REFINING CBS INEQUALITY FOR DIVISIONS OF MEASURABLE SPACE

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1 EFINING CBS INEQUALITY FO DIVISIONS OF MEASUABLE SPACE S. S. DAGOMI ; Abstract. In ths paper we establsh a re nement some reverses for CBS nequalty for the general Lebesgue ntegral on dvsons of measurable space. Applcatons for dscrete nequaltes weghted means of postve numbers are also gven.. Introducton The Cauchy-Bunyakovsky-Schwarz nequalty, or for short, the CBS nequalty, plays an mportant role n d erent branches of Modern Mathematcs ncludng Hlbert Spaces Theory, Probablty & Statstcs, Classcal eal Complex Analyss, Numercal Analyss, Qualtatve Theory of D erental Equatons ther applcatons. Let (; ; ) be a measure space consstng of a set ; a -algebra of subsets of denoted by a countably addtve postve measure on wth values n [ fg Denote by L w (; ) the Hlbert space of all C-valued functons f de ned on that are -w-ntegrable on ;.e., w (x) jf (x)j d (x) < ; where w! [; ) s a gven -measurable functon on We wrte for smplcty w jfj d nstead of w (x) jf (x)j d (x) The followng nequalty s well known n the lterature as the ntegral Cauchy- Bunyakovsky-Schwarz nequalty (CBS) w jfj d w jgj d provded that f; g L w (; ) We say that the famly of measurable sets F n () = f g f;;ng s a n-dvson for f = S n, \ j = ; for any ; j f; ; ng wth 6= j ( ) > for any f; ; ng In ths stuaton, f f L w (; ) then f L w ( ; ) for any f; ; ng fwd = P n fwd. Also, wd = P n wd wth wd > for any f; ; ng For a gven n we denote by D n () the set of all n-dvsons of consder the functonal (jfj ; jgj ; ) D n ()! de ned by (.) (jfj ; jgj ; F n ()) = w jfj d w jgj d ; where f; g L w (; ) 99 Mathematcs Subject Class caton. Prmary 6D5; Secondary 8A5. Key words phrases. CBS nequalty, Integrable functons, Lebesgue ntegral, Weghted means.

2 S. S. DAGOMI ; In ths paper we establsh some nequaltes concernng the functonal (jfj ; jgj ; ) that provde re nements reverses for the CBS ntegral nequalty (CBS). Applcatons for dscrete nequaltes weghted means of postve numbers are also gven. For recent papers on CBS nequalty, see [], [], [8], [9], [], [3], [4], [5], [6], [8] the references theren.. The Man esults We state the followng re nements of the CBS nequalty Theorem. For f; g L w (; ) F n () D n () we have (.) (jfj ; jgj ; F n ()) w jfj d w jgj d Proof. Let F n () = f g f;;ng be a n-dvson for We have by CBS ntegral nequalty n L w ( ; ) ; f; ; ng that w jfgj d = w jfgj d w jfj d w jgj d = (jfj ; jgj ; w) ; whch proves the rst nequalty n (.). By the CBS dscrete nequalty we also have (jfj ; jgj ; w) = w jfj d w jgj d! 3 4 w jfj d 5 4 " X n # " X n = w jfj d w jgj d = w jfj d w jgj d ; whch proves the second nequalty n (.). We can gve now some lower bounds for (jfj ; jgj ; ) The followng result holds # w jgj d! 3 Theorem. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exsts k; K; l; L > wth the property (.) < k w jfj d K < (.3) < l w jgj d L < 5

3 EFINING CBS INEQUALITY 3 for each f; ; ng Then (.4) w jfj d w jgj kl + KL d p klkl (jfj ; jgj ; F n ()) (.5) w jfj d (jfj ; jgj ; F n ()) (jfj ; jgj ; F n ()) w jgj d r K l r! k L Proof. We use the Pólya-Szegö nequalty that states that [7] (see also [4, p. 74]), f < a a A < < b b B < ; f; ; ng then (.6) P n P n a b ( P n a b ) (ab + AB) 4abAB Now, f we take a = w jfj d ; b = a = k; A = K; b = l B = L; then by (.6) we get!! (.7) w jfj d w jkj d (kl + KL) 4klKL, w jgj d f; ; ng,! w jfj d w jgj d the nequalty (.4) s proved. We use now the Shsha-Mond nequalty [9] (see also [4, p. 8]) that says that (.8) a b! a b r A b r! a X n a b B X n b provded < a a A < < b b B < ; f; ; ng, Now, f we take a = w jfj d ; b = w jgj d f; ; ng, a = k; A = K; b = l B = L; then by (.8) we get! w jfj d w jgj d w jfj d w jgj d r r! K k X n X n w jfj d w jgj d w jgj d; l L whch s equvalent to (.5). Corollary. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exsts m; M > wth the property (.9) < m w M < -a.e. on

4 4 S. S. DAGOMI ; Then (.) (.) w jfj d w jgj d mf g + MF G m M p f g F G (jfj ; jgj ; F n ()) w jfj d (jfj ; jgj ; F n ()) (jfj ; jgj ; F n ()) w jgj d s M F m f A m g M G where (.) f = g = mn jfj d f;;ng mn f;;ng ; F = jgj d ; G = max jfj d f;;ng max f;;ng Proof. Let F n () = f g f;;ng be a n-dvson for From (.9) we have < m jfj d w jfj d M jfj d < jgj d for any f; ; ng ; gvng that < m mn jfj d w jfj d M max f;;ng f;;ng jfj d < ; whch s equvalent to < m mn jfj d w jfj d f;;ng M max jfj d < f;;ng, smlarly < m mn jgj d w jgj d f;;ng M max jgj d < f;;ng for any f; ; ng Now, f we apply Theorem for k = m f ; K = M F ; l = m g L = M G we get the desred nequaltes (.) (.). Corollary. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exst a; b; A; B > wth the property (.3) < a jfj A < < b jgj B < -a.e. on

5 EFINING CBS INEQUALITY 5 Then (.4) (.5) w jfj d w jgj d abw + W AB p (jfj ; jgj ; F n ()) ; w W abab w jfj d (jfj ; jgj ; F n ()) (jfj ; jgj ; F n ()) w jgj d r AW bw r! aw ; BW where (.6) w = mn f;;ng wd W = max f;;ng w d Proof. From (.3) we have a wd w jfj d A wd < ; whch mples that aw w jfj d AW < for any f; ; ng Smlarly, we have bw w jgj d BW < ; for any f; ; ng Now, f we apply Theorem for k = aw ; K = AW ; l = bw L = BW we get the desred nequaltes (.4) (.5). The followng result holds Theorem 3. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exst k; K; l; L > wth the property (.) (.3) for each f; ; ng Then (.7) w jfj d w jgj d (jfj ; jgj ; F n ()) (.8) 3 n (KL kl) w jgj d + ll w jfj L d kk k + l (jfj ; jgj ; F n ()) K Proof. We use the followng Ozek s type nequalty [] (.9) a b! a b 3 n (AB ab) provded < a a A < < b b B < ; f; ; ng.

6 6 S. S. DAGOMI ; Now, f we take a =, w jfj d ; b = w jgj d f; ; ng, a = k; A = K; b = l B = L; then by (.9) we get (.) w jfj d w jgj d! w jfj d w jgj d 3 n (KL kl) ; whch proves (.7). Further, we recall Daz-Metcalf s nequalty [3] (see also [4, p. 3]) (.) b + bb B a aa a + b X n a b A provded < a a A < < b b B < ; f; ; ng. If we take a = w jfj d ; b = w jgj d, f; ; ng, a = k; A = K; b = l B = L; then by (.) we get w jgj d + bb (.) w jfj d aa B a + b X n w jfj d w jgj d A that s equvalent to (.8). Corollary 3. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exsts m; M > wth the property (.9). Then (.3) w jfj d w jgj d (jfj ; jgj ; F n ()) (.4) 3 n (MF G mf g ) w jgj d + g G w jfj d f F M G + m g (jfj ; jgj ; F m f M n ()) ; F where f ; F ; g G are de ned by (.). Proof. Follows by the nequaltes (.7) (.8) for or k = m f ; K = M F ; l = m g L = M G We also have

7 EFINING CBS INEQUALITY 7 Corollary 4. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exsts a; b; A; B > wth the property (.3). Then (.5) w jfj d w jgj d (jfj ; jgj ; F n ()) (.6) 3 n ABW abw w jgj d + bb w jfj BW d + bw (jfj ; jgj ; F n ()) ; aa aw AW where w W are de ned by (.6). Proof. Follows by the nequaltes (.7) (.8) for the choces k = aw ; K = AW ; l = bw L = BW The followng result holds Theorem 4. Let f; g L w (; ) F n () D n () ; F n () = f g f;;ng be such that there exsts p; P > wth the property d w jfj (.7) < p w jgj d P < for each f; ; ng Then (.8) w jfj d w jgj d (.9) (.3) (p + P ) 4pP (jfj ; jgj ; F n ()) ; w jfj d w jgj d (jfj ; jgj ; F n ()) (P p) w jgj d 4 (p + P ) 4 (P w jfj d w jgj d (jfj ; jgj ; F n ()) p) w jgj d Proof. We use the followng Cassels nequalty [] (see also [4, p. 7]) (.3) that holds provded P n P n a b ( P n a b ) (c + C) 4cC (.3) < c a b C < for any f; ; ng

8 8 S. S. DAGOMI ; If we take a =, w jfj d ; b = w jgj d f; ; ng, c = p C = P, then by (.3) we get P n w jfj d P n w jgj d Pn w jfj d w jgj d (p + P ) 4pP ; whch proves (.8). Further, we use the followng Shsha-Mond nequalty [9] (see also [4, p. 8])!! X n (.33) a b (C c) a b b 4 (c + C) that holds provded the condton (.3) s vald. Then by takng a = w jfj d ; b = c = p C = P n (.33) we get the desred result (.9). We use the followng reverse of CBS nequalty [4, p. 78] a b! a b 4 (C X n c), w jgj d f; ; ng, provded the condton (.3) s vald, for the choce a = w jfj d ; b =, w jgj d f; ; ng, c = p C = P Smple calculaton yelds the b desred nequalty (.3). In [5] (see also [7, p. 4]) we have shown amongst other that, f u; v L w (; ) there are a; A C wth e (aa) > such that (.34) w e [(Au v) (v au)] d ; then (.35) juj wd jvj wd e Now, f we replace u wth u n (.34), namely, we assume that (.36) w e [(Au v) (v au)] d ; then we have the followng nequalty of nterest (.37) juj wd jvj wd e We observe that f (.38) e [(Au v) (v au)] -a.e. on A + a wvud p e (aa) A + a wvud p e (aa) then (.36) s vald for any w -a.e. on Moreover, f A > a > u; v are real valued such that Au v au -a.e. on ; then (.38) holds true for any w -a.e. on

9 EFINING CBS INEQUALITY 9 Theorem 5. Let f; g L w (; ) F n () D n () be such that there exsts a; A C wth e (aa) > (.39) w e (Ag f) f ag d for any f; ; ng ; then we have (.4) (jfj ; jgj ; F n ()) e A + a p e (aa) ja + aj p e (aa) Proof. Let F n () D n () ; F n () = f g f;;ng If the condton (.39) s true, then by (.37) we have (.4) jfj wd jgj wd e for any f; ; ng If we sum over from to n then we get jfj w d jgj w d e ja + aj p e (aa) whch proves the rst nequalty n (.4). Snce e A + a = e A + a A + a = A + a = ja + aj then the second nequalty also holds. The last part s obvous. w jfgj d A + a p e (aa) A + a P n p ; e (aa) ; Corollary 5. If A > a > ; F n () D n () f; g are postve valued such that (.4) w [(Ag f) (f ag)] d ; for any f; ; ng ; then (.43) (jfj ; jgj ; F n ()) A + a p wf gd Aa We have the followng result as well Theorem 6. Let f; g L w (; ) ; F n () D n () be such that there exsts a; A C wth a + A 6= (.39) s vald, then we have a + A (jfj ; jgj ; F n ()) e + ja aj (.44) w jgj d ja + Aj 4 ja + Aj + ja aj w jgj d 4 ja + Aj

10 S. S. DAGOMI ; Proof. Let F n () D n () ; F n () = f g f;;ng We use the followng nequalty obtaned n [6] (see also [7, p. 3]) (.45) jfj `d jfj `d e a + A `fgd ja + Aj + ja aj ` jgj d; 4 ja + Aj where ` n -ntegrable, ` -a.e. on w e (Ag f) f ag Now, f we wrte the nequalty (.45) on wth f; ; ng ; then we get (.46) jfj wd jgj wd a + A e + ja aj w jgj d; ja + Aj 4 ja + Aj for any f; ; ng If we sum over from to n n (.46), then we get (.47) jfj w d jgj w d " a + A # e + ja aj w jgj d ja + Aj 4 ja + Aj that proves the rst nequalty n (.44). The second nequalty follows by the fact that a + A e a + A ja + Aj ja + Aj = a + A ja + Aj = Corollary 6. If A > a >, F n () D n () f; g are postve valued such that (.4) s vald for any f; ; ng ; then (.48) (jfj ; jgj ; F n ()) + (A a) wg d 4 a + A 3. Dscrete Inequaltes For a nonempty nte famly of ndces J postve weghts w j ; j J we denote W J = P jj w j Assume that, for n ; the famly J of ndces contanng more than n elements F n (J) = fj g f;;ng s a n-dvson for J; namely J = S n J J \ J j = ; for any ; j f; ; ng wth 6= j

11 EFINING CBS INEQUALITY For a gven n we denote by D n (J) the set of all n-dvsons of J consder the functonal (x; y; ) D n (J)! de ned by (3.) (x; y; F n (J)) X w j jx j j A jj where x = fx j g jj y = fy j g jj C. From (.) we have (3.) X w j x j y j (x; y; F n X w j jx j j A jj X w j jy j j A X w j jy j j A jj If F n (J) = fj g f;;ng s a n-dvson for J there exsts k; K; l; L so that (3.3) < X w j jx j j A jj K < ; (3.4) < X w j jy j j A jj L < for each f; ; ng ; then by Theorem we have X w j jx j j X w j jy j j A jj jj kl + KL p X w j jx j j A X w j jy j j A jj PjJ (3.6) w j jx j j (x; y; F n (J)) (x; y; F n (J)) PjJ w j jy j j r K l r! k L From Theorem 3 we also have (3.7) X jj w j jx j j X jj w j jy j X w j jx j j A jj 3 n (KL X w j jy j j A jj C A

12 S. S. DAGOMI ; (3.8) X jj w j jy j j + ll kk X w j jx j j jj L k + l X X w j jx j j A K X w j jy j j A jj If F n (J) = fj g f;;ng s a n-dvson for J there exsts a; A C wth e (aa) > X (3.9) w j e [(Ay j x j ) (x j ay j )] jj for any f; ; ng ; then by Theorem 5 we X (3.) w j jx j j X w j jy j j A jj jj e ja + aj p e (aa) A + a P jj w jx j y j p e (aa) X w j x j y j jj ja + aj X p w j jx j y j j e (aa) If F n (J) = fj g f;;ng s a n-dvson for J there exsts a; A C wth a + A 6= (3.9) s vald, then by Theorem 6 we also X w j jx j j X (3.) w j jy j j A jj jj jj 3 e 4 a + A X w j x j y j 5 + ja aj X w j jy j j ja + Aj 4 ja + Aj jj jj X w j x j y j jj + ja aj X w j jy j j 4 ja + Aj jj eferences [] J. M. Aldaz, Strengthened Cauchy-Schwarz Hölder nequaltes. J. Inequal. Pure Appl. Math. (9), no. 4, Artcle 6, 6 pp [] N. S. Barnett, S. S. Dragomr I. Gomm, On some ntegral nequaltes related to the Cauchy-Bunyakovsky-Schwarz nequalty. Appl. Math. Lett. 3 (), no. 9, 8. [3] J. B. Daz F. T. Metcalf, Stronger forms of a class of nequaltes of G. Pólya-G. Szegö L.V. Kantorovch, Bull. Amer. Math. Soc., 69 (963), [4] S. S. Dragomr, A survey on Cauchy-Bunyakovsky-Schwarz type dscrete nequaltes. J. Inequal. Pure Appl. Math. 4 (3), no. 3, Artcle 63, 4 pp. [5] S. S. Dragomr, everses of Schwarz, trangle Bessel nequaltes n nner product spaces. J. Inequal. Pure Appl. Math. 5 (4), no. 3, Artcle 76, 8 pp. [6] S. S. Dragomr, New reverses of Schwarz, trangle Bessel nequaltes n nner product spaces. Aust. J. Math. Anal. Appl. (4), no., Art., 8 pp.

13 EFINING CBS INEQUALITY 3 [7] S. S. Dragomr, Advances n Inequaltes of the Schwarz, Grüss Bessel Type n Inner Product Spaces. Nova Scence Publshers, Inc., Hauppauge, NY, 5. v+49 pp. ISBN [8] S. S. Dragomr, e nements of the CauchyBunyakovsky-Schwarz nequalty for functons of selfadjont operators n Hlbert spaces. Lnear Multlnear Algebra 59 (), no. 7, [9] M. Hajja, A generalzed Cauchy-Schwarz nequalty. Math. Inequal. Appl. 8 (5), no. 3, [] N. J. A. Harvey, A generalzaton of the Cauchy-Schwarz nequalty nvolvng four vectors. J. Math. Inequal. 9 (5), no., [] A. Ibrahm S. S. Dragomr, A survey on CauchyBunyakovsky-Schwarz nequalty for power seres. n Analytc Number Theory, Approxmaton Theory, Specal Functons, 47 95, Sprnger, New York, 4. [] S. Izumno J. Peµcarć, A weghted verson of Ozek s nequalty, Sc. Math. Japoncae, 56(3) (), [3] M. Jeong, Inequaltes va power seres Cauchy-Schwarz nequalty. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 9 (), no. 3, [4] C. Lupu D. Schwarz, Another look at some new Cauchy-Schwarz type nner product nequaltes. Appl. Math. Comput. 3 (4), [5] M. Masjed-Jame N. Hussan, More results on a functonal generalzaton of the Cauchy- Schwarz nequalty. J. Inequal. Appl., 39, 9 pp. [6] I. Pnels, On the Hölder Cauchy Schwarz nequaltes. Amer. Math. Monthly (5), no. 6, [7] G. Pólya G. Szegö, Problems Theorems n Analyss, Volume Seres, Integral Calculus, Theory of Functons (n Englsh), translated from german by D. Aeppl, corrected prntng of the revsed translaton of the fourth German edton, Sprnger Verlag, New York, 97. [8]. Sharma,. Bhar M. Gupta, Inequaltes related to the Cauchy-Schwarz nequalty. Sankhya A 74 (), no.,. [9] O. Shsha B. Mond, Bounds on d erences of means, Inequaltes, Academc Press Inc., New York, 967, pp [] G. S. Watson, G. Alpargu G. P. H. Styan, Some comments on sx nequaltes assocated wth the ne cency of ordnary least squares wth one regressor, Lnear Algebra ts Appl., 64 (997), Mathematcs, College of Engneerng & Scence, Vctora Unversty, PO Box 448, Melbourne Cty, MC 8, Australa. E-mal address sever.dragomr@vu.edu.au UL http//rgma.org/dragomr School of Computatonal & Appled Mathematcs, Unversty of the Wtwatersr, Prvate Bag 3, Johannesburg 5, South Afrca