J. Austal. Math. Soc. Se. B 40(1998), 80 85 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 2 2 2 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, 41000 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, 41000 ageb, Coatia. c Austalian Mathematical Society, 1998, Seial-fee code 0334-2700/98 80
[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)
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] Jensen s inequality fo distibutions possessing highe moments 83 4. 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.
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] 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) 81 98. [3] A. E. Eckbeg, Shap bounds on Laplace-Stieltjes tansfoms, with applications to vaious queueing poblems, Math. of Ope. es. 2 (1977) 135 142. [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) 526 528. [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) 91 94. [7] T. Popoviciu, Les fonctions convees (Hemann, Pais, 1945). [8] T. olski, Some inequalities fo GI=M=n queues, astos. Mat. 13 (1972) 43 47.