JENSEN S INEQUALITY FOR DISTRIBUTIONS POSSESSING HIGHER MOMENTS, WITH APPLICATION TO SHARP BOUNDS FOR LAPLACE-STIELTJES TRANSFORMS

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1 J. Austal. Math. Soc. Se. B 40(1998), JENSEN S INEQUALITY FO DISTIBUTIONS POSSESSING HIGHE MOMENTS, WITH APPLICATION TO SHAP BOUNDS FO LAPLACE-STIELTJES TANSFOMS B. GULJAŠ 1,C.E.M.PEACE 2 and J. PEČAIĆ 3 (eceived 7 May 1996; evised 23 Octobe 1996) Abstact A new vesion of Jensen s inequality is established fo pobability distibutions on the nonnegative eal numbes which ae chaacteized by moments highe than the fist. We deduce some new shap bounds fo Laplace-Stieltjes tansfoms of such distibution functions. 1. Intoduction In a pevious aticle [4] we established the following vaiant of Jensen s inequality. Fo an ealie discussion of this theme and eamples of applications see Pittenge [6]. THEOEM A. Suppose f./= is a positive, conve function on.0; 1/ and a pobability measue on [0; 1/, not consisting entiely of an atom at the oigin, whose second moment eists. Then f./ ½ Ð f : If f./= is stictly conve then stict inequality applies unless the suppot of intesects.0; 1/ in a single point. This esult may be put to use to give a tanspaent deivation of the following well-known inequality in the teletaffic liteatue elating to the G=M=n queue (see, fo eample, olski [8]). 1 Mathematics Depatment, Univesity of ageb, Bijenička Cesta 30, ageb, Coatia 2 Depatment of Applied Mathematics, The Univesity of Adelaide, Adelaide SA 5005, Austalia. 3 Faculty of Tetile Technology, Univesity of ageb, Pieottijeva 6, ageb, Coatia. c Austalian Mathematical Society, 1998, Seial-fee code /98 80

2 [2] Jensen s inequality fo distibutions possessing highe moments 81 THEOEM B. Let be a pobability measue with nonnegative suppot and positive moments m i D t i.t/.i D 1; 2/. Then the functional e st achieves its supemum uniquely at D ½ 2, whee the measue ½ 2 is given by d½ 2.t/ D.1 m 2 =m 1 2/Ž.t/ dt C.m 2 =m 1 2/Ž.t m 2 =m 1 / dt and whee as usual Ž.Ð/ epesents the Diac delta. A systematic povision of candidates fo applications of Theoem A emeges fom the notion of n-conveity.n ½ 2/. See Popoviciu [7], Aumann and Haupt [1], Bullen [2] and Pečaić, Poschan and Tong [5] fo a discussion of n-conve functions. We note in paticula that this useful class of functions can be chaacteized by the popety that, fo n ½ 2, f is n-conve if and only if f.n 2/ eists and is conve (see [1, p. 286]). Thus 2-conveity is just odinay conveity. We have the following theoem. THEOEM C. Suppose f is an n-conve function on.0; 1/ with f.i/.0/ D 0 (0 i < n 2). Then the map:! f./= n 2 is conve on.0; 1/. Thus we have that f.3/ ½ 0on.0; 1/ implies that the map:! [ f./ f.0/]= is conve. In ou ealie pape use was made of the homely paticula case f./ D e s. In this note we pusue the foegoing appoach to deive some new shap bounds fo the Laplace-Stieltjes tansfom of a pobability distibution on [0; 1/ chaacteized by highe moments. In Section 2 we pesent a moe geneal vesion of Theoem A fo highe moments. The equiements on the deivatives of f in Theoem C ae athe estictive fom the viewpoint of some pobabilistic applications and it tuns out to be pefeable to poceed diectly fom the esults of elementay calculus. These ae codified in Section 3 as Poposition 1. This agees with Theoem C fo n D 3 but offes futhe scope fo applications when n > 3. In Section 4 we may the esults of Sections 2 and 3 to engende a genealization of Theoem B. Finally, in Section 5, we illustate by an eample based on Section 4 the advantages that Poposition 1 can offe ove Theoem C. 2. Jensen s inequality THEOEM 1. Suppose that is nonnegative, that the map:! G./ D f./= is positive and conve on.0; 1/ and that is a pobability measue on [0; 1/ possessing an. C 1/-st moment and not consisting simply of an atom at the oigin. Then f./ ½ Ð C1 C1 Ð f C1 : (2.1)

3 82 B. Guljaš, C. E. M. Peace and J. Pečaić [3] If G is stictly conve then the inequality is stict unless the suppot of intesects.0; 1/ in a single point. POOF. Let X be a andom vaiable with pobability measue ¹ given by d¹.t/ D t.t/./ ; so that E.X/ D d¹ D C1 : By conveity, Jensen s inequality yields E[G.X/] ½ G.E.X//; o G./ ½ C1 ½ G ; whence we have (2.1). The statement on stict inequality is inheited fom the coesponding esult fo Jensen s inequality. 3. An analogue fo Theoem C POPOSITION 1. Suppose that f is a function on.0; 1/ with a second deivative and that is a positive intege. A necessay and sufficient condition that the map: t! f.t/=t be conve is that POOF. The esult is immediate fom h.t/. C 1/ f.t/ 2tf 0.t/ C t 2 f 00.t/ ½ 0: d 2 =dt 2 [ f.t/=t ] D t 2 h.t/: COOLLAY 1. If h is diffeentiable, its nonnegativity is guaanteed by the conditions h.0/ ½ 0 and h 0 ½ 0 on [0; 1]. Now h 0.t/ D. 1/ f 0.t/ 2. 1/tf 00.t/ C t 2 f 000.t/: Fo D 1, ou conditions educe to f.0/ ½ 0 and f 000 ½ 0, which hold automatically fo any nonnegative 3-conve function. Similaly the conditions ae satisfied tivially fo > 1 by any function f with f.0/ ½ 0 fo which f 0 and f 000 ae nonnegative and f 00 is nonpositive.

4 [4] Jensen s inequality fo distibutions possessing highe moments Bounds fo Laplace-Stieltjes tansfoms We now poceed to a genealization of Theoem B. THEOEM 2. Let be a pobability measue with nonnegative suppot not consisting puely of an atom at the oigin and with given positive moments m j D t i.t/. j D ; C 1/. Then the functional.s/ D e st.s½0/ achieves its supemum uniquely at D ½ C1, whee the measue ½ C1 is given by d½ C1.t/ D 1 m C1 =m / dt: m C1 m C1 ½ Ž.t/ dt C mc1 Ž.t m C1 POOF. By Coollay 1, Poposition 1 applies fo f./ D 1 e s.s > 0/ fo all positive integal. Hence this choice of f satisfies the conditions of Theoem 1 and fom (2.1) we have 1 e s ½ mc1 ð 1 ep. smc1 =m m / Ł : (4.1) C1 The fundamental inequality fo L p noms gives 1=p 1=q jj p jj q if 0 < p < q fo a pobability measue, with stict inequality if does not consist of a single atom. Theefoe m C1 < m,sothat½ C1 C1 is a pope two-point pobability measue and (4.1) may be cast as.s/ e s d½ C1./: A simple calculation shows that ½ C1 has -th and. C 1/-st moments m and m C1 espectively. This gives the main pat of the enunciation. Uniqueness follows fom the final statement in Theoem 1, since thee is a unique measue on [0; 1/ with the two given moments whose suppot intesects.0; 1/ in a single point. This esult appeas to be new fo > 1. Fo D 2 it takes m 2 and m 3 as given and povides an inteesting complement to a esult of Eckbeg [3]. Eckbeg showed that if all thee moments m 1 ; m 2 ; m 3 ae given, then.s/ 1 m 2 m 2 1 C m 2 e sm2=m1 ; m 2 1 that is, the same uppe bound applies as when only m 1 ; m 2 ae given. Eckbeg emaked that the uppe bound needs an infinitesimal mass at 1" to achieve the coect thid moment. This last esult etends to ou geneal contet.

5 84 B. Guljaš, C. E. M. Peace and J. Pečaić [5] COOLLAY 2. The uppe bound given in Theoem 2 applies if m C2 is also given. POOF. The pobability measue ½ C1 has. C 2/-nd moment C2 Qm C2 D mc1 mc1 Ð D m2 C1 : m C1 m m By Cauchy s theoem m C2 ½ mc1 2 =m fo any pobability measue,sothatm C2 ½ Qm C2. Thee is nothing to pove if equality holds, so suppose m C2 > Qm C2. This enables us to constuct (fo some positive intege K ) a sequence.¼ k / k½k of pobability measues whose moments of odes ; C 1; C 2 ae espectively m ; m C1 ; m C2 with ¼ k conveging weakly to ½ C1 as k!1. Since the Laplace-Stieltjes tansfom of ¼ k conveges to that of ½ C1 we shall then have the desied esult. The constuction may be implemented as follows. Set m D m C1 =m and define ž 2;k D.m C2 Qm C2 /=[.k Þ C m/.k Þ C 1=k/k Þ ] ; ž 3;k D.m C2 Qm C2 / k= [.m 1=k/.k Þ C 1=k/] ; ž 1;k D.m C2 Qm C2 / k= [m k Þ ] ; ž 0;k D ž 2;k C ž 3;k ž 1;k ; fo all k ½ K.HeeKischosen sufficiently lage that m 1=K > 0, ž 3;K >ž 1;K, ž 0;K < 1 m C1 =m and ž C1 1;K < m C1 =m C1 and Þ is chosen sufficiently lage that [1 1=.mK/] > 1 C K 1 Þ. We eadily veify that, fo k ½ K, the measue ¼ k given by ½ m C1 d¼ k.t/ D 1 ž 0;k Ž.t/ dt C ž 3;k Ž.t.m 1=k// dt m C1 ½ m C1 C ž m 1;k Ž.t m/ dt C ž 2;k Ž.t.k Þ C m// dt C1 is a pobability measue with moments as stated that conveges weakly to ½ C1. 5. Theoem C and Poposition 1 It is inteesting to compae the analysis of Theoem 2, based on Poposition 1, fo D 2 with a paallel development using Theoem C with n D 4. The coesponding natual choice with the latte is then f./ D e s 1 C s, the last tem being foced on us by the equiement that f 0.0/ be zeo. We have at once that f./= 2 is conve. Theoem 1 leads to.s/ 1 C sm 1 ½ m3 ð 2 e sm 3=m 2 Ł 1 C sm m 2 3 =m 2 3

6 [6] Jensen s inequality fo distibutions possessing highe moments 85 o.s/ ½ 1 m 3 2 m 2 3 ½ C m3 2 e sm3=m2 s ð m m 2 1 m 2 =m 2 3Ł : 3 By Cauchy s inequality, the last tem in backets in nonnegative. If it is stictly positive, as must happen fo 6D ½ 3, then we obtain a vey poo lowe bound fo, since the last tem on the ight is unbounded fo s!1. Moeove, we appea to lack an appopiate pobabilistic intepetation fo this esult. efeences [1] G. Aumann and O. Haupt, Einfühung in die eelle Analysis, Band II (Walte de Guyte, Belin, 1938). [2] P. S. Bullen, A citeion fo n-conveity, Pacific J. Math. 36 (1971) [3] A. E. Eckbeg, Shap bounds on Laplace-Stieltjes tansfoms, with applications to vaious queueing poblems, Math. of Ope. es. 2 (1977) [4] C. E. M. Peace and J. E. Pečaić, An integal inequality fo conve functions, with application to teletaffic congestion poblems, Math. of Ope. es. 20 (1995) [5] J. E. Pečaić, F. Poschan and Y. L. Tong, Conve functions, patial odeings and statistical applications, Mathematics in Science and Engineeing, Vol. 187 (Academic Pess, San Diego, 1992). [6] A. O. Pittenge, Shap mean-vaiance bounds fo Jensen-type inequalities, Pobability and Statistics Lettes 10 (1990) [7] T. Popoviciu, Les fonctions convees (Hemann, Pais, 1945). [8] T. olski, Some inequalities fo GI=M=n queues, astos. Mat. 13 (1972)

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