Milovanović, Gradimir V.

Milovanović, Igor Ž.

English

Milovanović Gradimir V.

Milovanović Igor Ž.

University of Szeged

Acta Sei. Math. 47 (1984), 3^1—403 Some discrete inequalities of Opial's type GRADIMIR V. MILOVANOVlC and IGOR 1. MILOVANOVlC 1. Introduction Let us given an index set 1= {1, 2 , . . . , n) and weight sequences t—(rk)kei=^ = l / i , - , r „ ) and p=(pk)kiI=(Pi, •••,/>„)• For a sequence x=(xk)k(I=(x1, ...,x„) (1) l|x||r = ( J 1 ^xl)1/2 ic=i and (2) (x ,Vx)= ¿PkxkVxk, . . n = l where the sequence Vx is given by V x = ( x l 5 x 2 — . . . , x „ — x „ _ x ) . If we put x0=0 and Vxk=xk—xk-i (k=l, ..., ri), then the sequence Vx can be expressed in the form Vx=(Vx l5 Vx2 , . . . , Vx„). In this paper we determine the best constants A„ and B„ in the inequalities (3) A | | x | | ? S ( x , V x ) ^ i y x | | r V which are a. discrete analogue of inequalities of Opial's type (see, for example, [1, pp. 154—162]). The idea for this paper came from the papers [2] and [3]. 2. Main results Theorem. Define a sequence (Qk(x)) of polynomials for the given weight sequences r and p using the recursive relation (4) *&- i (* ) "= hQk (x)+akQk-l(x) + bk-1Qk.2(x) (ic = 1,2, ...), 8o(x) = <2o ̂  o. 8-iW=fo, . Received February 16, 1983 414 Gradimir V. Milovanovii and Igor 2. Milovanovii where (5) ak = (p,Jrk) (fc - 1, ..., n) and bk = 2 (k = 1, ..., « - 1 ) . For each sequence x=(xk)kiI of real numbers the inequalities (3) hold, where A„ and B„ are the minimum and the maximum zeros of polynomial Q„(x), respec-tively. Equality holds in the left-hand (right-hand) inequality in (3) if and only if xk= =(C/Y~rk)Qk_1(X) (k=l,...,«), where X=An (X=Bn) and C is an arbitrary real constant different from zero. Proo f . Let X be an «-dimensional euklidean space with scalar product (z, w)= n — 2zkwk, where z=[z1,..., 2„]T and vv=[w1,..., vf„] . Let, further, a = ( a l 9 ... k= 1 ..., an), b=(bi, ..., 6„_i), and define a three-diagonal matrix by «1 h 0 .. ,. 0 0 h a2 b2 ' 0 0 0 0 0 a„-i h-x 0 0 0 a„ Introducing zk=Yrkxk (k=1, . . . ,n), from (1) and (2) we get 11*11? = 2 r k x l = 2 4 = & z), . fc = l k=l and (x, Vx) = 2 PkXk^k = 2 ( P k z k / f c ) v ( z k / f i \ ) = *=1 k=l = (Pi4/ri) + 2 (pk Zk/rk f r ^ (Yr^l Zk - z k _ i). k=2 Thus by (5), (x,Vx) = (tf„(a,b)z,z). On the other hand, let us consider the sequence (Qk(x)) of polynomials defined by (4). For k= 1 ,2 , . . . , « , we obtain from (4) the equality (6) xv = //„(a, b)v + bnQn(x)e, where v=[Qo(x), &.(*), . . . , £?„_i(;c)]T and ?=[0,0, ...,0, i f . Setting x=X in (6), we conclude: If A is such that Q„(X)=0, then A is an eigenvalue of the mat-rix Hn(a, b) and v=[Q0{X), Qx(X),..., <2„-i(A)]T is tlie corresponding eigenvector of the matrix Hn(a, b), and conversely, according to (6), if X is an eigenvalue of the matrix H„(a, b), then Qn(X)=0, i.e. A is a zero of the polynomial Qn(x). Some discrete inequalities of Opial's type 415 Thus, the eigenvalues of the matrix Hn(a, b) are exactly the zeros the of poly-nomial Qn(x). Since Hn( a, b) is a three-diagonal matrix (&?> 0, i=\,...,n—1) all its eigenvalues Xt (/ = 1, . . . ,«) are real and distinct, and An(z,z)m(Hn(a,b)z,z)^Bn(z,z) hold, with equality for eigenvectors corresponding to the eigenvalues A„=mmXi, Bn=maxA(. This completes the proof of the theorem. Coro l l a ry 1. Let the sequences r and p be given recursively by rk+i = {4k(k+s)l(2k+s+i)% (k = 1, ..., n-1), pk = (2k + s-l)rk (k = .1, ..., n), . • with rx= 1 and j> —1. Then for every sequence of real numbers the inequalities (3) hold, where A„ and B„ are the minimal and the maximal zeros of the normalized generalized Laguerre polynomials Dn(x)=Lsn(x)/\\Ls„\\. Here L'n{*)= ¿ i r ^ k - ^ M and ll^ll - Vr(n+s+l)/n\. m=o \n — m/ ' • • Equality holds in the left-hand (right-hand) inequality in (3) if and only if xk = = (Ck/Yrk)L:!k_1(X) (k=l,..., «), where X = A„ (X=Bn) and C ( ^ 0 ) is an arbitrary constant. • P roof . For the proof of this result it is enough to show that in this case (4) reduces to the recurrence relation for generalized Laguerre polynomials. Since ak = (pjr^ = 2k + s-l and bk = ~(pk+i/2]/rkrk+1) = ~yk(k + s), (4) becomes xQk-iW = -yk(k+s)Qk(x)+(2k+s-i)Qk.1(x)-Hk-m + s-l)Qk^(x\ which is the recurrence relation for normalized generalized Laguerre polynomials (iQk(x)=Dk(x)). In the special case pk=rk = 1 (k=l, ...,ri), we have the following result: Coro l l a ry 2. For every sequence \ = (xk)ki, of real numbers and for x 0=0, the inequalities (7) 2 sin2 (71/2(n + l)j ¿xl ^ 2 xk(xk-xk^) ^ 2 cos2 (tt/2(n +1)) fc = l » = 1 k = 1 are valid. ' : " -416 Gradimir V. Milovanovic and Igor 2 . Milovanovid Equality holds in the left-hand inequality if and only if xk=C sin (kn/(n+lj) (k=l, •••,«), where C=const¿¿0, and in the right-hand inequality if and only if x t=(—1)* - 1C sin (knfcn+l)), (k=l, ..., w), where C=const¿¿0. P r o o f . In this case, we haiye ak=1, bk= —1/2 and (8). xQk^{x) = -( l/2)0 t(*)+.&_ 1(*)-( l/2)G l_ 2(x) , where Qo(x) can be Qo(x)=l. If we put t=l— x, one can easily obtain the solu-tion of the difference equation (8), for example for | / |<1, i. e. 0 < x < 2 , (9) Qk (x) = (sin (k +1) 0/sin 9) (k = 1, ..., n), where ei9=t+i^l-t2. Then, from Qn(x)=0 it follows A*=2sin2 (kn/2(n+l)) (k=l,...,«), implying A„ = min Xk = 2 sin2 (7r/2 (n +1)) and B„ = max Xk = 2 cos2 (n/2 (n +1)). Using (9) the conditions for equality are simply obtained. Also we note that the inequalities (7) can be written in the form - COS (n/(n + 1)) 2 xl^ 2 xkxk-l — C°S (JTl(n + 1)) 2 X\, k=l k=a *=1 i.e., . . ' (10) | 2 *»*n-i| ^ cos [nl(n + \)) 2 xl k=2 k = 1 R e m a r k . The inequality (10) is related to an extremal problem occurring in the investigation of approximative properties of positive polynomial operators. Namely, let Cm be the class of all nonnegative trigonometric polynomials of order m (11) Tm{t) = l+2a x cos ... +2am cos mt. The problem is to determine a polynomial T*£Cm which has the greatest coefficient ax (see, for example, [4, pp. 113—115]). If the polynomial (11) is written in the form Tm(t) = |x1+x2e"+••• +xm+1e im , | = 2 * * l + 2 f ^ x^x*^) cos k=l k=2 where xk (Jc=l, ..., m+1) are real numbers, the determination of T* is reduced to finding m+l m+1 SUpfli = SUp 2 XkXk-l, 2 x\ = 1. k=2 k=1• Putting n=m+l in (10), we have sup ax=cos (n/{m+2)). Some discrete inequalities of Opial's type 417 References [1] D. S. MiTRiNovii and P. M. VASIC, Analytic inequalities, Grundlehren der Math. Wiss., Band 165, Springer-Verlag (Berlin—Heidelberg—New York, 1970) . [2] K . FAN, O . TAUSSKY and J . TODD, Discrete analogs of inequalities of Wirtinger, Monatsh. Math., 5 9 (1955) , 7 3 — 7 9 . [3] G. V. MILOVANOVIC and I. 2. MILOVANOVID:, On discrete inequalities of Wirtinger's type, J. Math. Anal. Appl., 8 8 (1982) , 3 7 8 — 3 8 7 . [4] V. M. TIHOMIROV, Some problems of approximation theory, Univ. Moscow (1976) . (Russian) FACULTY OF ELECTRONIC ENGINEERING DEPARTMENT OF MATHEMATICS BEOGRADSKA 14 18 000NIS, YUGOSLAVIA 

Some discrete inequalities of Opial's type

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Some discrete inequalities of Opial's type

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