The Australian Journal of Mathematical Analysis and Applications

Transcription

1 The Austrlin Journl of Mthemticl Anlysis nd Applictions Volume 13, Issue 1, Article 1, pp. 1-9, 016 SOME NEW GENERALIZATIONS OF JENSEN S INEQUALITY WITH RELATED RESULTS AND APPLICATIONS STEVEN G. FROM Received 13 Jnury, 016; ccepted 0 April, 016; published 17 My, 016. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEBRASKA AT OMAHA, OMAHA, NEBRASKA sfrom@unomh.edu ABSTRACT. In this pper, some new generliztions of Jensen s inequlity re presented. In prticulr, upper nd lower bounds for the Jensen gp re given nd compred nlyticlly nd numericlly to previously published bounds for both the discrete nd continuous Jensen s inequlity cses. The new bounds compre fvorbly to previously proposed bounds. A new method bsed on series of loclly liner interpoltions is given nd is the bsis for most of the bounds given in this pper. The wide pplicbility of this method will be demonstrted. As by-products of this method, we shll obtin some new Hermite-Hdmrd inequlities for functions which re 3-convex or 3-concve. The new method works to obtin bounds for the Jensen gp for non-convex functions s well, provided one or two derivtives of the nonliner function re continuous. The men residul life function of pplied probbility nd relibility theory plys prominent role in construction of bounds for the Jensen gp. We lso present n exct integrl representtion for the Jensen gp in the continuous cse. We briefly discuss some inequlities for other types of convexity, such s convexity in the geometric men, nd briefly discuss pplictions to relibility theory. Key words nd phrses: b-entropy, convex ccording to the geometric men, convex function, decresing men residul life DMRL, Hermite-Hdmrd inequlity, Jensen gp, Jensen s inequlity, liner interpoltion, moments, men residul life function MRL, moment-generting function, new better thn used in expecttion NBUE, rel nlytic, Riemnn-Stieltes Sums. 000 Mthemtics Subect Clssifiction. Primry 6, 8, 60. Secondry 6A, 6D.. ISSN electronic: c 016 Austrl Internet Publishing. All rights reserved.

2 STEVEN G. FROM 1. INTRODUCTION It is well-known tht the discrete Jensen s inequlity sttes tht if f is convex function on [, b], p i > 0, i = 1,..., n, n =1 p = 1, nd x i [, b], i = 1,..., n, then 1.1 D = p i fx i f p i x i 0. The most generl form of Jensen s inequlity sttes tht if Ω, F, µ is mesure spce with µω = 1, fx is convex, nd gx is µ-intergrble rel-vlued function, then 1. D = f gdµ f gdµ 0. Ω Ω In this pper, gx = x, very importnt specil cse. We shll minly be concerned with the discrete cse nd pss to the limit to the possibly continuous cse. Mny ppers hve been written on Jensen s inequlity nd its mny compnion inequlities. See [9] nd [4], for exmple. In this pper, we shll obtin new inequlities for the Jensen gp, D, in 1.1 nd 1.. These will llow us to obtin new compnion inequlities, including new Hermite-Hdmrd type inequlities for 3-convex functions. We re concerned with obtining bounds for the Jensen gp D in 1.1. In continuous cses where there exists probbility density function bsolutely continuous with respect to Lebesgue mesure hx, then 1. becomes, in the continuous cse: 1.3 D = fxhxdx f x hxdx. We will obtin upper nd lower bounds for D in 1.1 nd 1.3 even if fx is not convex/concve, provided f hs one or two continuous derivtives. We focus first on the discrete cse nd present new bounds nd compre them nlyticlly nd numericlly to previously published bounds. Lter, we obtin, s limiting cse, bounds for D in the continuous cse. We shll obtin both n exct integrl representtion nd n exct infinite series representtion for the Jensen gp D in 1.3 for the specil cse where hx = 0, if x [, b], tht is, when the support of hx is [, b], where < b re rel numbers. Then it is simple mtter to extend to the cses where b or. In pplied probbility nd relibility theory res, the function gx below is of gret importnce: gx = t xhtdt x htdt, x < b x 1.4 gb = 0. The function gx is known s the men residul life function. We shll see tht the Jensen gp bounds re strongly relted to gx s is the exct single integrl representtion of D to be given lter in Theorem 4.3. A new method is given to obtin the inequlities in this pper. This method is widely pplicble to obtin new Jensen type inequlities. First, let s discuss the finite n discrete cse. The following bounds on the Jensen gp D hve previously been published. First, let s discuss some globl bounds for D. Severl globl bounds for the Jensen gp D hve been published. Two of the most prominent re given below., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

3 GENERALIZATIONS OF JENSEN S INEQUALITY 3 Theorem A. Drgomir [1]. Suppose p i > 0, n =1 p = 1. If f is differentible convex mpping on [, b], then D = p i fx i f p i x i b f b f D f, b. The next theorem ws given in [7]. Theorem B. Under the conditions of Theorem A bove, D = p i fx i f p i x i b f + fb f S f, b. Theorem C. Theorem 1 of [15]. Let f : [, b] R be differentible convex function on, b. Let x i, b, i = 1,,..., n. Let p i 0, i = 1,,..., n nd n p i = 1. Let x = n p ix i. Then D = p i fx i f x 1.7 p i fx i f x p i x i x f x L DS. Theorem D.. Theorems 1 nd of [14]. Let f : [, b] R be differentible convex function on, b, x i, b, p i 0, i = 1,,..., n, n p i = 1. Let D nd x be s given in Theorem C bove. Then 1.8 D p i x i f x i x p i f x i U D,1 nd 1.9 b D p i x i f x i + f f 1 p i f x i f 1 p i f x i p i f x i f x U D, n=1 U D,1, where f 1 is the inverse function of the derivtive f. Next, let s present some of the best locl bounds for Jensen gp D. We shll compre these to the new bounds to be discussed in lter sections., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

4 4 STEVEN G. FROM Theorem E. Theorem 1 of []. Let f : I R where I is n intervl. Let x i I, i = 1,,..., n, p i 0, i = 1,,..., n with n p i = 1. Let M = mx{x 1, x,..., x n }, m = min{x 1, x,..., x n }. If f is differentible nd f is strictly incresing, then 1.10 D = p i fx i f p i x i λ, where 1.11 λ = fm fm fm fm f 1 + M m M m fm fm f f 1 U BP P 1 M m Mfm mfm M m where f 1 is the inverse function of f. Theorem F. Theorem 11 of []. Let f : [m, M] R be convex function on [m, M], p i > 0, i = 1,,..., n, n p i = 1, n nd x i [m, M], i = 1,..., n, with m x 1 < x < < x n M. Then 1.1 D = p i fx i f p i x i mx 1 k n {p kfm + 1 p kfm fp km + 1 p km} U BP P where p k = k p i, k = 1,,..., n. Theorem G. Theorem 1.3 of [17]. Let f[, b] R be continuous, twice differentible on, b. Let x i [, b], p i 0, i = 1,,..., n with n p i = 1. Suppose m = inf{f x : x [, b]} nd M = sup{f x : x [, b]} exist. Then 1.13 D = p i fx i f p i x i m p i x i p i x i F L nd 1.14 D M p i x i p i x i F U. The following theorem is specil cse of more generl result given in [10]., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

5 GENERALIZATIONS OF JENSEN S INEQUALITY 5 Theorem H. Let f : [, b] R be differentible convex function. Let x i [, b], p i 0, i = 1,..., n, n =1 p = 1. Then 1.15 D = p i fx i f p i x i 1/ p i f x i p i x i p i x i 1/ U DG.. NEW BOUNDS - DISCRETE CASE To present new bounds for D, we shll first need the following lemms. Lemm.1. Let ft be rel-vlued function of t on [c, d], such tht ft nd its first two derivtives f t nd f t re ll continuous throughout the intervl c t d. Let Lt denote the interpolting polynomil of degree 1 or less pssing through the points c, fc nd d, fd. Then for ech t 0 in [c, d], there is number ξt 0 in c, d such tht.1 ft 0 = Lt 0 + f ξt 0 t 0 ct 0 d, where fd fc. Lt 0 = fc + t 0 c, c < d. d c Thus, the pproximtion ft 0 Lt 0 hs liner interpoltion error.3 Et 0 ft 0 Lt 0 = f ξt 0 t 0 ct 0 d. Proof. See [5], p Lemm.. Let ft be rel-vlued function of t on [c, d]. Suppose ft nd its first three derivtives f t, f t nd f 3 t re continuous throughout [c, d]..4.5 If f t 0 nd f 3 t 0, c t d, then for ech t 0 in [c, d], the pproximtion error Et 0 in the pproximtion ft 0 Lt 0, where Lt 0 is given by., stisfies [d cf d fd + fc] t 0 ct 0 d d c Et 0 [fd fc d cf c] t 0 ct 0 d d c, where Et 0 = ft 0 Lt 0. b If f t 0 nd f 3 t 0, then.4 holds with the inequlity reversed, tht is, for c < d, Proof. See [19], p [fd fc d cf c] t 0 ct 0 d d c Et 0 [d cf d fd + fc] t 0 ct 0 d d c., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

6 6 STEVEN G. FROM Now, we re redy to present some min results on bounds for the Jensen gp D in the discrete cse. Theorem.3. Let ft be rel-vlued function on [, b]. Suppose tht f nd its first two derivtives f nd f re continuous on [, b]. Suppose tht x 1 x x n b. Suppose p i 0, i = 1,,..., n, with n p i = 1, p 1 > 0, p n > 0. Let.6 = p i + p i p n = p, i = 1,,..., n,.7 x i = p i+1 x i p n x n = nd.8 x i = = pi pi x i + Ri+1 =i n L=i+1 p Lx L +1, i = 1,,..., n 1, x i x i + 1 p i x i, i = 1,,..., n. Then there exist rel numbers θ 1, θ,..., θ n 1 with x i < θ i < x i, i = 1,,..., n 1 such tht D = p i fx i f p i x i.9 Proof. Define A i by.10 A i = = = = i 1 p L fx L + fx i L=1 i 1 p L fx L + f L=1 f θ i x i x i x i x i f θ i p i 1 p i pi x i x i. x i + 1 p i x i,, i = 1,,..., n, where ny impossible sum is defined to be zero. Then.11 f p i x i p i fx i = A 1 A n = A A +1. Now simple lgebr gives x +1 = x nd.1 A A +1 = R fx [ p R =1 fx + 1 p ] fx R = 1,,..., n 1. If x = x, A A +1 = 0. If x < x, then the expression in brckets of.1 is the linerly interpolted vlue of fx on the intervl x t x. Letting c = x, d = x, t 0 = x in, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

7 GENERALIZATIONS OF JENSEN S INEQUALITY 7 Lemm.1, we obtin this vlue s Lx = p R fx + Thus there exists rel number θ in x, x such tht 1 p fx R..13 fx Lx = f θ x x x x. Thus,.1 gives f θ A A +1 = R x x x x, 1 n 1, which is true lso when x = x, since x = x = x in this cse. Summing over gives, using.13: f θ.14 f p i x i p i fx i = x x x x R. Also, simple lgebr gives nd from which we obtin x i x i =.15 x i x i x i x i = p i Thus,.14 gives.16 nd Theorem.3 is proved. =1 1 p i x i x i x i x i = p i x i x i, i = 1,,..., n 1, D = = 1 p i x i x i, i = 1,,..., n 1. p i fx i f p i x i f θ i p i 1 p i x i x i Remrk.1. Note tht x i x i x i nd x i x i x i x i 0, i = 1,,..., n 1. Thus, if ft is convex on [, b], then.16 bove is nonnegtive s required by Jensen s inequlity. From Theorem.3, we obtin the following corollry. The proof is omitted since it follows immeditely from Theorem.3. Corollry.4. Under the ssumptions of Theorem.3, we hve 1 p i.17 L J 1 m ip i 1 M ip i 1 p i x i x i D x i x i U J, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

8 8 STEVEN G. FROM where m i = inf{f t : x i t x i } nd M i = sup{f t : x i t x i }, i = 1,,..., n 1. If f is 3-convex on [, b], tht is, the third derivtive f 3 x 0 on [, b], then m i = f x i nd M i = f x i. If f 3 x 0 on [, b] insted, then m i = f x i nd M i = f x i, i = 1,..., n. Theorem.5 below improves on the bounds on D, but ssumes more conditions on f nd its derivtives. Theorem.5. Let ft be rel-vlued function of t on [, b]. Suppose tht f nd its first three derivtives f, f nd f 3 re continuous on [, b]. Suppose tht x 1 < x < < x n b. Suppose p i 0, i = 1,,..., n with n p i = 1, p 1 > 0, p n > 0. Suppose tht f t 0 nd f 3 t 0, t b. Let D be the difference D = p i fx i f p i x i. Then.18 h 1 =1 W p 1 p D R =1 where x, x nd R re given in Theorem.3 bove, nd V p 1 p h, R V = x x f x fx + fx,.19 W = fx fx x x f x, = 1,,..., n 1. b Suppose f t 0 nd f 3 t 0 on [, b]. Then the reverse inequlity holds, tht is.0 h D h 1. Proof of. The proof proceeds in the sme wy s the proof of Theorem.3. Then, s before, using the definitions of A given in.10 bove,.1 [ A A +1 = R fx p fx + 1 p ] fx R R, = 1,,..., n 1. Insted of using Lemm.1, we use Lemm., prt with c = x, d = x. Reclling tht Lx = p R fx + 1 p R fx, we obtin tht A A +1 = 0, if x = x. If x < x, then we obtin x x x x W x x fx Lx x x x x V x x 1 n 1., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

9 GENERALIZATIONS OF JENSEN S INEQUALITY 9 Multiplying by R, using.11 nd summing over gives, s done in the proof Theorem.3, =1 x x x x R W x x D =1 From 1, we obtin, upon cncelltion of x x terms =1 This completes the proof of prt. W p 1 p D R =1 R V x x x x x x. V p 1 p. R The proof of b uses Lemm., prt b insted nd is omitted. Remrk.. If f t 0 on [, b] nd either f 3 t 0 on [, b] or f 3 t 0 on [, b], then we my pply Theorem.5 to ft insted to get bounds on D, hence on D. Remrk.3. The quntities h 1 nd h re nerly Riemnn sums for vrious integrls when pssing from the finite n cse to the continuous cse s n nd will be discussed lter. At lest one of these bounds for lrge n is usully very good. This will be seen in Theorem 4.3 lter. Remrk.4. The bounds given in Theorems.3 nd.5 use series of locl liner interpoltions moving forwrd from x = x 1 to x = x n from left to right. However, moment s reflection revels tht we could lso perform the sequence of locl liner interpoltions in reverse order. Let f x = f+b x, where = x 1, b = x n, x b. Then f +b x = fx, x b nd n p ifx i = n p n+1 if + b x n+1 i. Replcing p i by p n+1 i, x i by + b x n+1 i nd f by f in Theorems.3 nd.5 llows us to obtin reverse order versions of L J, J J, h 1 nd h. Cll these L J, U J, h 1 nd h, respectively. These new reverse order bounds sometimes do improve on L J, U J, h 1, h bounds. Next, we stte the fmous Hermite-Hdmrd inequlity. This will be needed in the proofs of severl new results to be discussed in this pper. In ddition, we shll present refinement of this inequlity lter for convex functions which re lso 3-convex on [, b]. Hermite-Hdmrd inequlity. Let f be convex function on [, b], where < b. Then + b. f 1 f + fb fxdx. b Next, we show tht the bounds of Theorem.5 re better thn Corollry.4 bounds, if we hve the ssumptions of Corollry.4 met. Theorem.6. Suppose tht the ssumptions of Theorem.5 hold. If f t 0 nd f 3 t 0 on [, b], then L J h 1 D h U J. b If f t 0 nd f 3 t 0 on [, b], then L J h D h 1 U J. Proof. We shll prove only. The proof of b is very similr nd is omitted. From Theorem.5, it remins only to prove L J h 1 nd h J. We shll prove h U J only; the proof tht, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

10 10 STEVEN G. FROM L J h 1 is very similr nd is omitted. Then h = V p 1 p R =1 = [x x f x fx + fx ] p 1 p.3. =1 By the Men Vlue Theorem, there is number λ 0, 1 such tht.4 f x + λ x x = fx fx x 1 x x = f tdt x x. But f 3 t 0, so f is convex on [, b]. By the Hermite-Hdmrd inequlity pplied to f, 1 x x + x.5 x x f tdt f = f x + 1 x x x. From.4 nd.5, we get f x + λ x x f x + 1 x x. Since f is incresing, λ 1 must hold. Thus, fx fx f x + 1 x x x x nd.3 gives h [f x f x + 1 ] x x x x p 1 p. R =1 Applying the Men Vlue Theorem gin to the expression in brckets, we cn find number λ in 0, 1/ such tht f x f x + 1 x x = f λ 1 x x. Thus, h =1 =1 1 f λ p 1 p x x R 1 M p 1 p x x = U J, R where M = sup{f t : x t x }. This completes the proof of. How do the new bounds given in this pper compre to previously published bounds? For lrge n, the new bounds re quite good, bsed upon mny numericl comprisons done nd not reported here Next, let s compre some of the new bounds to the previously proposed bounds discussed in previous works, both nlyticlly, when it is possible to do so, nd numericlly. In most cses, comprisons re somewht delicte since different bounds hve slightly different ssumptions, nd hve greter degrees of computtionl complexities so tht we re compring pples nd ornges. However, such comprisons give some ide bout utility nd reltive merits of bounds. From the results of numericl comprisons, it is the cse tht most of the bounds, both old nd new, re the best for some choices of p i nd x i nd n, but the new bounds compre very x R, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

11 GENERALIZATIONS OF JENSEN S INEQUALITY 11 fvorbly to previously published bounds. Moreover, some of the new bounds re still vlid for non-convex/non-concve choices of fx nd led to some nice pplictions. The following theorem sttes tht bounds L J nd U J re t lest s good s the bounds F L nd F U discussed [17] nd given in Theorem G erlier. It should be mentioned, however, tht the F L nd F U bounds do not require continuity of f, wheres the new bounds L J nd U J of Theorem.3 do require it. Hence, the bounds of [17] re slightly more generlly pplicble. Theorem.7. Suppose tht Theorem.3 ssumptions hold. Let L J nd U J be the new bounds of Corollry.4. Let F L nd F U be the bounds discussed in Theorem G. Let D = p i fx i f p i x i. Then F L L J D b D U J F L. Proof. We shll prove prt. The proof of prt b is similr nd is omitted. The inequlity L J D ws proven in Corollry.4. First, replcing fx by f A x = x in Theorem.3, we obtin p i x i p i x i = p 1 p x x. R Then, with m = inf{f t : t b}, m i = inf{f t : x i t x i }, we get F L = m p i x i p i x i = m p 1 p x x R =1 m p 1 p R This completes the proof of prt. I=1 =1 I=1 x x = L J. From numericl comprisons done, it ppers tht the following generl conclusions re vlid. 1 The new bounds bsed on h 1 nd h of Theorem.5 re often the best, but these re not s widely pplicble s the other bounds, both old nd new. The new lower bounds L J nd L J re not s good s L DS when the vrince of x i is lrge, unless n is lrge. Then they re significntly better. 3 Among the upper bounds, U J nd U J re usully better thn U D,1, U D,, U BP P 1, U BP P, U DG, if the vrince of the x i vlues is not too lrge. But for lrge n, U J nd U J re substntilly better regrdless of vrince. 4 For non-convex/non-concve choices for fx with f x continuous, only the new bounds given in Theorem.3 nd Corollry.4 nd the F L, F U bounds of [17] re pplicble. Of course, in this cse, Theorem.7 stted tht the new bounds re t lest s good. 5 Most of the bounds discussed in this pper re the best or nerly so, for some choices of p i nd x i. Given this fct, how menble re the bounds, both old nd new, for convenient pplictions? It will be seen tht the new bounds cn be used to refine vrious =1, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

12 1 STEVEN G. FROM inequlities, such s the Hermite-Hdmrd inequlity nd to obtin brnd new inequlities in probbility nd relibility theory. We briefly discuss some other pplictions first, which re more of clssicl nture. Remrk.5. If fx is log-convex on [, b], then we my pply Theorems.3 nd.5 to Lnfx insted of fx to obtin bounds on D = n p ilnfx i Lnf n p ix i. Then exponentition will provide bounds on the rtio n e D = fx i p i f n I=1 p ix i. If fx = Ln [ 1 x r ] x, r > 0, 0 < x < 1/, then we obtin new nd improved inequlities of Ky-Fn type. In [11], upper bounds re given for e D. See their Theorem 1, p. 5 nd Proposition 5.6, p. 61. Numericl comprisons of these bounds to the new bounds suggest tht the new bounds derived from Theorem.5 re lwys better nd tht the Corollry.4 bounds for e D re better if either the vrince of the x i vlues is not too gret, or if n is lrge. Theorems.3 nd.5 cn be used to obtin new refinements of the clssicl rithmetic men-geometric men AM-GM inequlity. These new refinements compre fvorbly to mny previously discussed bounds for the difference nd rtio of rithmetic men nd geometric mens. Theorems.3 nd.5 hve been pplied to obtin mny new bounds in informtion theory such s b-entropy bounds discussed in Theorem 4.3 of [11]. These new bounds, in numericl comprisons done hve been found to compre very fvorbly to other bounds in this cse s well. The new bounds cn lso be widely pplied to obtin bounds for quntities of interest in informtion theory. We consider here one such ppliction. Suppose X is discrete rndom vrible with support {x 1, x,..., x n } where 0 < x 1 < < x n. Let p i = ProbX = x i, i = 1,,..., n. Suppose p i > 0, i = 1,,..., n. Then the b-entropy of X is, using the nottion of Drgomir nd Goh 1996: 1 H b x = p i Log b, b > 1. p i I=1 In Drgomir nd Goh 1996, the following theorem is given. Theorem I. Theorem 4.3 of Drgomir nd Goh [ ].6 0 Log b n H b x 1 n p i 1 U H. Ln b Equlity holds if nd only if p i = 1, i = 1,,..., n. n If we tke fx = Log b x in Theorem.5, prt b, we immeditely obtin the following corollry, which gives both n upper nd lower bound for Log b n H b x. Corollry.8. Suppose tht p > 0, = 1,,..., n, n =1 p = 1. Let P 1 P P n denote the p vlues ordered from lrgest to smllest. Let x = 1 P, = 1,,..., n. Let fx = Log b x. Then.7 I L Log b n H b x I U, where I L is the vlue of h in Theorem.5 with P replcing p everywhere in Theorems.3 nd.5 nd I U is the vlue of h 1 with these sme replcements., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

13 GENERALIZATIONS OF JENSEN S INEQUALITY 13 Proof. The results is immedite, except we note here tht since it ws ssumed tht x 1 x x n, nd we re choosing x i = 1 p i, we must order the p vlues before we cn pply Theorem.5, prt b. This is no problem since entropies re invrint to permuttions. Of course, if p = 1 n for ll, then we obtin I L = J U = 0. Numericl comprisons of the upper bound U H given in Theorem 4.3 of [10] with the new bound I U of Corollry.4 found tht the new upper bound I U is lwys t lest s good s U H, but no proof could be found. However, Corollry.4 lso provides very good lower bound I L. Here we present smll numericl comprison. Suppose n = 3, P 1 =.5, P =.3, P 3 =. Then we tke b = e here, but the choice of the bse b > 1 is irrelevnt for purposes of comprison. We obtin Log e n H e X = , U H = , I L = , I U = , L 1 = , U 1 = The bound I U improves on U H especilly if the p i vlues re not extremely diverse. The lower bound I L is lso quite good. A theoreticl reson for this will be given lter. The U H bound is, however, esier to compute thn the new bounds. U 1 is better thn U H when the p i vlues re not too diverse. Otherwise, U H is better thn U 1. Here, both U H nd U 1 re the sme. 3. INTEGRAL REPRESENTATIONS AND BOUNDS Theorem 3.1 below extends the bounds to the continuous cse. We present bounds for the continuous Jensen gp D. In the continuous cse, we ssume tht Hx in Theorem 3.1 below is continuous on [, b], with H = 0, Hb = 1. Thus, Hx is continous distribution function on [, b]. In most, but not ll cses, there will exist probbility density function bsolutely continuous with respect to Lebesgue mesure, hx with H x = hx on [, b], but we do not hve to mke this ssumption in Theorem 3.1. For this reson, we shll use Riemnn-Stieltes integrls in Theorem 3.1 to obtin slightly more generl result. All integrls below re of Lebesgue-Stieltes type. Since we re intergrting with respect to bounded continuous monotonic function, they cn be considered Riemnn-Stieltes integrls s well. The function gx given in 3.1 in Theorem 3.1 is clled the men residul life function nd plys prominent role in pplied probbility, relibility theory nd sttistics. See [18], for exmple. First, we consider the bounded support cse. Next, we consider the continuous version of Jensen s gp utilizing Lebesgue/Riemnn Stieltes integrls. Theorem 3.1. Suppose tht f is continuous on [, b]. Let Hx be bounded, continuous nondecresing function on [, b] with H = 0 nd Hb = 1. Suppose 0 < Hx < 1 on, b. Let 3.1 gx = Let t xdht x 1 Hx lim x b gx = 0,, x < b, x = b. 3. q 1 x = inf{f t : x t x + gx}, nd 3.3 q x = sup{f t : x t x + gx}., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

14 14 STEVEN G. FROM Let 3.4 L 1 = 1 nd 3.5 U 1 = 1 Then 3.6 L 1 q 1 xgx dhx, q xgx dhx. fxdhx f xdhx U 1. The proof of Theorem 3.1 will utilize Lebesgue-Stieltes sum pproximtions. Since Ht is bounded nondecresing function, this is equivlent to Riemnn-Stieltes sum pproximtions. Given ny prtition P = {x 0, x 1,..., x n } of [, b] with = x 0 < x 1 < < x n = b, we shll let P denote the norm or mesh of the prtition. Then P = mx{x i+1 x i : 0 i n 1}. We shll pply Theorems.3 nd.5 to prtition P with suitbly smll enough P. We shll ssume x i+1 x i = P = x, i = 0, 1,..., n 1 throughout. Let p = Hx Hx 1, = 1,,..., n. Then p 0, =,..., n 1, p 1 > 0, p n > 0 nd n =1 p = 1. The Riemnn-Stieltes sums involved with utilize integrnd vlues t x = x = 1,,..., n. Let, x i nd x i be given by.7.8 of Theorem.3 for these choices of x nd p, = 1,,..., n. Then = 1 Hx i, i = 1,,..., n 1. To prove Theorem 3.1, we need the following lemms. Lemm 3.. There exists constnt K 1 > 0 such tht 3.7 g x i x i x i K 1 x, i = 1,,..., n. Proof. Let Then B i = 1 Hx i, C i = t x i dht, x i Ĉ i = x L+1 x i Hx L+1 Hx i. L=i gx i x i x i = C i Ĉi B i = Now t x L+1 x, L = i,..., n 1. Thus, n 1 xl+1 L=i x L n 1 xl+1 L=i x L t x L+1 dht. dht 3.8 gx i x i x i x, i = 1,,..., n 1. Also, clerly we hve 3.9 gx i + x i x i sup{gt : t b} + mx{x i x i, i = 1,,..., n 1} Multipliction of 3.8 nd 3.9 produces b + b = b. g x i x i x i b x, i = 1,,..., n. Letting K 1 = b, the proof of Lemm 3. is complete., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

15 GENERALIZATIONS OF JENSEN S INEQUALITY 15 Lemm 3.3. Under the conditions of Lemm 3., we hve the following. Let Then Z i = sup { f t f x i : x i t x i + gx i }, if x i x i + gx i, Z i = sup { f t f x i + gx i : x i + gx i t x i }, if x i + gx i x i, i = 1,,..., n 1. sup{f t : x i t x i } sup{f t : x i t x i + gx i } Z i, i = 1,,..., n 1. Proof. Consider the cse x i x i + gx i. Then for x i t x i + gx i, we hve f t = f x i + f t f x i sup{f t : x i t x i } + f t f x i sup{f t : x i t x i } + Z i, since x i x i x i + gx i. Since this holds for ll t with x i t x i + gx i, we obtin sup{f t : x i t x i + gx i } sup{f t : x i t x i } + Z i. Since [x i, x i ] [x i, x i + gx i ], subtrction proves the lemm. The proof of the other cse is similr nd is omitted. Proof of Theorem 3.1. First, ssume tht Hx stisfies Hx k+x, x b or some k+b integer k. We shll then prove the generl cse. We shll prove only the upper bound right hlf of 3.6 in Theorem. 3.1 The proof of the other hlf is very similr nd is omitted. Let s show tht there exists positive number M such tht for every ɛ > 0, we hve 3.10 fxdhx f xdhx U 1 + ɛm, where U 1 = 1 q xgx dhx. Since f is uniformly continuous on [, b], there is δ 1 > 0 such tht if P = {x 1, x,..., x n } is prtition of [, b] with xdhx n p ix i < δ1, then f xdhx f p i x i < ɛ. Since n p ix i is Riemnn-Stieltes sum for xdhx, there is δ > 0 such tht P < δ implies xdhx n p ix i < δ1 nd thus 3.11 f xdhx f p i x i < ɛ. Similrly, there is δ 3 > 0 such tht P < δ 3 implies 3.1 fxdhx p i fx i < ɛ., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

16 16 STEVEN G. FROM Tking δ 4 = minδ 1, δ 3, we see tht if P < δ 4, then fxdhx f xdhx 3.13 p i fx i f p i x i + ɛ. Now pply Theorem.3 to the right-hnd side of 3.13 utilizing Lemms 3. nd 3.3 to show tht 3.10 holds. To this end, define Let Y 1,i = sup{f t : x i t x i } Y,i = sup{f t : x i t x i + gx i }, i = 1,,..., n 1 = q x i. Q 1 = 1 Q = 1 Q 3 = 1 Y 1,i p i 1 p i Y,i p i 1 p i Y 1,i p i 1 p i Q 4 = 1 Y,i p i gx i. x i x i, gx i, gx i, Note tht Y 1,i = M i in Corollry.4 nd Q 1 is the upper bound on D = p i fx i f p i x i given there. Then 3.14 Now, clerly, we hve p i fx i f p i x i Q Q 1 U 1 Q 1 Q 3 + Q 3 Q + Q Q 4 + Q 4 U 1. Then 3.16 Q 1 Q 3 1 Y 1,i p i 1 p i x i x i gx i. Since f is bounded, Lemm 3. shows tht Q 1 Q 3 is O x, tht is, there is constnt K > 0 such tht Q 1 Q 3 K x. By Lemms 3. nd 3.3, from the fct tht Y 1,i Y,i Z i nd the uniform continuity of f on [, b], given ɛ > 0, we my tke x smll enough so tht 3.17 Q 3 Q ɛ sup{gt : t b} ɛ b., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

17 GENERALIZATIONS OF JENSEN S INEQUALITY 17 Since Q 4 is Riemnn sum pproximtion to U 1, given ɛ > 0, we my choose x smll enough so tht 3.18 Q 4 U 1 ɛ. It remins to show tht Q Q 4 cn be mde rbitrrily smll s P = x 0. By the boundedness of f on [, b], there exists constnt K such tht f x K, x b. Then Q Q 4 1 K p i gx i. Clerly, gx i b x i, i = 1,..., n. Also, = 1 Hx i, i = 1,,..., n 1. Let Sx = b x k+x, x < b. Since Hx, Sx k + b b x, x < b. Let 1 Hx k+b Sb lim x b Sx = 0. Then Sx is bounded on [, b]. Since 0 gx i Sx i, there is constnt K 3 such tht 0 gx i Now p i K 3. Thus, Q Q 4 1 K K 3 p i. p i p i sup Hx i Hx i 1 1 i n which cn be mde rbitrrily smll s n, by the uniform continuity of H on [, b], nd since n p i = 1. Thus, we my tke P = x smll enough so tht 3.19 Q Q 4 ɛ. Since ll four bsolute differences in 3.15 cn be mde rbitrrily smll s x = P 0, we cn mke Q 1 U 1 rbitrrily smll. So we cn choose x smll enough so tht, from , we obtin Q 1 U 1 K x + ɛ b + ɛ. Choosing x ɛ, then Q 1 U 1 K + 1 b + ɛ. In prticulr, 3.0 Q 1 U 1 + K + 1 b ɛ. From the definition of Q 1 nd Corollry.4, nd 3.0, we obtin for x smll enough, fxdhx f xdhx p i fx i f p i x i Q 1 U 1 + Mɛ, i 1 where M = K + 1 b +., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

18 18 STEVEN G. FROM Thus, 3.10 holds for ll ɛ > 0 nd Hx with Hx k+x k+b suppose the generl cse for Hx. Let for some integer k. Now H k x = Hx + x k 1 + b k, x b, k = 1,, 3,.... Then H k x k + x, x b, k = 1,, 3,.... k + b Thus, the conclusion of Theorem 3.1 is vlid for H k x in plce of Hx. Now lim k H k x = Hx, x b, H k x 1, x b, k 1. Let g k x, q,k x, L 1,k nd U 1,k be the vlues of gx, q x, L 1 nd U 1 given in Theorem 3.1 with H k x replcing Hx throughout, respectively. Then 3.1 where fxdh k x f x dh k x U 1,k, U 1,k = 1 q,k xg k x dh k x, x g k x = t xdh kt, x < b, g k b = 0, nd 1 H k x q,k x = sup{f t : x t x + g k x}, Clerly, q,k x sup{ f t : t b}. Also, g k x b, x b, k 1. Also, lim k q,k x = q x. In ddition, 3. lim k fxdh k x = = fxdhx + lim k b k + b fxdhx. fxdx Similrly, by the continuity of f, 3.3 lim k xdh k x = lim k f xdhx = f xdhx, 1 b b k + b nd lim g kx = k lim k t xdht + b t x b dt x x k+b 1 Hx = gx, x < b., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

19 GENERALIZATIONS OF JENSEN S INEQUALITY 19 This holds for x = b, lso. Also, by the Dominted Convergence Theorem, lim U 1 b 1,k = lim q,k xg k x dhx + q,k xg k x b k k k + b dx 3.4 = 1 q xgx dhx = U 1. nd the proof of Theorem 3.1 is complete. Remrk 3.1. In few results to come lter, we shll need to ssume the existence of density function hx = H x on [, b]. This will be needed in some pplictions to be discussed next. In this cse, the men residul life function is gx = t xhtdt x htdt, x < b x 3.5 gb = EXACT INTEGRAL REPRESENTATIONS AND APPLICATIONS Next, we present vrious pplictions of some theorems in Section 3. They re improvements on the clssicl Hermite-Hdmrd inequlity. Mny such ppers hve been published nd they re too numerous to cite. However, see [1], [3], [] nd [3]. Theorem 4.1 below is refinement of the Hermite-Hdmrd inequlity for convex functions which re lso 3-convex on [, b], nd is n ppliction of Theorem 3.1. Theorem 4.1. Let f be convex function on [, b]. Suppose tht f is lso 3-convex on [, b], tht is, the third derivtive f 3 x 0 on [, b]. Then N 1 1 b where 4. N 1 = 1 [ b f nd 4.3 N = 4.4 b nd 1 b tht is, N +f +b + b fxdx f N f + b b + b f + 1 fb f 4 ] f + b + b f + fb fxdx N + f, is t lest s good n upper-bound s f+fb is for 1 b fxdx., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

20 0 STEVEN G. FROM Proof of. We pply Theorem 3.1 to obtin N 1 from L 1 in Theorem 3.1. Then hx = 1 x b, gx = b x, nd q 1x = f x, since f 3 0. Then 1 b + b fxdx f b L 1 = 1 q 1 xhxgx dx = 1 f 1 b x x b dx. Integrting by prts twice, we obtin 4.5 L 1 = = b f [ b 1 b b f 4 b x b f xdx f + 1 ] fxdx. Applying the Hermite-Hdmrd inequlity to the integrl in 4.5, we obtin: 1 b b L 1 [ f f + 1 ] + b b 4 b f = 1 [ b f 1 4 f + 1 ] + b f = N 1. This proves the first non-trivil left hlf of 4.1. To prove the other hlf involving N, we proceed similrly with few modifictions. Using q x = f x + gx = f x+b, Theorem 3.1 gives U 1 = 1 = 1 q x hx gx dx x + b f 1 b b x dx. Mking the substitution u = x+b nd gin integrting by prts twice, we obtin 4.6 U 1 = = = +b 1 b f u b u du 1 u bf udu b +b + b b f + fudu +b b + b f 4 b + b f 4 Applying the Hermite-Hdmrd inequlity on [ +b, b] to the integrl in 3.6, we obtin: b + b + b U 1 f f b f + fb 4 b + b = f 1 + b 4 f + 1 fb = N. This complets the proof. b,, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

21 GENERALIZATIONS OF JENSEN S INEQUALITY 1 Proof of b. Now simple lgebr gives + b N + f if nd only if + b b f f which is equivlent to +b f b tdt But f is convex, so f t f +b, t +b. So +b b f tdt holds nd the proof of b is complete. f + fb + b f, + b f. f + b Remrk 4.1. Upper bound 4.6 is better upper bound thn N nd is exct for qudrtic choices of fx. Similrly, 4.5 is better lower bound thn N 1 nd is lso exct for qudrtic choices for fx. Using 4.5 nd 4.6, we could, if desired, get even better refinements by pplying the Theorem 4.3 to the integrls in 4.5 nd 4.6 nd iterting to the limit on successive intervls hlf s lrge t ech itertion. Of course, this would give much longer expressions for bounds nd involve evlution of f nd f t more nd more points in [, b]. Theorem 4.1 required convexity of f in the form of f x 0 on [, b]. But, s pointed out in Fink nd Páles 007, we my often replce this condition by less restrictive ssumptions on fx. In ny cse, the next theorem, Theorem 4., gives Hermite-Hdmrd type of bound requiring only 3-convexity or 3-concvity on [, b], nd not convexity of f itself. Theorem 4.. Suppose f 3 x is continuous on [, b]. If fx is 3-convex on [, b], tht is, f 3 x 0 on [, b], then fxdx + b b + b 4.7 f + f b f b 1 nd fxdx + b b + b 4.8 f + f f. b 1 b If fx is 3-concve on [, b], then inequlities 4.7 nd 4.8 hold with the inequlity signs reversed. Proof. We shll prove only 4.7 prt. The proof of 4.8 of prt follows upon considering the function f x = f + b x nd noting tht f x is 3-convex, if fx is 3-concve nd vice-vers. Also, fxdx = f xdx nd f +b = f +b. Prt b follows from prt using fx insted of fx in prt. To prove, pply Theorem 3.1 with Hx = x, b x b. Then gx = b x, x b. Since f is 3-convex, f is nondecresing in its rgument. Clerly gx = b x is nonincresing in x. Then Theorem 3.1 gives fxdx + b f + 1 f x + gx g 1 x b b dx., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

22 STEVEN G. FROM An ppliction of the Chebychev-Gruss inequlity gives: fxdx + b f + 1 x + b b x f 1 b b dx + b f + 1 x + b f 1 b dx b x 1 b dx + b b + b = f + f b f. 1 The proof of 4.7 in prt is complete. Next, let s consider bounds for b fxhxdx f b x hxdx = D. From Theorem.5, we obtined in.18 h 1 = W p 1 p p i fx i f p i x i R =1 V p 1 p = h R or its reversl.0. We shll now show tht the bound h is best possible bound s n when considering p i = Hx Hx i 1. Even if f nd f 3 re not of one sign, it will be proven tht h pproches D s n, so tht the integrl or continuous nlogue for h will be n exct representtion. Thus h behves like Riemnn sum for D. We hve the following theorem. =1 Theorem 4.3. Suppose f nd h re continuous on [, b]. Let x = fx + gx fx gxf x nd 4.10 x = gxf x + gx fx + gx + fx. Then 4.11 D fxhxdx f xhxdx = xhxdx H. b If f 3 x is continuous on [, b], f x 0 nd f 3 x 0 on [, b], then 4.1 H 1 1 xhxdx D = xhxdx H. c If f 3 x is continuous on [, b], f x 0 on [, b] nd f 3 x 0 on [, b], then 4.13 H = D H 1. d Under the ssumptions of prt b, H 1 is better lower bound for D thn is L 1 given in Theorem 3.1, tht is, L 1 H 1. e Under the ssumptions of prt c, H 1 is better upper bound for D thn is U 1 given in Theorem 3.1, tht is, U 1 H 1., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

23 GENERALIZATIONS OF JENSEN S INEQUALITY 3 Proof of. Recll tht gx = for differentiting n integrl, we obtin 4.14 g x = hxgx Hx where Hx = x h = = htdt. Thus xhxdx = R b x t xhtdt R b, upon differentition of gx using Leibnitz s rule x htdt gxf x + gxhxdx 1, gxf x + gx fx + gx + fxhxdx fx + gxhxdx + fxhxdx. Mking the substitution w = wx = x + gx nd using 4.14 to obtin w x, we obtin h = f wxw x Hxdx fx + gxhxdx + Integrting by prts, nd using Hb = 0, H = 1, we obtin h = Hfw + fwxhxdx fxhxdx, fxhxdx. fwxhxdx upon ppliction of 4.14 bove. Since w = + g = xhxdx, we obtin 4.16 h = f x hxdx + fxhxdx, since the expression in prentheses in 4.15 equls zero. Thus, h = D nd prt is proven. Prts b nd c explin why the bound h is so good for lrger n when bounding n p ifx i f n p ix i nd hs been verified in mny numericl comprisons done. Proof of b. Since the proof of the left hlf of 4.1 is very similr to the proof of Theorem 3.1 given erlier, we merely sketch its proof. Now choosing the sme prtition of [, b] s used for Theorem 3.1, we hve p h 1 = W p W, R where from.19. =1 W = fx fx x x f x fx + gx fx fx f x, since x x + gx ws shown in Lemm 3. erlier. Also, n 1 =1 =1 W p R cn be shown to be O x. Now n 1 =1 W p is Riemnn sum for D = xhxdx. Similrly, for sketch of proof for prt c. The proofs of prts d nd e re similr to the proof of Theorem 3.1, prt given erlier nd re omitted., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

24 4 STEVEN G. FROM Remrk 4.. The results given on bounds for the Jensen s gp in the continuous cse on n intervl [, b] re esily extended to certin types of improper integrls. For mny pplictions in pplied probbility, the choice = 0, b = is very importnt. For exmple, if = 0 nd b =, Theorems 3.1 nd 4.3 re vlid in this cse provided the improper integrls exist, nd f is continuous on [0,. For purposes of comprison, let s determine the vlues of L 1, U 1 in Theorem 3.1 nd H 1 nd H in Theorem 3.1 nd compre them to the lower bound given in Wlker [8] for the cse where fx = e x/ nd hx = e x, x > 0. As discussed in Remrk 4., Theorem 3.1 is vlid for improper integrls s well since convergence of ll integrls holds. These choices for fx nd hx were lso used in [8]. There, the uthor obtined D =.00 nd Wlker s lower bound on D ws given s The new lower bounds on D using = 0, b = in Theorems 3.1 nd 4.3, per Remrk 4., re L 1 = nd H 1 = Thus, H 1 improves on the lower bound of Wlker [8], but L 1 does not. However, Theorems 3.1 nd 4.3 provide upper bounds s well. They re U 1 =.06 nd H =.000. Per Theorem 4.3, prt b, H is exct in this cse. The computer lgebr pckge MAPLE esily computed ll necessry integrls. The bounds of Wlker [8] ssumes fx hs power series representtion of the form fs = n=0 ω ns n for 0 < S < R, where R > 0, nd ssumes fx is convex. The new bounds L 1 nd U 1 mke no convexity ssumption nd require only continuity of f x. The H 1 nd H bounds do, however, require convexity nd either 3-convexity or 3-concvity. The bounds of Wlker [8] re given in terms of n rbitrry probbility mesure on 0,, so it is vlid, in prticulr, in the discrete nd continuous cses; one does not need seprte formuls for these two cses, unlike the bounds of Theorem 4.3 given in this pper. However, the new bounds presented in this pper cn hndle cses tht previously proposed bounds cn not. Moreover, they led to extensions of vrious well-known results, such s Levinson s inequlities nd other types of convexity, such s convexity in the geometric nd hrmonic men. They lso hve pplictions to pplied probbility nd relibility theory some of which ws discussed in [1]. Next, we demonstrte how Theorem 4. cn be used, long with Theorem 4.3 to get more inequlities of Hermite-Hdmrd type for 3-convex/concve functions. Note tht convexity of f is not required, ust convexity of f. Theorem 4.4. Let f 3 be continuous on [, b]. If f 3 x 0 on [, b], then fxdx + b + b + 3b f f + f + b + 3b f b f b b 3 f 3 + 5b 4 3 f b 3 + 5b 3 + b f f If f 3 x 0 on [, b], then the reverse inequlity holds. Proof. We shll prove only the 3-convexity cse. Theorem 4.3, prt gives, with hx = 1 b nd gx = b x, x b 4.18 fxdx f b Now + b = [ b x b + x f fx = b + x f f +x x f tdt. b + x ] + fx 1 b dx., Vol. 13, No. 1, Art. 1, pp. 1-9, 016

25 GENERALIZATIONS OF JENSEN S INEQUALITY 5 Since f is convex, the Hermite-Hdmrd inequlity pplied to f gives b + x b + 3x b x 4.19 f fx f. 4 Then 4.18 nd 4.19 nd integrtion by prts gives 4.0 fxdx + b + b f f + f b+x dx b b Also, 4.1 f b+3x 4 b x dx b = 3 + b 3 f Applying Theorem 4., prt to the functions f 1 x = f b+x 4.0, we obtin, since f 1 nd f re 3-convex, 4. f b+x dx + 3b b f + b 4 4 nd 4.3 f b+3x 4 b x dx b x b + 3x f dx. 4 nd f x = f b+3x 4 + 3b f b f 4 b + 3x 3 + 5b b 3 + 5b 3 + b f dx f + f f From , we obtin upon ddition nd subtrction, the desired result. Remrk 4.3. Numericl investigtions suggest 4.17 improves on 4.7 in Theorem 4., but no proof hs been found. Also, the bounds given in Theorem 4.4 re quite good. If fx is 3-convex, then we my pply the bove theorem to f x = f + b x insted in the obvious R b wy to obtin lower bound for fxdx f +b b, or n upper bound for this difference, if f is 3-concve. Mny more such inequlities of Hermite-Hdmrd type cn be obtined using the methods given in this pper nd will be discussed in forthcoming pper. Next, let s consider some pplictions to relibility theory nd pplied probiblity. First, let s obtin more convenient representtions of the Jensen gp D = b fxhxdx f b xhxdx in the form of possibly infinite series. Assume for the moment tht fx is rel nlytic on some open set contining [, b. In relibility theory, 0 < b, so we ssume this here lso. Then by Theorem 4.3, prt, with the extension to possibly improper integrl with gx < on [, b, we obtin 4.4 D = [gxf x + gx fx + gx + fx] hxdx. Since f hs Tylor series expnsion t x, we my write f x 4.5 fx + gx = fx + gx,! ssuming tht the intervl of convergence centered t x hs rdius of convergence greter thn gx. This would be true, for exmple, if f is polynomil or exponentil function. Similrly, 4.6 f f x x + gx = 1! gx 1. =1 =1. in, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

26 6 STEVEN G. FROM From bove, we obtin D = f x gx hxdx.! = Eqution 4.7 gives nice nd convenient representtion of the Jensen gp D in terms of the men residul life function gx for purposes of obtining numerous new inequlities in relibility theory. It cn lso be used to obtin mny more new inequlities of Hermite-Hdmrd type. Note tht 4.7 is finite series for polynomil choices of fx nd is n esily mngeble infinite series for exponentil functions, two very importnt specil choices in pplied probbility nd relibility theory, since moments bout the origin nd moment probbility generting functions re importnt in these res. In [18], the concept of men residul life MRL function nd decresing men residul life DMRL is discussed. The theoreticl spects, importnce, nd wide rnge of pplictions of these re surveyed nd discussed. In [4], the incresing filure rte IFR, incresing filure rte verge IFRA nd new better thn used in expecttion NBUE nonprmetric clsses of life distributions re discussed. See the bove two references for definitions of these subclsses, the IFs the smllest, with IFR IFRA NBUE. Also, IFR DMRL NBUE. It is lso well known tht if X is lifetime rndom vrible nd is member of ny of the bove four subclsses, then the n th moments bout the origin of X, µ n = EX n = x n hxdx, 0 n = 0, 1,,..., stisfy the inequlity 4.8 µ n n! µ n, where µ = µ 1 = EX. This bound is shrp, since the exponentil distribution with hx = λe λx, x > 0, λ > 0, is member of ll four subclsses. See [4], p. 116, for exmple. First, let s give nother proof of 4.8 for the NBUE clss. In the sequel, let fx = x n, n = 1,, 3,.... We ssume without loss of generlity, tht = 0, b = below, since we my tke gx = 0 for x > b, if b is finite. Theorem 4.5. Suppose rndom vrible X is NBUE, tht is, gx µ on [0,. Then 4.8 holds, tht is, µ n n!µ n. Proof. The result is trivilly true for n = 1. We shll use mthemticl induction. Supposing tht µ n n!µ n holds for some positive integer n, 4.7 gives n+1 1 µ n+1 = EX n+1 = f xgx hxdx + µ n+1! n+1 = 0 1! = 0 n + 1! n + 1! xn+1 µ hxdx + µ n+1. But by the induction hypothesis, µ n+1 n + 1!µ n+1, so n+1 1 µ n+1 µ n + 1!! n + 1! n + 1!µn+1 + µ n+1 = n+1 = n + 1!µ n+1 = n + 1!µ n+1. This completes the proof. = 1! = n + 1!µ n µ n+1 n + 1!, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

27 GENERALIZATIONS OF JENSEN S INEQUALITY 7 Inequlity 4.8 gives bounds on µ n in terms of only µ. If the vrince of X or, equivlently, µ is vilble s well, then it is cler tht we cn improve on these bounds, using representtion 4.7 in recursive fshion. In future pper, we shll discuss the detils. Also, lower bounds for D cn be obtined s well. The bove theorem cn be generlized to more generl clss thn the NBUE. If there exists constnt M with gx M on [, b, where M µ, then it is esily shown tht µ n n!m n+1. Remrk 4.4. Either representtion 4.7 or Theorem 3.1 given erlier quickly yields the result 4.9 σ = VrX = gx dhx = gx hxdx, if hx exists, where σ denotes the vrince of X. If this representtion of the vrince s the men of the squred MRL function is lredy known, it is unknown by this uthor. In ny cse, representtion 4.7 is of much more generl use. Using 4.7, it cn be shown, for exmple, tht if X is NBUE, then EX 3 6µσ, which is n improvement on 4.8 for n = 3, since σ µ holds in the NBUE cse. Bounds for EX n for n 4 cn lso be obtined in recursive mnner using 4.7 in conunction with the Chebychev-Gruss inequlity which lso improve on 4.8. We omit the detils here. More inequlities of type similr to those in [6], [7] nd [8] cn be obtined, if the eqution for hx is completely known, to obtin bounds on moment-generting functions nd moments, in prticulr. Finlly, we discuss the extensions of results given in this pper to other kinds of convexity. In Niculescu [1], the following definition is given. Definition. Let I, J be subintervls of 0,. Suppose tht t 1, t I nd p 0, 1. Let f : I J. Then f is multiplictively convex on I, if 4.30 ft p 1t 1 p ft 1 p ft 1 p. This is type of convexity ccording to the geometric men, insted of the rithmetic men. From 4.9, it follows tht if t i I, p i 0, 1 with n p i = 1, then, if f is multiplictively convex on I, then n n 4.31 f ft i p i. t p i i As discussed in [1], if f : I 0, is multiplictively convex function nd if we define F by F = log f exp : logi R, where denotes functionl composition, then F is convex function. We cn rewrite 4.31 in terms of F s: F p i Logt i p i F Logt i., Let D GM denote the Jensen gp D GM = = n n ft i p i f t p i i p i F x i F p i x i, Vol. 13, No. 1, Art. 1, pp. 1-9, 016

28 8 STEVEN G. FROM where x i = Logt i, i = 1,,..., n. Assuming x 1 x x n, we cn pply Theorems.3 nd.5 nd corollries to obtin bounds on D GM. We my lso pply Theorems 3.1 nd 4.3 to the continuous nlogue of D GM s well, except we would substitute F x for fx throughout. REFERENCES [1] A. ABRAMOVICH, J. BARIĆ, nd J. PEČARIĆ, Feer nd Hermite-Hdmrd type inequlities for superqudrtic functions, J. Mth. Anl. Appl., , pp [] M. K. BAKULA, J. PEČARIĆ, nd J. PERIĆ, On the converse Jensen inequlity, Appl. Mth. Comp., 18 01, pp [3] A. BARANI, S. BARANI nd S. S. DRAGOMIR, Refinements of Hermite-Hdmrd inequlities for functions when power of the bsolute vlue of the second derivtive is P -convex, J. Appl. Mth., 01, Art. ID , p. 10. [4] R. E. BARLOW nd F. PROSCHAN, Sttisticl Theory of Relibility nd Life Testing, [5] R. L. BURDEN nd J. D. FAIRES, Numericl Anlysis, Sixth Ed., Brooks/Cole, [6] P. CERONE, Specil functions: Approximtions nd Bounds, Appl. Anl. nd Discr. Mth., 1 007, pp [7] P. CERONE, On n identity for the Chebychev functionl nd some rmifictions, J. Ineq. Pure nd Appl. Mth., 3 00, pp [8] P. CERONE nd S. S. DRAGOMIR, On some inequlities rising from Montegormer s identity, J. Cmput. Anl. Appl., 5 003, No. 4, pp [9] D. CASTORELLI nd R. SPIGLER, How shrp is the Jensen inequlity? J. Inequl. Appl., , p. 10. [10] S. S. DRAGOMIR nd C. J. GOH, A counterprt of Jensen s discrete inequlity for differentible convex mppings nd pplictions in informtion theory, Mthl. Cmput. Modeling, , No., pp [11] S. S. DRAGOMIR nd B. MOND, A refinement of Jensen s inequlity for log-convex mppings nd pplictions, Bull. Allhbd Mth. Soc., 10/ /1996, pp [1] S. S. DRAGOMIR, A converse result for Jensen s inequlity vi Gruss inequlity nd pplictions in informtion theory, Anlele Univ. Orde. Fsc. Mth., , p [13] S. S. DRAGOMIR nd C. E. M. PEARCE, Selected Topics on Hermite-Hdmrd Inequlities, RGMIA Monogrphs, Victori University, 000. [14] S. S. DRAGOMIR, On converse of Jensen s inequlity, Univ. Beogrd. Publ. Elektrotehn. Fk., 1 001, pp [15] S. S. DRAGOMIR nd F. P. SCARMOZZINO, A refinement of Jensen s discrete inequlity for differentible convex functions, Est Asin Mth. J., 1 004, pp [16] A. M. FINK nd Z. PÁLES, Wht is Hdmrd s inequlity? Appl. Anl. nd Disc. Mth., 1 007, pp [17] A. E. FRISSI nd Z. LAREUCH, Jensen type inequlities for twice differentible functions, J. Nonliner Sci. Appl., 5 01, pp [18] F. GUESS nd F. PROSCHAN Men Residul Life: Theory nd Applictions, Hndbook of Sttistics, 05/1985. [19] P. M. HUMMEL, The ccurcy of liner interpoltion, Amer. Mth. Monthly, , pp , Vol. 13, No. 1, Art. 1, pp. 1-9, 016

29 GENERALIZATIONS OF JENSEN S INEQUALITY 9 [0] D. S. MITRINOVIC, J. PEČARIĆ, A. M. FINK, Clssicl nd New Inequlities in Anlysis Mthemtics in its Applictions, Springer, 199. [1] C. P. NICULESCU, Convexity ccording to the geometric men, Mth. Inequl. nd Appl., 3 000, No., pp [] C. P. NICULESCU, Old nd new on the Hermite-Hdmrd inequlity, Rel Anl. Exchnge, 9 003/004, No., pp [3] C. P. NICULESCU, The Hermite-Hdmrd inequlity for log-convex functions, Nonliner Anl., 75 01, pp [4] J. PARK, Hermite-Hdmrd-like type inequlities for twice differentible convex functions, Int. J. Mth. Anl., 8 014, No. 59, pp [5] J. E. PEČARIĆ, F. PROSCHAN, Y. L. TONG, Convex Functions, Prtil Orderings nd Sttisticl Applictions, Vol. 187, Mthemtics in Science nd Engineering, Acdemic Press, 199. [6] F. QI, Z. L. WEI nd Q. YANG, Generliztions nd refinements of Hermite-Hdmrd inequlity, Rocky Mountin J. Mth., , No. 1, pp [7] S. SIMIC, On n upper bound for Jensen s inequlity, JIPAM, J. Inequl. Pure nd Appl. Mth., , No., Article 60, p. 5. [8] S. G. WALKER, On lower bound for the Jensen inequlity, SIAM J. Mth. Anl., , No. 5, pp [9] S. ZLOBEC, Jensen s inequlity for nonconvex functions, Mth. Commun., 9 004, No., pp , Vol. 13, No. 1, Art. 1, pp. 1-9, 016