.9) holds if and only if xk = C(−1)k sin kπ , n+1 k = 1, , n, where C is an arbitrary constant Proof For pk = rk = we obtain αk = −2 and βk = for each k Consequently, the recurrence relation (3.5) becomes πk+1 (t) = (t + 2)πk (t) − πk−1 (t), π0 (t) = 1, π−1 (t) = Putting t + = 2x and πk (t) = Sk (x), this relation reduces to the three-term recurrence for Chebyshev polynomials of the second kind Sk+1 (x) = 2xSk (x) − Sk−1 (x), S0 (x) = 1, S1 (x) = 2x Thus, we have (cf Milovanovi´c [17, pp 143–144]) (3.10) πk (t) = Sk (x) = sin(k + 1)θ , sin θ cos θ = x = t+2 , and therefore the zeros of πn (t) are (in an increasing order) (3.11) τν(n) = −4 sin2 θν , θν = (n + − ν)π , n+1 ν = 1, , n Thus, the best constants in (3.9) are An = −τn(n) = sin2 π 2(n + 1) and (n) Bn = −τ1 = sin2 nπ π = cos2 2(n + 1) 2(n + 1) π/2 for each k, using (3.10) and (3.11) we find the extremal Since Sk = sequences for the left and the right inequality in (3.9) For example, for the right inequality we have (n) cos kπ nπ kπ sin kθ1 πk−1 (τ1 ) =− = sin k sin = , πk−1 sin θ1 sin θ1 n+1 sin θ1 n+1 DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 299 from which follows xk = C(−1)k sin kπ n+1 (k = 1, , n), where C is an arbitrary constant Remark 3.1 Theorem 2.2 is contained in Corollary 3.3 In a more general case we can take pk = (a + bk)2 and rk = (a + bk)(a + b(k + 1)), with a, b ≥ When b = we obtain Corollary 3.3 However, if b = 0, because of homogeneity in (3.8), it is enough to put b = In that case, we obtain the same polynomials as in Corollary 3.3 Corollary 3.4 For each sequence of the real numbers x0 (= 0), x1 , , xn , xn+1 (= 0), the following inequalities (3.12) sin2 π 2(n + 1) n n k=1 (k + a)2 x2k ≤ k=0 (k + a)(k + a + 1)(xk − xk+1 )2 ≤ cos2 π 2(n + 1) n (k + a)2 x2k k=1 hold, where a ≥ Equality in the left inequality (3.12) holds if and only if xk = kπ C sin , k+a n+1 k = 1, , n, where C is an arbitrary constant Equality in the right inequality (3.12) holds if and only if xk = kπ C(−1)k sin , k+a n+1 k = 1, , n, where C is an arbitrary constant Remark 3.2 The corresponding inequalities for a = were considered in [19] Corollary 3.5 For each sequence of the real numbers x0 (= 0), x1 , , xn , xn+1 (= 0), we have n (3.12) An k=1 n x2k ≤ k=0 n k(xk − xk+1 )2 ≤ Bn x2k , k=1 where An and Bn are minimal and maximal zeros of the monic Laguerre polynomial Ln (x), respectively 300 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC Equality in the left (right ) inequality (3.12) holds if and only if xk = CLk−1 (x)/(k − 1)! (k = 1, , n), where x = An (x = Bn ) and C is an arbitrary constant In this case we have αk = −(2k + 1) and βk = k , so that the relation (3.5) becomes πk+1 (t) = (t + 2k + 1)πk (t) − k πk−1 (t) Putting t = −x and πk (−x) = (−1)k Lk (x), this relation reduces to one, which corresponds to the monic Laguerre polynomials orthogonal on (0, +∞) with respect to the measure dµ(x) = e−x dx The norm of Lk (x) is given by Lk = k! In a more general case we can take (3.13) r0 = 0, rk = , B(s + 1, k) pk = (k + s)B(s + 1, k) (k ≥ 1), where s > −1 and B(p, q) is the beta function (B(p, q) = Γ(p)Γ(q)/Γ(p + q), Γ is the gamma function) Then we have αk = −(2k + s + 1) and βk = k(k + s), and the corresponding recurrence relation, after changing variable t = −x and πk (−x) = (−1)k Lsk (x), becomes (3.14) Lsk+1 (x) = (x − (2k + s + 1))Lsk (x) − k(k + s)Lsk−1 (x), where Lsk (x), k = 0, 1, , are the generalised monic Laguerre polynomials orthogonal on (0, +∞) with respect to the measure dµ(x) = xs e−x dx Thus, we have the following result: Corollary 3.6 Let s > −1 and let r = (rk )k∈N0 and p = (pk )k∈N be given by (3.13) For each sequence of real numbers x0 (= 0), x1 , , xn , xn+1 (= 0), we have n (3.15) An k=1 n pk x2k ≤ k=0 n rk (xk − xk+1 )2 ≤ Bn pk x2k , k=1 where An and Bn are minimal and maximal zeros of the monic generalised Laguerre polynomial Lsn (x), respectively Equality in the left (right ) inequality (3.15) holds if and only if xk = CLsk−1 (x) (k − 1)!Γ(k + s) (k = 1, , n), where x = An (x = Bn ) and C is an arbitrary constant Case 2◦ (x1 = 0) Here, in fact, we consider the inequalities n−1 n (3.16) An k=1 pk x2k ≤ k=1 n rk (xk − xk+1 )2 ≤ Bn pk x2k , k=1 DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 301 for any sequence of the real numbers x1 (= 0), x2 , , xn Using Lemma 3.1 (Part 2◦ ) we put N = n − 1, (3.17) rk+1 rk + rk+1 , βk = pk+1 pk+1 pk+2 √ = −ak , βk = −bk (k ≥ 1) Taking (k ≥ 1), αk−1 = − and also αk−1 π ∗ (t) = [ π0∗ (t) π1∗ (t) T ∗ πn−2 (t) ] and en−1 = [ 0 T 1] , where πk∗ (t) = πk (t)/ πk , we have, as in the previous case, tπ ∗ (t) = Jn−1 π ∗ (t) + ∗ (t)en−1 , βn−1 πn−1 but now Hn−1 (a, b) = −Hn−1 (−a, −b) = −Jn−1 − rn Dn−1 , pn where Dn−1 = diag (0, , 0, 1) So, we obtain that ∗ βn−1 πn−1 (t) − Hn−1 (a, b)π ∗ (t) + tπ ∗ (t) = rn ∗ π (t) en−1 , pn n−2 from which we conclude that the eigenvalues of Hn−1 (a, b), in notation λν = −τν , ν = 1, , n − 1, are the zeros of the polynomial ∗ βn−1 πn−1 (t) − (3.18) rn ∗ π (t) pn n−2 The corresponding eigenvectors are π ∗ (τν ) βn−1 , the polynomial (3.18) can be reduced to one repSince πn−1 = πn−2 resented in terms of the monic polynomials, (3.19) Rn−1 (t) = πn−1 (t) − rn πn−2 (t) pn Theorem 3.7 Let p = (pk )k∈N and r = (rk )k∈N be two positive sequences, αk−1 and βk (k ≥ 1) be given by (3.17), and let (πk ) be a sequence of polynomials satisfying (3.5) Then for any sequence of real numbers x1 (= 0), x2 , , xn , inequalities (3.16) hold, with An = min{−τν } Bn = max{−τν }, where τν , ν = ν ν 1, , n − 1, are zeros of the polynomial Rn−1 (t) given by (3.19) Equality in the left (right ) inequality (3.16) holds if and only if x1 = 0, C πk−2 (t) , xk = √ · pk πk−2 k = 2, , n, where t = −An (t = −Bn ), πk is given by (3.6) and C is an arbitrary constant Some corollaries of this theorem are the following results: ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 302 Corollary 3.8 For each sequence of real numbers x1 (= 0), x2 , , xn , the following inequalities hold : (3.20) sin2 n π 2(2n − 1) k=2 n−1 x2k ≤ k=1 (xk − xk+1 )2 ≤ cos2 π 2n − n x2k k=2 Equality in the left inequality (3.20) holds if and only if xk = C sin (k − 1)π , 2n − k = 1, , n, where C is an arbitrary constant Equality in the right inequality (3.20) holds if and only if xk = C(−1)k sin 2(k − 1)π , 2n − k = 1, , n, where C is an arbitrary constant Here we have (as in Corollary 3.3) that πk (t) = Sk (x) = sin(k + 1)θ , sin θ and Rn−1 (t) = Sn−1 (x) − Sn−2 (x) = and therefore τν = −4 sin2 νπ , 2n − t + = 2x, cos((2n − 1)θ/2) , cos(θ/2) ν = 1, , n − Corollary 3.9 Let s > −1 and let r = (rk )k∈N and p = (pk )k∈N be given by (3.21) r1 = 0, rk+1 = 1 , pk+1 = B(s + 1, k) (k + s)B(s + 1, k) (k ≥ 1) For each sequence of real numbers x1 (= 0), x2 , , xn , we have n−1 (3.22) k=1 n rk (xk − xk+1 )2 ≤ Bn pk x2k , k=2 where Bn is a maximal zero of the monic generalised Laguerre polynomial Ls+1 n−2 (x) Equality in (3.22) holds if and only if (3.23) x1 = 0, xk = C(−1)k Lsk−2 (Bn ) , Γ(k + s − 1) k = 2, , n, DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 303 where C is an arbitrary constant Proof Taking πk (−x) = (−1)k Lsk (x), with (3.21) we obtain the recurrence relation (3.14), so that the polynomial (3.19) becomes Rn−1 (t) = πn−1 (t) − (n + s − 1)πn−2 (t) = (−1)n−1 Lsn−1 (−t) + (n + s − 1)Lsn−2 (−t) = (−1)n tLs+1 n−2 (−t) Thus, Bn is a maximal zero of the monic generalised Laguerre polynomial Ls+1 n−2 (x) Evidently, An = Since (k + s − 1)Γ(s + 1)(k − 2)! · Γ(k + s) πk−2 (−Bn ) = √ · pk πk−2 = (−1)k (−1)k−2 Lsk−2 (Bn ) (k − 2)!Γ(k + s − 1) Γ(s + 1) s L (Bn ), Γ(k + s − 1) k−2 we obtain the extremal sequence (3.23) for which the equality is attained in (3.22) Remark 3.3 A few members of the monic generalised Laguerre polynomials Ls+1 k (x) are Ls+1 (x) = 1, Ls+1 (x) = x − (s + 2), Ls+1 (x) = x − 2(s + 3)x + (s + 2)(s + 3), Ls+1 (x) = x − 3(s + 4)x − (s + 3)(s + 12)x − (s + 2)(s + 3)(s + 4) It is not difficult to show that B3 = s + 2, B4 = s + + √ s + Remark 3.4 For s = the inequality (3.22) reduces to (see [19]) n−1 X n X k=1 k=2 (k − 1)(xk − xk+1 )2 ≤ Bn x2k , where Bn is a maximal zero of the monic generalised Laguerre polynomial L1n−2 (x) Remark 3.5 If for every k we take xk = (−1)k ak the inequalities (3.1) become An n X k=1 pk |ak |2 ≤ n X k=0 rk |ak + ak+1 |2 ≤ Bn n X k=1 Moreover, these inequalities are valid for complex numbers too pk |ak |2 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC 304 At the end of this section we mention some results of Losonczi [15] He considered inequalities of the form n α± i (3.24) k=0 |xk |2 ≤ i n |xk ± xk+m |2 ≤ βi± k=0 |xk |2 , where x0 , x1 , , xn are real or complex numbers, ≤ m ≤ n, summation symbols defined by: n−m = , k=0 n = k=0 n−m = k=−m n = k=−m with xn+1 = · · · = xn+m = 0, with x−m = · · · = x−1 = 0, with x−m = · · · = x−1 = = xn+1 = · · · = xn+m , ± α± i , βi (i = 1, 2, 3, 4) are constants and either the + or the − sign is taken It is easy to see that the cases i = and i = are the same apart from the notation of the variables xk Hence there are different cases in (3.24) corresponding to i = 1, or i = 3, and the + or − sign Losonczi found the best constants α± i and βi± in all cases and it was based on the determination of eigenvalues of some suitable Hermitian matrices Theorem 3.10 Let n and m be fixed natural numbers (1 ≤ m ≤ n) and r = [n/m] The inequalities (3.24) hold for every real or complex numbers x0 , x1 , , xn , with the best constants: π ; 2(r + 1) π − + − α+ , = α2 = α3 = α3 = sin 2(2r + 3) π β2+ = β2− = β3+ = β3− = cos2 ; 2r + π π − , β4+ = β4− = cos2 α+ = α4 = sin 2(r + 2) 2(r + 2) − α+ = α1 = 0, β1+ = β1− = cos2 Remark 3.6 In connection with extremal properties of nonnegative trigonometric polynomials Szeg˝ o [33] and Egerv´ ary and Sz´ asz [9] proved that for every complex numbers x0 , x1 , , xn the inequalities (3.25) −γ n X k=0 |xk |2 ≤ n−m X k=0 (xk x ¯k+m + x ¯k xk+m ) ≤ γ n X k=0 |xk |2 DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 305 holds, with the best constant γ = cos(π/(r + 2)), where r = [n/m] The case m = was previously proved by Fej´er [11] It is clear that the inequalities (3.25) are related to (3.24) Inequalities for Higher Differences In this section we give a short account on generalisations of Wirtinger’s type inequalities to higher differences The first results for the second difference were proved by Fan, Taussky and Todd [10]: Theorem 4.1 If x0 (= 0), x1 , x2 , , xn , xn+1 (= 0) are given real numbers, then n−1 (4.1) k=0 (xk − 2xk+1 + xk+2 )2 ≥ 16 sin4 with equality in (4.1) if and only if xk = A sin an arbitrary constant π 2(n + 1) n x2k , k=1 kπ , k = 1, 2, , n, where A is n+1 Theorem 4.2 If x0 , x1 , , xn , xn+1 are given real numbers such that x0 = x1 , xn+1 = xn and (2.3) holds, then n−1 (4.2) k=0 (xk − 2xk+1 + xk+2 )2 ≥ 16 sin4 π 2n n x2k k=1 The equality in (4.2) is attained if and only if xk = A cos (2k − 1)π , 2n k = 1, 2, , n, where A is an arbitrary constant A converse inequality of (4.1) was proved by Lunter [16], Yin [36] and Chen [7] (see also Agarwal [1]) Theorem 4.3 If x0 (= 0), x1 , x2 , , xn , xn+1 (= 0) are given real numbers, then n−1 (4.3) k=0 (xk − 2xk+1 + xk+2 )2 ≤ 16 cos4 π 2(n + 1) with equality in (4.3) if and only if xk = A(−1)k sin A is an arbitrary constant Chen [7] also proved the following result: n x2k , k=1 kπ , k = 1, 2, , n, where n+1 306 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC Theorem 4.4 If x0 , x1 , , xn , xn+1 are given real numbers such that x0 = x1 and xn+1 = xn , then n−1 k=0 (xk − 2xk+1 + xk+2 )2 ≤ 16 cos4 π 2n n x2k , k=1 with equality holding if and only if xk = A(−1)k sin (2k − 1)π , n k = 1, 2, , n, where A is an arbitrary constant Proof In this case, the n × n symmetric matrix corresponding to the quadratic form n−1 F2 = k=0 is (xk − 2xk+1 + xk+2 )2 = (Hn,2 x, x)  −3  −3 −4     −4  −4     =   −4 −4     −4 −3  −3  Hn,2 This matrix is the square of the n × n matrix  −1  −1 −1   −1 (4.4) Hn = Hn,1 =     −1 −1        −1  −1 The eigenvalues of Hn are λν = λν (Hn ) = cos2 (n − ν + 1)π , 2n ν = 1, , n, and therefore, the largest eigenvalue of Hn is λn (Hn ) = cos2 π > λn−1 (Hn ) 2n The corresponding eigenvector is xn = [ x1n xνn = (−1)ν sin (2ν − 1)π , 2n x2n xnn ]T , where ν = 1, 2, , n DISCRETE INEQUALITIES OF WIRTINGER’S TYPE 307 Thus, the largest eigenvalue of Hn,2 is λn (Hn,2 ) = 16 cos4 π > λn−1 (Hn,2 ), 2n and the associated eigenvector is xn Remark 4.1 Notice that the minimal eigenvalue of the matrix Hn (and also Hn,2 ) is λ1 = Therefore, the condition (2.3) must be included in Theorem 4.2 and the best constant is the square of the relevant eigenvalue λ2 = cos2 (n − 1)π π = sin2 2n 2n For any n-dimensional vector x = [x1 x2 xn ]T , Pfeffer [30] introduced a periodically extended n-vector by setting xi+rn = xi for i = 1, 2, , n and r ∈ N, and used the mth difference of x given by x(m) = [∆m x1 ∆m x2 ∆m xn ]T , where m m xi−[m/2]+r , ≤ i ≤ n, ∆m xi = (−1)m−r r r=0 in order to prove the following result: Theorem 4.5 If x is a periodically extended n-vector and (2.3) holds, then (x(m) , x(m) ) ≥ sin2 π n m (x, x), with equality case if and only if x is the periodic extension of a vector of the form C1 u + C2 v, where u = [ u1 u2 T un ] v = [ v1 and v2 have the following components uk = cos 2kπ , n vk = sin 2kπ , n k = 1, , n, and C1 and C2 are arbitrary real constants Recently we have studied inequalities of the form (see [21]) um n (4.5) An,m k=1 x2k ≤ n k=lm (∆m xk ) ≤ Bn,m x2k , k=1 where lm = − [m/2], um = n − [m/2] and m (−1)i m ∆ xk = i=0 m xk+m−i i T ] 308 ´ AND I Z ˇ MILOVANOVIC ´ G V MILOVANOVIC um The quadratic form Fm = (∆m xk ) for m = reduces to k=lm n−1 F1 = x21 + k=2 n−1 2x2k + x2n − xk xk+1 , k=1 with corresponding tridiagonal symmetric matrix Hn = Hn,1 given by (4.4) Under conditions xs = x1−s , xn+1−s = xn+s (lm ≤ s ≤ 0) we proved that the corresponding matrix of the quadratic form Fm is exactly the mth power of the matrix Hn = Hn,1 so that 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