INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS

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1 INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS S. S. DRAGOMIR Abstrct. Some inequlities for two inner products h i nd h i which generte the equivlent norms kk nd kk with pplictions for invertible bounded liner opertors, positive de nite self-doint opertors, integrl nd discrete inequlities re given.. Introduction Let (H h i) be n inner product over the rel or complex number eld K. The following inequlity (.) hx yi x y H is well known in the literture s Schwrz s inequlity. It plys n essentil role in obtining vrious results in the Geometry of Inner Product Spces s well s in its pplictions in Opertor Theory, Approximtion Theory nd other elds. Due to the fct tht (.) x y H we cn introduce the ngle between the vectors x y denoted by xy through the formul (.3) cos xy := x y 6= 0: As observed by Krein in 969, [6] (see lso [5, p. 56]), the following interesting inequlity holds: xz xy yz for ny x y z Hn f0g : We now recll some inequlities in which the quntity hx yi = () for di erent vectors is involved: hx yi hy zi hz xi 3 kxk kzk hx yi hy zi kzk hz xi (.4) kzk kxk hx yi hy zi hz xi kxk kzk [, p. 37] Dte: 9 Februry, Mthemtics Subect Clssi ction. Primry 47A05, 47A Secondry 6D5. Key words nd phrses. Inner product, Schwrz inequlity, Bounded liner opertors, Integrl inequlities, Discrete inequlities.

2 S. S. DRAGOMIR (.5) (.6) (.7) hx yi hx zi hz yi kzk hx zi hz yi kzk hx yi [, p. 38] hx yi hy zi kxk kzk hx zi kxk kzk (Buzno s inequlity, [, p. 49]) hx yi hy zi kxk kzk hx zi kxk kzk p p [, p. 5] Re where K nd = Re 6= nd hx yi hy zi kxk kzk hx yi hy zi hx zi kxk kzk kxk kzk hx zi (.8) kxk kzk hx zi kxk kzk [, p. 5], where x y z Hn f0g : We notice tht (.8) is re nement of Buzno s inequlity (.6). For other inequlities of this type, see [], [4], [7], [8] nd [9]. Motivted by the bove results, the min im of the present pper is to compre the quntities hx yi hx yi nd kxk kxk in the cse when the inner products h i nd h i de ned on H generte two equivlent norms, i.e., we recll tht kk nd kk re equivlent if there exists the constnts m M > 0 such tht (.9) m kxk M kxk for ny x H: Applictions for invertible bounded liner opertors, positive de nite self-doint opertors, integrl nd discrete inequlities re lso given. The following result my be stted.. The Results Theorem. Assume tht the inner products h i i i f g on the rel or complex liner spce H generte the norms kk i i f g which stisfy the following condition: (.) m kxk M kxk for ny x H where 0 < m M < re given constnts. If x y Hn f0g stisfy the condition 0 then (.) m M kxk M m :

3 INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS 3 If < 0 then (.3) m M kxk Proof. For ny inner product h i on H we hve (.4) = x y kxk x y Hn f0g : Utilising the ssumption (.), we hve successively: = x y M (.5) Now, if 0 then which implies tht = M M " kxk kxk m m = M m (.6) I M m M kxk Utilising (.5) nd (.6) we deduce M m : x y # =: I: M kxk = M m kxk : M m kxk which produces the second inequlity in (.). By (.4) nd (.) we lso hve (.7) m x y " = m kxk kxk m M M = m M Due to the fct tht 0 we hve =: J: # m kxk

4 4 S. S. DRAGOMIR which implies tht (.8) J m M m = m kxk M : kxk By mking use of (.7) nd (.8) we get m M kxk which is clerly equivlent to the rst inequlity in (.). Finlly, if < 0 then > 0 nd writing the inequlity (.) for y insted of y we esily deduce (.3). Corollry. Let A B (H) be n invertible opertor on the Hilbert spce (H h i) : If x y Hn f0g re such tht 0 then: (.9) kak A If x y Hn f0g nd < 0 then (.0) kak A Proof. Since A B (H) is invertible, then Re hax Ayi kaxk kayk kak ka k : Re hax Ayi kaxk kayk kak ka k : ka kxk kaxk kak kxk for ny x H: k Applying Theorem for hx yi := hax Ayi hx yi := hx yi nd m = ka k M = kak nd doing the necessry clcultions, we deduce the desired result. Corollry. Let A B (H) be self-doint opertor on the Hilbert spce (H h i) which stis es the condition (.) I A I in the opertor order of B (H) where 0 < < re given. If x y Hn f0g re such tht 0 then (.) Re hax yi [hax xi hay yi] If x y Hn f0g re such tht < 0 then (.3) Proof. From (.) we hve Re hax yi [hax xi hay yi] hx xi hax xi hx xi x H which implies tht p kxk [hax xi] = p kxk x H: : :

5 INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS 5 Now, if we pply Theorem for hx yi := hax yi hx yi := hx yi x y H nd m = p M = p then we obtin the desired result. The following lemm is of interest in itself. Lemm. Assume tht the inner products h i i i f g de ned on H stisfy the condition (.). Then for ny x y H we hve (.4) nd (.5) respectively. m [kxk hx yi ] hx yi M [kxk hx yi ] m [kxk ] M [kxk ] Proof. We use the following result obtined by Drgomir nd Mond in [3] (see lso [, p. 9]): If [ ] [ ] re two hermitin forms on H with [x x] = [x x] = for ny x H then (.6) [x x] = [y y] = [x y] [x x] = [y y] = [x y] nd (.7) [x x] = [y y] = Re [x y] [x x] = [y y] = Re [x y] for ny x y H: Now, if we pply (.6) nd (.7) rstly for [ ] := h i [ ] = m h i nd then for [ ] = M h i [ ] := h i we deduce the desired results (.4) nd (.5). The following result my be stted s well. Theorem. Assume tht the inner products h i i i f g stisfy the condition (.). Then for ny x y Hn f0g we hve the inequlities: (.8) m M hx yi hx yi kxk hx yi M m : Proof. Dividing the inequlity (.4) by 6= 0, we obtin m hx yi hx yi (.9) M hx yi for ny x y Hn f0g : Observe, by (.) tht: kxk m nd hx yi Utilising the second inequlity in (.9), we deduce hx yi M m hx yi kxk hx yi M kxk :

6 6 S. S. DRAGOMIR which is equivlent with the second inequlity in (.8). In ddition, we hve M kxk nd hx yi m kxk hx yi which together with the rst inequlity in (.9) produces the rst prt of (.8). Remrk. On utilising the inequlity (.5) nd n rgument similr to the one in the proof of Theorem, we cn reobtin the inequlities (.) nd (.3) from Theorem. The detils re omitted. Corollry 3. Let A B (H) be n invertible opertor on the Hilbert spce (H h i) : Then (.0) kak A hx yi hax Ayi kaxk kayk hx yi kak ka k for ny x y Hn f0g : The proof follows from Theorem on choosing hx yi := hax Ayi hx yi := hx yi x y H nd m = ka k M = kak : Corollry 4. Let A B (H) be self-doint opertor stisfying the condition (.). Then (.) for ny x y Hn f0g : hx yi hax yi [hax xi hay yi] hx yi The proof follows by Theorem on choosing hx yi := hax yi hx yi := hx yi x y H nd m = p M = p : 3. Applictions for Integrl Inequlities Assume tht (K h i) is Hilbert spce over the rel or complex number eld K. If : [ b] R![0 ) is mesurble function then we my consider the spce L ([ b] K) of ll functions f : [ b]! K tht re strongly mesurble nd R b (t) kf (t)k < : It is well known tht L ([ b] K) endowed with the inner product h i de ned by nd generting the norm hf gi := kfk := is Hilbert spce over K. The following proposition cn be stted. (t) hf (t) g (t)i (t) kf (t)k!

7 INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS 7 Proposition. Let : [ b]! [0 ) be two mesurble functions with the property tht there exists 0 < < so tht (3.) (t) for.e. t [ b] : (t) Then for ny f g L ([ b] K) we hve the inequlities: R b (t) hf (t) g (t)i (3.) R b (t) kf (t)k R b (t) kg (t)k R b (t) hf (t) g (t)i R b (t) kf (t)k R b (t) kg (t)k R b (t) hf (t) g (t)i R b (t) kf (t)k R b (t) kg (t)k Proof. From (3.) we hve p (t) kf (t)k! p (t) kf (t)k! (t) kf (t)k Applying Theorem for h i i = h i i i f g nd H = L ([ b] K) = L ([ b] K) we deduce the desired result. Remrk. A similr result cn be stted if one uses Theorem. The detils re omitted. 4. Applictions for Discrete Inequlities Assume tht (K h i) is Hilbert spce over the rel or complex number eld nd p = (p i ) in with p i 0 i N nd P i= p i < : De ne ( ) X `p (K) := x := (x i ) in xi K i N nd p i kx i k < : It is well known tht `p (K) endowed with the inner product nx hx yi p := p i hx i y i i nd generting the norm is Hilbert spce over K. kxk p := i= X i= p i kx i k! i= :! :

8 8 S. S. DRAGOMIR Proposition. Assume tht p = (p i ) in q = (q i ) in re such tht p i q i 0 i N, P i= p i P i= q i < nd (4.) nq i p i Nq i for ny i N where 0 < n N < : Then we hve the inequlity P n N i= p i hx i y i i (4.) P i= p i kx i k P i= p i ky i k P i= q i hx i y i i P i= q i kx i k P i= q i ky i k P i= p i hx i y i i P i= p i kx i k P i= p i ky i k N n A similr result cn be stted if one uses Theorem. However the detils re omitted. References [] M.L. BUZANO, Generlizzione dell diseguglinz di Cuchy-Schwrz (Itlin), Rend. Sem. Mt. Univ. e Politech. Torino, 3 (97/73), [] S.S. DRAGOMIR, Advnces in Inequlities of the Schwrz, Tringle nd Heisenberg Type in Inner Product Spces, Nov Science Publishers, NY, 007. [3] S.S. DRAGOMIR nd B. MOND, On the superdditivity nd monotonicity of Schwrz s inequlity in inner product spces, Mkedon. Akd. Nuk. Umet. Oddel. Mt.-Tehn. Nuk. Prilozi 5 (994), no., 5 (996). [4] S.S. DRAGOMIR nd J. SÁNDOR, Some inequlities in prehilbertin spces, Studi Univ. Bbeş-Bolyi, Mth., 3() (987), [5] K.E. GUSTAFSON nd D.K.M. RAO, Numericl Rnge, Springer, 997. [6] M. KREIN, Angulr locliztion of the spectrum of multiplictive integrl in Hilbert spce, Funct. Anl. Appl., 3 (969), [7] M.H. MOORE, An inner product inequlity, SIAM J. Mth. Anl., 4(3) (973), [8] T. PRECUPANU, On generliztion of Cuchy-Bunikowski-Schwrz inequlity, Anl. St. Univ. Al. I. Cuz Işi, () (976), [9] U. RICHARD, Sur des inéglités du type Wirtinger et leurs ppliction ux équtiones différentible ordinires, Coll. of Anl. Rio de Jneiro, August, 97, School of Computer Science nd Mthemtics, Victori University, PO Box 448, Mebourne City 800, VICTORIA, Austrli. E-mil ddress: sever.drgomir@vu.edu.u URL: :