A NEW (PROBABILISTIC) PROOF OF THE DIAZ METCALF AND PÓLYA SZEGŐ INEQUALITIES AND SOME APPLICATIONS

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1 Teor Imov r.tmtem.sttist. Theor. Probbility nd Mth. Sttist. Vip. 70, 2004 No. 70, 2005, Pges 3 22 S ) Article electroniclly published on August 2, 2005 A NEW PROBABILISTIC) PROOF OF THE DIAZ METCALF AND PÓLYA SZEGŐ INEQUALITIES AND SOME APPLICATIONS UDC 59.2 TIBOR K. POGÁNY Abstrct. The Diz Metclf nd Póly Szegő inequlities re proved in the probbilistic setting. These results generlize the clssicl cse for both sums nd integrls. Using these results we obtin some other well-known inequlities in the probbilistic setting, nmely the Kntorovich, Rennie, nd Schweitzer inequlities.. Introduction The celebrted Cuchy Bunykovskiĭ Schwrz inequlity cn be written in the probbilistic setting s follows: ) E ξη 2 E ξ 2 E η 2 where ξ nd η re rndom vribles defined on some probbility spce Ω, F, P). An inequlity, converse to ), is lso true but only in exceptionl cses known s the Diz Metclf nd Póly Szegő inequlities. Some prtil cses of the ltter two results re known s Kntorovich, Rennie, nd Schweitzer inequlities see [6, 2.], [7]). Below we prove new inequlities tht cn be viewed s inverses to ) for lmost surely bounded rndom vribles ξ nd η. These inequlities involve two first moments E ξ nd E ξ 2 nd the upper nd lower bounds m nd M of the rndom vrible ξ, thtis,we ssume tht P{m ξ M} =. We provide necessry nd sufficient conditions for ll cses under considertion. These results generlize the clssicl Diz Metclf nd Póly Szegő inequlities s well s other inequlities mentioned bove for the probbilistic setting. The proof below is given for both discrete nd continuous cses. We lso consider inequlities for rndom vectors nd inequlities with weights. The methods we use re elementry nd re bsed on the properties of the opertor E. We write ξ ψ if the rndom vrible ξ hs the distribution/density ψ. The symbols I A nd χ S t) stnd for the indictor of rndom event A nd the chrcteristic function of set S, respectively; N 0 denotes the set of nonnegtive integers, δ λµ is the Kronecker delt. Finlly L 2 ϕ[a] denotes the spce of functions {h: A ht) 2 ϕt) dt < } such tht supph) ={t: ht) 0} Mthemtics Subject Clssifiction. Primry 26D5, 60E5. Key words nd phrses. Almost surely bounded rndom vrible, Diz Metclf inequlity, discrete inequlity, integrl inequlity, Kntorovich inequlity, mthemticl expecttion, Póly Szegő inequlity, Rennie inequlity, Schweitzer inequlity. 3 c 2005 Americn Mthemticl Society License or copyright restrictions my pply to redistribution; see

2 4 TIBOR K. POGÁNY 2. Diz Metclf inequlity Below we consider rel, lmost surely bounded rndom vribles ξ for which there re rel constnts m nd M, m M, such tht P{m ξ M} =. The moments E ξ r, r>0, cn be estimted in the cse of m 0bym r E ξ r M r. The vrince stisfies the inequlity Vr ξ M m) 2 /4. In wht follows we obtin more sophisticted results. First we provide n uxiliry result used then to derive conditions where the Diz Metclf inequlity becomes n equlity. This result is communicted to the uthor by O. I. Klesov [5]. Equlity Theorem. Let X be rndom vrible for which there re two rel numbers m nd M, m < M, such tht Pm X M) =. The following two conditions re equivlent: 2) EX m)m X) =0 nd there exist rndom events A nd B such tht 3) X = mi A MI B.s., 4) PA B) =, 5) PA B) =0. Here.s. stnds for lmost surely. Proof. Put δ =X m)m X). Conditions 3) 5) imply 2), since δ = 0 lmost surely in this cse. To prove the converse put A = {ω : Xω) =m} nd B = {ω : Xω) =M}. Then 2) implies PX R \{m, M}) =0, whence conditions 3) nd 4) follow. Since m < M, condition 5) lso holds. Now we re redy to prove the Diz Metclf inequlity for bounded rndom vribles. Theorem. Let ξ nd η be rel rndom vribles defined on the sme probbility spce Ω, F, P). Assume tht P{m ξ M } =, P{m 2 η M 2 } =, m M, m 2 M 2,ndm 2 > 0. Then 6) E ξ 2 m M E η 2 m M ) E ξη. m 2 M 2 The inequlity becomes n equlity if nd only if either i) m /M 2 = M /m 2,orii) m /M 2 <M /m 2 nd 7) P Proof. It is esy to see tht 8) The opertor E is monotone, thus M η 9) E { ξ η m, M }) =. m 2 m M 2 ξ η M m 2. ) ξ ξ m ) η 0. M 2 License or copyright restrictions my pply to redistribution; see

3 PROBABILISTIC PROOF OF INEQUALITIES 5 Now 6) follows from 9), since E is liner opertor. It remins to tret the cse of n equlity in 6), tht is, in 9). Since the cse i) is obvious, we consider the cse ii). Let m /M 2 <M /m 2 nd condition 7) hold. We hve in the cse of ii) tht Pm 2 ξ = M η M 2 ξ = m η)= nd Pm 2 ξ = M η M 2 ξ = m η)=0. Thus inequlity 9) becomes n equlity if 8) holds. On the other hnd, if 9) becomes n equlity, then 0) P{M 2 ξ = m η m 2 ξ = M η} =. Put A = {ω Ω: M 2 ξ = m η} nd B = {ω Ω: m 2 ξ = M η}. We pply the Equlity Theorem to X = ξ/η, m = m /M 2, M = M /m 2 nd complete the proof of 7). Corollry. Weighted Diz Metclf type inequlity in the discrete cse). Assume tht m x k M, k =,...,n. Let 0 <m 2 y l M 2, l =,...,m, p kl 0, nd n n k= l= p kl =.Then ) x 2 kq k m M m y 2 m l r l M ) n m x k y l p kl m 2 M 2 where k= m p kl = q k, l= l= k =,...,n, p kl = r l, k= Inequlity ) becomes n equlity if nd only if i) either m /M 2 = M /m 2, ii) or m /M 2 <M /m 2 nd 2) p kl =, k,l I x,y where I x,y := {k, l): x k /y l {m /M 2,M /m 2 }}. k= l= l =,...,m. Proof. Let ξ,η) be rndom discrete vector whose coordintes re defined on probbility spce Ω, PΩ), P). Let η be nonnegtive rndom vrible nd PΩ) = {S : S Ω}. Let ξ,η) P{ξ = x k,η = y l } = p kl, k =,...,n, l =,...,m,nd n k= m l= p kl =. Theorem nd inequlity 6) prove ). The cse of n equlity in ) follows from the corresponding prt of Theorem. Remrk. Putting m = n nd p kl = δ kl /n, k, l =,...,n, in ) note tht q k = r l =/n in this cse) we obtin the originl Diz Metclf inequlity: x 2 k m M y 2 m l M ) n x k y k m 2 M 2 k= l= see [2]). This inequlity becomes n equlity if nd only if x k /y l {m /M 2,M /m 2 } for ll k =,...,n. This follows from 2), since p kl =0forllk l nd p kk =/n. A generliztion of the ltter inequlity is given in [2] for complex numbers see lso the list of references in [6, pp ]). k= License or copyright restrictions my pply to redistribution; see

4 6 TIBOR K. POGÁNY Corollry.2 Weighted Diz Metclf inequlity for integrls). Let f nd g be Borel functions such tht m fx) M nd 0 <m 2 gy) M 2 3) for lmost ll x, y) [, b] [c, d]. Let wx, y) be nonnegtive function such tht suppw) =[, b] [c, d] nd Put Then 4) d c d wx, y) dy = w x), f 2 x)w x) dx m M m 2 M 2 m M ) c wx, y) dx dy =. d c d Inequlity 4) becomes n equlity if nd only if either i) m /M 2 = M /m 2,or ii) m /M 2 <M /m 2 nd I x,y wx, y) dx dy = where I x,y := {x, y): fx)/gy) {m /M 2,M /m 2 }}. c wx, y) dx = w 2 y). g 2 y)w 2 y) dy fx)gy)wx, y) dx dy. Proof. Let ξ,η) berndom vectorwiththedensity wx, y) whose support is [, b] [c, d]. It is cler tht w x) ndw 2 y) re the densities of the rndom vribles ξ nd η, respectively. Since f nd g re Borel functions, they belong to L 2 w [, b] ndl 2 w 2 [c, d], respectively. Let m,m 2 nd M,M 2 be rel numbers such tht P{m fξ) M } = P{m 2 gη) M 2 } =. It follows from Theorem tht 5) E f 2 ξ) m M m 2 M 2 E g 2 η) m M ) E fξ)gη) which is equivlent to 4). The cse of equlity is obvious in the cse of i). In the cse of ii) we hve E f 2 ξ) m M E g 2 m η) M ) E fξ)gη) m 2 M 2 = E fξ) M ) gη) fξ) m ) gη) m 2 M 2 nd thus the sttement on the equlity follows from the Equlity Theorem. Remrk 2. If ξ,η) wx, y) =b ) χ [,b] 2x, y)δ xy, then Corollry.2 coincides with the clssicl Diz Metclf inequlity for integrls: f 2 x) dx m M g 2 m x) dx M ) fx)gx) dx, m 2 M 2 since the densities of the coordintes in this cse re given by w x) =b ) χ [,b] x) =w 2 x). License or copyright restrictions my pply to redistribution; see

5 PROBABILISTIC PROOF OF INEQUALITIES 7 The equlity holds if nd only if either i) m /M 2 = M /m 2, or ii) m /M 2 <M /m 2 nd I x w x) dx =, where I x I x,x, x [, b] see Corollry.2 nd [2]). Other relted results cn be found in [6, p. 64]. 3. Póly Szegő inequlity The Póly Szegő inequlity is published in [6, pp. 57 nd 23 24] for both sums nd integrls. Below we provide probbilistic inequlity for moments covering this clssicl result. Theorem 2. Let ξ nd η be rel rndom vribles defined on probbility spce Ω, F, P). If P{m ξ M } =nd P{m 2 η M 2 } =for 0 <m j M j, j =, 2, then E ξ 2 E η 2 6) E ξη) 2 ) 2 m m 2 M M 2. 4 M M 2 m m 2 Inequlity 6) becomes n equlity if nd only if 7) ξ = m I A ω)m I Ω\A ω) nd η = m 2 I Ω\A ω)m 2 I A ω).s. for some rndom event A F such tht M m 2 8) PA) =. m M 2 M m 2 Proof. Assume tht ξ nd η re lmost surely nonnegtive bounded rndom vribles. Applying rithmetic-men geometric-men inequlity to the left-hnd side of the Diz Metclf inequlity 6) we obtin 9) E ξ 2 m M E η 2 m M 2 E ξ m 2 M 2 E η 2. 2 Now it is esy to get the desired result. To prove the necessity in the sttement on equlity let ξ nd η be defined by 7) nd 8), respectively. It is esy to see tht E ξ 2 = m M m m 2 M M 2 ), m M 2 M m 2 E η 2 = m 2M 2 m m 2 M M 2 ), m M 2 M m 2 E ξη = 2m m 2 M M 2, m M 2 M m 2 whence we obtin n equlity in 6). Inequlity 6) becomes n equlity if inequlities 6) nd 9) become equlities. Assume tht m m 2 <M M 2 the theorem is obvious otherwise). Then { ξ P η m, M }) ) =, b) E ξ 2 = m M m 2 M 2 ) E η 2. Reltion ) holds if nd only if there exist events A, B F, A B =Ω,A B =, such tht 20) ξ = m I A ω)m I B ω), η = m 2 I B ω)m 2 I A ω). License or copyright restrictions my pply to redistribution; see

6 8 TIBOR K. POGÁNY Substituting 20) into b) we obtin PA) M /m = PB) M 2 /m 2, tht is, M m 2 M 2 m PA) =, PB) =. m M 2 M m 2 m M M m 2 Thus reltions 7) nd 8) re proved. Corollry 2. Weighted Póly Szegő type inequlity for sums). Assume tht 0 <m x k M, k =,...,n, 0 <m 2 y l M 2, l =,...,m. Also let p kl 0 nd n n k= l= p kl =.Then m 2) x 2 kq k yl 2 r l ) 2 n ) 2 m m 2 M M 2 m x k y l p kl 4 M M 2 m m 2 where k= l= m p kl = q k, l= k =,...,n, p kl = r l, k= k= l= l =,...,m. Inequlity 2) becomes n equlity if nd only if m = n nd 22) M m 2 n κ = N 0, m M 2 M m 2 23) x i = = x iκ = m, x iκ = = x in = M, y i = = y iκ = M 2, y iκ = = y in = m 2, i j {,...,n}, 24) p kl =0, k, l) {i,...,i κ } {i κ,...,i n } {i κ,...,i n } {i,...,i κ }. Proof. Let rndom vector ξ,η) be defined on probbility spce Ω, PΩ), P) nd 0 < ξ,η) P{ξ = x k,η = y l } = p kl. Applying 6) we esily prove 2). The sttement on equlity esily follows from conditions 22) 24), since m = n. The Kntorovich inequlity is prticulr cse of the weighted Póly Szegő inequlity for sums see [4]). Putting m = n, x 2 j = γ j, yk 2 = γ k,weobtin0< µ γ k M where m = µ nd M = M. If 25) p kl = u 2 k n δ kl, j= u2 j k,l =,...,n, where u k 0fortlestonek, then we get the Kntorovich inequlity from 2): γ j u 2 u 2 )2 j j µ γ j= j= j 4 M M n ) 2 u 2 j µ j= see lso [6, p. 6]). For m = n we define the weights p kl similrly to 25). Then x 2 ku 2 k yku 2 2 k ) 2 m m 2 M M 2 n 2 x k y k u 4 M M 2 m m k) 2, 2 k= k= k= which is exctly the Greub nd Rheinboldt result see [3] nd [6, p. 6]). License or copyright restrictions my pply to redistribution; see

7 PROBABILISTIC PROOF OF INEQUALITIES 9 Putting m = n nd p kl = n δ kl we obtin from 2) the clssicl weighted Póly Szegő inequlity for sums: 26) x 2 k yk 2 ) 2 m m 2 M M 2 n ) 2 x k y k 4 M M 2 m m 2 k= k= for ll positive rel numbers x k nd y k, k =,...,n, bounded s in Corollry 2. see for exmple, [6, pp. 60 6], [8]). The condition for the equlity is simpler in this cse s compred to Corollry 2.), nmely 26) becomes n equlity under conditions 22) 24). Corollry 2.2 Weighted Póly Szegő inequlity for integrls). Let f nd g be positive Borel functions such tht k= 27) 0 <m fx) M nd 0 <m 2 gy) M 2 for lmost ll x, y) [, b] [c, d]. Let wx, y) be nonnegtive weight function with suppw) =[, b] [c, d]. Put Then 28) d c wx, y) dy = w x), f 2 x)w x) dx d c g2 y)w 2 y) dy ) d 2 c fx)gy)wx, y) dx dy wx, y) dx = w 2 y). m m 2 4 M M 2 ) 2 M M 2. m m 2 Inequlity 28) becomes n equlity if nd only if [, b] [c, d] nd 29) fx) =m χ S x)m χ [,b]\s x), gy) =M 2 χ [,b]\s y)m 2 χ S y), for some Borel set S [, b] whose Lebesgue mesure is such tht 30) S = M m 2 b ), m M 2 M m 2 nd 3) wx, y) =0 for ll x, y) S 2 [, b] \ S) 2. Since the proof of inequlity 28) is similr to tht of Corollries.2 nd 2., we omit it. Remrk 3. If [, b] =[c, d] ndξ,η) wx, y) =b ) χ [,b] 2x, y)δ xy,then 32) f 2 x) dx g2 x) dx ) b 2 fx)gx) dx m m 2 4 M M 2 M M 2 m m 2 ) 2 for ll f,g L 2 [, b] such tht 0 <m fx) M, 0 <m 2 gx) M 2. If functions f nd g stisfy conditions 30) nd 3), then inequlity 32) becomes n equlity, nd vice vers. Inequlity 32) is proved by Póly nd Szegő see [8, pp. 7 nd , Problem 93]). Note tht the cse of equlity in 32) is not studied in [8]. License or copyright restrictions my pply to redistribution; see

8 20 TIBOR K. POGÁNY 4. Rennie nd Schweitzer inequlities In this section we give generliztion of the Rennie nd Schweitzer inequlities for the probbilistic setting. There re two methods of proof of these results, nmely ) the proof bsed on the linerity nd monotonicity of the opertor E, 2) the proof bsed on n ppliction of the Diz Metclf nd Póly Szegő inequlities. Let ζ be positive nd bounded rndom vrible, tht is, P{m ζ M} =, m > 0. Then Thus E Mζ ) ζ m) 0. E ζ mm E ζ m M. The ltter is Rennie type inequlity for positive bounded rndom vribles. According to the rithmetic-men geometric-men inequlity, the left-hnd side of the Rennie inequlity is greter thn or equl to 2 mm E ζ E/ζ). Hence which is Schweitzer type inequlity. E ζ E ζ m M)2 4mM, Theorem 3. Let ζ be n lmost surely positive bounded rndom vrible defined on probbility spce Ω, F, P), tht is, m>0 nd P{m ζ M} =.Then 33) E ζ mm E ζ m M where the inequlity becomes n equlity if nd only if ζ = mi A ω)mi Ω\A ω) for some rndom event A F. Moreover 34) E ζ E ζ m M)2 4mM where the inequlity becomes n equlity if nd only if ζ = mi A ω)mi Ω\A ω) for some rndom event A F such tht PA) = 2. Proof. It is necessry to put ξ 2 = ζ = η 2 in 6) nd 6). The following result is given in [9] without ny discussion of the cse of equlity see lso [6, p. 63]). Corollry 3. Rennie inequlity for sums). Let x j, j n, be positive rel numbers nd let π j 0 for ll j n nd n j= π j =.Then 35) x j π j mm j= j= π j x j m M, where m nd M re the miniml nd mximl terms of the sequence {x j }. Inequlity 35) becomes n equlity if nd only if x j {m, M}, j =,...,n. License or copyright restrictions my pply to redistribution; see

9 PROBABILISTIC PROOF OF INEQUALITIES 2 Proof. Consider rndom vrible ζ with the distribution P{ζ = x j } = π j. Applying 34) we get 35). Let I {,...,n}. Putx j = m for j I nd x j = M for j I. Since π j x j π j mm = m π j M π j mm π j ) π j = m M, x j= j= j m M j I j I j I j/ I Corollry 3. is proved. Corollry 3.2 Rennie inequlity for integrls). Let f,/f L w[, b] where wx) 0 nd wx) dx =.If0 <m fx) M for ll x [, b], then wx) 36) fx)wx) dx mm dx m M. fx) The inequlity becomes n equlity if nd only if fx) =mχ S x)mχ [,b]\s x) for some Borel set S [, b]. Corollry 3.3 Weighted Schweitzer inequlity for sums). Let 0 <m x j M, π j 0, j =,...,n, nd moreover n j= π j =.Then 37) n π j m M)2 x j π j x j 4mM. j= j= The inequlity becomes n equlity if nd only if n is even n =2λ), λ l= π l =/2, nd x j = = x jλ = m, x jλ = = x jn = M, j l {,...,n}. Corollry 3.4 Weighted Schweitzer inequlity for integrls). Let f L w[, b] nd /f L w[, b] where wx) 0 nd wx) dx =.If0 <m fx) M for ll x [, b], then wx) m M)2 38) fx)wx) dx dx fx) 4mM. The inequlity becomes n equlity if nd only if fx) =mχ S x)mχ [,b]\s x) for some Borel set S [, b] whose Lebesgue mesure S is equl to b )/2. The Schweitzer inequlity cn be found in [0] for both sums nd integrls. rndom vrible ξ P{ξ = x j } = π j =/n, j =,...,n, is defined on some probbility spce Ω, PΩ), P), then 34) implies tht n x j m M)2 n 2 x j 4mM, j= j= which coincides with the Schweitzer inequlity for sums. If ξ U[, b] in 39), tht is, wx) =b ) χ [,b] x), then dx m M)2 fx) dx fx) 4mM b )2. This is the originl Schweitzer inequlity proved in [0]. Other relted results cn be found in [6, pp. 60 6], [8, pp. 7 nd ]. If License or copyright restrictions my pply to redistribution; see

10 22 TIBOR K. POGÁNY Acknowledgements. The uthor is grteful to O. I. Klesov, Kiev, for suggesting the bove Equlity Theorem, which llows one to shorten the nlysis of the cses where the Diz Metclf nd Póly Szegő inequlities become equlities. The uthor lso thnks Tmás F. Móri, Budpest, for helpful discussions of the topic nd for the mnuscript of the pper [], where further generliztions of moment inequlities re proved. Bibliogrphy. V. Csiszár nd T. F. Móri, The convexity method of proving moment type inequlities, Stt. Probb. Lett ), no. 3, MR g:60032) 2. J. B. Diz nd F. T. Metclf, StrongerformsofclssofinequlitiesofG. Póly G. Szegő nd L. V. Kntorovich, Bull. Amer. Mth. Soc ), MR :3846) 3. W. Greub nd W. Rheinboldt, On generliztion of n inequlity of L. V. Kntorovich, Proc. Amer. Mth. Soc ), MR :3774) 4. L. V. Kntorovich, Functionl nlysis nd pplied mthemtics, Uspekhi Mtem. Nuk N.S.) 3 948), no. 628), Russin) MR :380) 5. O. I. Klesov, Letter to the uthor 2003). Unpublished) 6. D. S. Mitrinović, Anlitičke nejednkosti, Grdevinsk Knjig, Beogrd, 970. MR :4984) 7. D. S. Mitrinović nd J. E. Pečrić, Men Vlues in Mthemtics, Mtemtički problemi i ekspozicije, vol. 4, Nučn Knjig, Beogrd, G. Póly nd G. Szegő, Problems nd Theorems in Anlysis, Clssics in Mthemtics Series, vol. I, Springer-Verlg, New York, 976. MR B. C. Rennie, On clss of inequlities, J.Austrl.Mth.Soc.3 963), MR :3590) 0. P. Schweitzer, An inequlity concerning the rithmetic men, Mth.Phys.Lpok23 94), Hungrin) Fculty of Mritime Studies, University of Rijek, Studentsk 2, 5000 Rijek, Croti E-mil ddress: pognj@brod.pfri.hr Received 20/MAR/2002 Trnslted by THE AUTHOR License or copyright restrictions my pply to redistribution; see