Some discrete inequalities of Grüss type and their applications in estimating the p-moments of guessing mapping are given

Sever Silvestru Dragomir

Dragomir, Sever Silvestru

Hrčak - Portal of scientific journals of Croatia

UDK 517.51517.958Izvorni znanstveni ~lanak Prihva}eno za tisak: 3.10.2005.SOME DISCRETE INEQUALITIES OF GRÜSS TYPE ANDAPPLICATIONS IN GUESSING THEORYSever Silvestru DragomirAbstractSome discrete inequalities of Grüss type and their applications in estimat-ing the p-moments of guessing mapping are given.Key words and phrases: Discrete Inequalities, Guessing Mapping, p-Mo-ments1. IntroductionIn 1935, G. Grüss proved the following integral inequality which gives anapproximation of the integral of the product in terms of the product of the integralsas follows:(1.1)1 1 1b af x g x dxb af x dxb ag x dxababab ( ) ( ) ( ) ( )  14( )( )  where f, g: ¢a, b£ 	 R are integrable on ¢a, b£ and satisfy the condition(1.2)  f (x)   g (x) for each x  ¢a, b£ where    , are given real constants.Moreover, the constant 14is sharp in the sense that it can not be replaced by asmaller one.For a simple proof of (1.1) as well as some other integral inequalities of Grüssntype see Chapter X of the recent book ¢1£ by Mitrinovi}, Pe~ari} and Fink.In 1950, M. Biernacki, H. Pidek and C. Ryll-Nardzewski ¢1, Ch. X£ establishedthe following discrete version of Grüssn inequality:Theorem 1. Let a = (a1, ..., an), b = (b1, ..., bn) be two n-tuples of real numbers such that rai  R and s  bi  S for i = 1, ..., n. Then one has:(1.3)1 1 11 1 1na bnanbiini iiniin     13Date: March, 1999.1991 Mathematics Subject Classification. 26D15, 94Axx.   12112nnnnR r S s( )( )where ¢x£ is the integer part of x, x  R.A weighted version of Grüssn discrete inequality was proved by J.E. Pe~ari} in1979, ¢1, Ch. X£:Theorem 2. Let a and b be two monotonic n-tuples and p a positive one. Then(1.4)1 1 11 1 1Pp a bPp aPp bniini iniinini iin         a a b bP PPn nk nk kn1 1 1 112max ,where P p P P Pn i k n kin: ,    1 11.In 1981, A. Lupas¸ ¢1, Ch. X£ proved some similar results for the first difference ofa as follows:Theorem 3. Let a, b be two monotonic n-tuples in the same sense and p a positive n-tuple.Then(1.5) min min1 1 1 1 1 121 1         i n i i i n i ininiia a b bPi pPip112nin      1 1 11 1 1Pp a bPp aPp bni iinini iinni iin          max max1 1 1 1 1 121 1i n i i i n i ininiia a b bPi pPip112nin.If there exist the numbers a¯, a¯1, r, r1 (rr1 > 0) such that ak = a¯ + kr and bk = a¯1 + kr1, thenin (1.5) the equality holds.For some generalizations of Grüssn inequality for isotonic linear functionalsdefined on certain spaces of mappings see Chapter X of the book ¢1£ where furtherreferences are given.In the recent paper ¢2£, S.S. Dragomir and G.L. Booth obtained the followinginequality of Grüssn type:Theorem 4. Let i, xi be real numbers and pi  0 (i = 1, ..., n) so that piin 11.14Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theoryThen we have the inequality:(1.6) p x p p xi i i i iini iinin    1 11         max max1 1 1 1 1 121j n j j j n j j i iinx x i p ip  in12.The inequality (1.6) is sharp in the sense that the constant C = 1 in the right side can not bereplaced by a smaller one.Note that in paper ¢2£ they proved the inequality (1.6) in the general case ofnormed linear spaces.The main aim of this paper is to point out another inequality of Grüssn type andapply it to estimate the moments of guessing mappings as in the papers ¢5£-¢7£.The following results holds:Theorem 5. Let ai, bi (i = 1, ..., n) be a real number and pi (i = 1, ..., n) a probability distri-bution. Definea a a b b bij k kk jiij k k( ) ( ): , :      1111k ji 11for all 1  j  i  n and  > 1,1 1  = 1. Then we have the inequality(2.1) p a b p a p biini i i i i iinin   1 11     !     ( ) ( )( )i j p p a i j p p bj i ni j ij i j i11 jj i n( )    11   an nj i ni jb i j p p1 11( ) ( ) ( )  .The first inequality in (2.1) is sharp.Proof. It is well known that the following equality holds:p a b p a p bi i i i i i iininin 11115Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theory= p p a a b bi jj i ni j i j1    ( )( ).Using the simple observation which asserts thata a a a b b b bi jk jik k i j k kk ji         ( ), ( )11 11and the generalized triangle inequality, we get(2.2) p a b p a p biini i i i iinin   1111       p p a a b bi jj i nk k l lk l ji11 11( )( ),.Using Hölderns discrete ineequality for double sums, we have(2.3) ( )( ),a a b bk k l lk l ji    1 11     a a b bk kk l jik kk l ji11111, , 1   !   ( ) ( )i j a a i j b bk kk jik kk111111  ji11 ( ) ( ) ( )i j a bij ij for all 1  j  i  n.Now, by (2.2) and (2.3) and by Hölderns inequality, we have:p a b p a p bi i i i i i iininin 111    ( ) ( ) ( )i j p p a bi j ij ijj i n 116Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theory    !    ( ) ( )( ) (i j p p a i j p p bij i nj ij i j ij11   ) )11  j i nand the first inequality in (2.1) is proved.For the second inequality, let observe thata a b b j i nij n ij n( ) ( ) ( ) ( ) .       1 1 1and for allThen( ) ( )( )i j p p a i j p p bij i nj ij i j ij    !    11 ( )    11j i n  !   a i j p p b i j p pn ij i nj n i j1111( ) ( )( ) ( )11  j i n   a b i j p pn n ij i nj1 11( ) ( ) ( ) and the second inequality is also proved.For the sharpness of the first inequality in (2.1), let choose ai = a1 + (i – 1) a,bi = b1 + (i – 1)b with a, b > 0 and i =2, ..., n. Then we have:p a b p a p bi iini i iini iin     1 1 1   ab i j p pj i ni j( )21.Also, we havea a i j b b i jij ij( ) ( )( ) , ( )      1 1and then( ) ( )( )i j p p a i j p p bij i nj ij i j ij    !    11 ( )    11j i n17Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theory  !       a i j p p b i j p pij i nj i jj i n( ) ( )21121 1   ab i j p pij i nj( )21and the sharpness of the first inequality in (2.1) is proved.The following corollary holds:Corollary 1. With the above assumptions we have:(2.4)1 1 1111na bnanbi i i iininin      !      1211ni j a i j bj i nij ij( ) ( )( ) ( )  11  j i n  nna a b bk kknk kkn21211216 1.The first inequality in (2.4) is sharp.Proof. The first inequality is obvious by the above theorem.Let us compute( )i jj i n  1             ( ) ( ) ... ( )2 31 2 1 3 1j j n jj j j n                 2 2 1 2 3 3 1 2 3 1 21 22 2( ) ( ) ... ( ... ).n n n.. ( ) ( ) ... ( ... )           n n2 1 1 2 1 2 3 1 2    kk kk kn nknknknkn2112111212( ) ( 2 16 )and the corollary is thus proved.18Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theory3. Applications for the Moments of Guessing MappingJ.L. Massey in the paper ¢3£ considered the problem of guessing the value ofrealization of a random variable X by asking questions of the form: "Is X equal to x?"until the answer is "Yes".Let G (X) denote the number of guesses required by a particular guessingstrategy when X = x.Massey observed that E (G (X)), the average number of guesses is minimized bya guessing strategy that guesses the possible values of X in decreasing order ofprobability.We begin by giving a formal and generalized statement of the above problemby following E. Arikan ¢4£.Let (X, Y) be a pair of random variable with X taking values in a finite set " ofsize n, Y taking values in a countable set Y. Call a function G (X) of the randomvariable X a guessing function in X if G: " 	 #1, ..., n$ in one-to-one. Call a functionG (X%Y) a guessing function for X given Y if for any fixed value Y = y, G (X	y) is aguessing function for X. G (X	y) will be thought of as the number of guessingrequired to determine X when the value of Y is given.The following inequalities on the moments of G (X) and G (X	Y were provedby E. Arikan in the recent paper ¢4£.Theorem 6. For an arbitrary guessing function G (X) and G (X	Y) and any pwe have:(3.1) E G X n P xp pXxpp( ( ) ) ( ln ) ( ) 1111"and(3.2) E G X Y n P x yp pX Ypxy Y( ( ) ) ( ln ) ( , ),  1111"pwhere PX,Y and PX are probability distributions of (X, Y) and X, respectively.In paper ¢7£, S.S. Dragomir and J. van der Hoek have proved the followingestimation result for the moments of guessing mapping:Theorem 7. Assume thatPM : = max #pi % i = 1, ..., n$ an Pm : = min #pi % i = 1, ..., n$and PM Pm. Then we have the estimates:19Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theoryG n P nP PnPp mpM mp mp( )( )( ) 1 111 E (G (X)p)  G n P nP PnPp MpM mp Mp( )( )( )1 111for p  1, whereG nS nnppp( ):( )1andS n ippin( ):1.Corollary 2. With the above assumption, we have:1211 2 12 nP P n nPP Pm M mM m    E G XnP P n P nP Pm M MM m( ( ))1211 2 12.For other estimations of E (G (X)p) see the papers ¢5£-¢7£.Now, let us introduce the notationsh p k kijp pk ji( ) ( ): ( ( ) )     111where 1  j < i  n, p > 0,  > 1.The following proposition holds.Proposition 1. Let p, q > 0. Under the above assumptions, we have%E(G(X)p+q) - E(G(X)p)%     !    ( ) ( ) ( )( )i j p p h p i j p pij i nj ij i j11      h qijj i n( ) ( ) 11      ( ( ) ) ( ( ) )k k k kp pknq qkn1 12121( )i j p pi jj i n  120Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theorywhere ,  > 1 and1 11   .The proof follows by Theorem 5 applied for ai = ip, bi = iq (i = 1, ..., n). We shallomit the details.Now, starting to the probability distribution pi  0 (i =1, ...., n) we define:p p pij k kk ji( ) :    111where 1  j < i  n,  > 1.Using Corollary 1 we have the following estimation of the moments of guessingmapping:Proposition 2. Let p > 0. Then we have the estimate.%E(G(X)p) - Sn(p)%     !    111ni j h p i j pj i nij ij( ) ( ) ( )( ) (  )     11j i n   n k k p pp pknk kkn22112161( ( ) )  1.In particular, we have the following bound for the average number of guesses:E G Xn( ( ))12      11122 211nn ni j pijj i n( )( ) ( ) 1 ( )( )n np pk kkn211211 16 .21Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theoryReferences¢1£ D.S. MITRINOVI], J.E. PE^ARI] and A.M. FINK, Classical and New Inequalities inAnalysis, Kluwer Academic Publishers, Dordrecht, 1993.¢2£ S.S. DRAGOMIR and G.L BOOTH, On a Grüss-Lupas¸ type inequality and its applica-tions for the estimation of p-moments of guessing mappings, Mathematical Communi-cation, 5(2000), 117-126.¢3£ J.L. MASSEY, Guessing and entropy, Proc. 1994, IEEE Int. Symp. on Information Theory,Trodheim, Norway, 1994, p. 204.¢4£ E. ARIKAN, An inequality on guessing and its applications to sequential decoding, IEEETransactions on Information Theory, 42(1996), 99-105.¢5£ S.S. DRAGOMIR and J. VAN DER HOEK, Some new inequalities for the averagenumber of guesses, Kyungpook Mathematical Journal, 39(1)(1999), 11-17.¢6£ S.S. DRAGOMIR and J. VAN DER HOEK, New inequalitis for the moments of guessingmapping, East Asian Mathematical Journal, 14(1)(1998), 1-14.¢7£ S.S. DRAGOMIR and J. VAN DER HOEK, Some new analytic inequalities and theirapplications in guessing theory, J. Math. Anal. Appl., 225(1998), 542-556.Diskretne nejednakosti Grüssovog tipai primjene u tepriji odlu~ivanjaSever Silvestru DragomirSA@ETAKDokazane su neke nejednakosti Grüssovog tipa i dane su njihoveprimjene pri procjenjivanju p-momenata preslikavanja u teoriji odlu~ivanjaSever Silvestru DragomirSchool of Computer Science and Mathematics,Victoria University, PO Box 14428, Melbourne City, MC 8001 AustraliaE-mail address: severºmatilda.vu.edu.au22Rad Hrvat. akad. znan. umjet. 491. Matemati~ke znanosti 15 (2005), str 13-22.S.S. Dragomir Some discrete inequalities of grüss type and applications in guessing theory

Some discrete inequalities of Grüss type and applications in guessing theory

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia