RECURSIVELY DEFINED REFINEMENTS OF THE INTEGRAL FORM OF JENSEN S INEQUALITY

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1 M athematical I equalities & A pplicatios Volume 9, Number 4 (206, doi:0.753/mia-9-90 RECURSIVELY DEFINED REFINEMENTS OF THE INTEGRAL FORM OF JENSEN S INEQUALITY LÁSZLÓ HORVÁTH AND JOSIP PEČARIĆ (Commuicated by C. P. Niculescu Abstract. I this paper we establish ifiite chais of itegral iequalities related to the classical Jese s iequality by usig special refiemets of the discrete Jese s iequality. As applicatios, we itroduce ad study ew itegral meas (geeralized quasi-arithmetic meas, ad give refiemets of the left had side of Hermite-Hadamard iequality.. Itroductio The itegral form ad the discrete versio of Jese s iequality provide the startig poit for much of the discussio i this paper. They ca be stated as follows: THEOREM A. (classical Jese s iequality, see [7] Let g be a itegrable fuctio o a probability space (,A, μ takig values i a iterval I R. The gdμ lies i I. If f is a covex fuctio o I such that f g is itegrable, the f gdμ f gdμ. THEOREM B. (discrete Jese s iequality, see [7] Let C be a covex subset of a real vector space V, ad let f : C R be a covex fuctio. If p,...,p are oegative umbers with p i =, ad v,...,v C, the i= f ( i=p i v i i= p i f (v i. Jese obtaied his famous iequality i [6]. There is a extesive theory for the study of refiemets of the discrete Jese s iequality, see [5], but there are oly Mathematics subject classificatio (200: 26A5 (26D5. Keywords ad phrases: Classical ad discrete Jese s iequality, covex fuctio, geeralized quasiarithmetic meas, Hermite-Hadamard iequality. The research of the 2d author has bee fully supported by Croatia Sciece Foudatio uder the project c D l,zagreb Paper MIA

2 228 L. HORVÁTH AND J. PEČARIĆ few papers dealig with refiemets of the classical Jese s iequality, see Rooi [8], Horváth [9] adhorváth ad Pečarić [4]. I this paper we establish ifiite chais of itegral iequalities related to the classical Jese s iequality. The key of our treatmet is special refiemets of the discrete Jese s iequality which have bee developed i Horváth [3]. As a immediate applicatio, ew ifiite refiemets of the classical Jese s iequality are derived. We essetially follow the approach of Bretić, Pearce ad Pečarić[3], but our treatmet is applicable i a more geeral eviromet. I Sectio 3 we cosider our results i some iterestig special cases. I Sectio 4 some ew itegral meas (geeralized quasi-arithmetic meas are itroduced, ad their properties are studied. Sectio 5 is devoted to refiemets of the left had side of Hermite- Hadamard iequality. 2. Prelimiaries ad the mai iequalities N ad N + deote the set of oegative ad positive itegers, respectively. Before proceedig to the results we preset some hypotheses, ad a iequality from [3] which will be eeded. (H Let N + be fixed, ad deote { } S k := (i,...,i N + i j = + k (H 2 Let (a j (m m N+, j be strictly icreasig sequeces such that, k N +. ( α := a (=...= a ( > 0. (2 (H 3 Let p,...,p be oegative umbers with p j =. Uder the hypotheses (H ad(h 2, defie the fiite sequeces (u k (i,...,i (i,...,i S k, k N + recursively by u (,..., := α, (3 ad for every (i,...,i S k+ (see ( u k+ (i,...,i := {l {,...,} i l } + a l(i l a l (i l a l (i l + j l a j(i j a j(i j + a j(i j a l (i l a l (i l a l (i l u k (i,...,i l,i l,i l+,...,i. Now we state oe of the mai results i [3].

3 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 229 THEOREM. Assume (H H 3. Let C be a covex subset of a real vector space, ad {x,...,x } be a fiite subset of C. If f : C R is a covex fuctio, the f ( p j x j for each k N + = T... T k T k+... T k = T k, (x,...,x ; p,...,p ;a,...,a := u k (i,...,i ( a j (i j p j f (i,...,i S k p j f (x j, a j (i j p j x j. a j (i j p j We follow this sectio by itroducig some otatios. Let ( i,b i, μ i (i T l := {,...,l} be probability spaces for some l N +, l 2. The σ -algebra i T l :=... l geerated by the projectio mappigs pr i :... l i, pr i (x,...,x l =x i (i =,...,l is deoted by B T l. μ T l meas the product measure o B T l : this is the oly measure o B T l (the measures are σ -fiite which satisfies μ T l (B... B l =μ (B...μ l (B l, B i B i, (i =,...,l. The l -fold product of the probability space (,B, μ is deoted by ( l,b l, μ l. The followig abbreviatios will be used: dμ T l (x := dμ T l (x,...,x l ad dμ l (x := dμ l (x,...,x l. λ l is always meas the Lebesgue measure o the Borel sets of R l. Our first purpose is to obtai a exteded ad refied versio of the classical Jese s iequality. THEOREM 2. Assume (H H 3. Suppose the followig hypotheses are also hold (H 4 Let( i,b i, μ i (i =,..., be probability spaces. (H 5 Foreachi=,...,, let g i be a μ i -itegrable fuctio o i takig values i a iterval I R. (H 6 Let f be a covex fuctio o I such that f g i is μ i -itegrable o i (i =,...,. The (a f p i g i dμ i T... T k T k+... i= i i= p i i f g i dμ i, (4

4 230 L. HORVÁTH AND J. PEČARIĆ T k = T k, ( f ;g i ; μ i ; p i ;a i := u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j g j (x j f dμt (x, k N +. (5 T a j (i j p j (b For each k N + ad all t [0,] f p i g i dμ i H k (0 H k (t T k, (6 i= i H k (t =H k, (t; f ;g i ; μ i ; p i ;a i := u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j g j (x j f t +( t T a j (i j p j a j (i j p j g j dμ j j dμ T (x. (7 a j (i j p j REMARK. By (H 3 H 6, the hypotheses of Lemma 2. i [8] are all satisfied, ad so it yields that all the itegrals i (5 ad(7 existadfiite. ad Assume (H H 6. (a If =, the S k = {k} ad u k (k= a (k for all k N +, ad therefore H k (t=h (t= f T k = T = f g dμ, k N +, tg (x +( t g dμ dμ (x, t [0,], k N +.

5 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 23 We ca see that for = (4 is trivial (the classical Jese s iequality, while (6 gives f g dμ = H (0 H (t H (= f g dμ, t [0,]. (b Suppose 2. The S = {(,...,}, ad hece T = T f ( p j g j (x j dμ T (x, ad H (t= T f t p j g j (x j +( t p j g j dμ j dμ T (x, t [0,]. j Now we summarize the essetial properties of the fuctio H k defiedi(7. THEOREM 3. Assume (H H 6. The for each k N + (a H k is covex ad icreasig. (b H k (0 f p i g i dμ i, H k (=T k. i= i (c H k is cotiuous o [0,[. (d If f is cotiuous, the H k is cotiuous o [0,]. It is easy to costruct examples which show that H k is ot cotiuous at i geeral. The followig refiemets of the classical Jese s iequality are immediate cosequeces of Theorem 2. THEOREM 4. Assume (H H 3. The hypotheses (H 4 H 6 are replaced by (Ĥ 4 Let(,B, μ be a probability space. (Ĥ 5 Letgbea μ -itegrable fuctio o takig values i a iterval I R. (Ĥ 6 Let f be a covex fuctio o I such that f gisμ-itegrable o. The (a f gdμ T... T k T k+... f gdμ,

6 232 L. HORVÁTH AND J. PEČARIĆ T k = T k, ( f ;g; μ; p i ;a i = u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j g(x j f dμ (x, k N +. a j (i j p j (b For each k N + ad all t [0,] f gdμ = H k (0 H k (t T k, H k (t =H k, (t; f ;g; μ; p i ;a i = u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j g(x j f t +( t gdμ dμ (x. a j (i j p j 3. Results whe recursio is explicitly represeted We first recall the followig example from [3]. EAMPLE. Assume (H ad(h 3. Let α > 0, a 0adb j R ( j such that the umbers a + b j are all positive. Defie the sequeces (a j (m m N+ by a j (m := α m i= The these sequeces are strictly icreasig ad thus they satisfy (H 2. ( +, m N +, j. (8 ai + b j α = a (=...= a ( > 0,

7 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 233 I this case it ca be proved that for every k N + u k (i,...,i = α k + a( + j + b l l= ( i j (am + b j m= (k! (i!...(i!, (i,...,i S k. (9 As illustratios, we just cosider Theorem 4 i two special cases of the previous example. The first part of the ext result ca be cosidered as the itegral versio of Theorem (ai[2]. COROLLARY. Assume (H, (H 3, ad (Ĥ 4 Ĥ 6. By choosig α = a = ad b j = 0 ( j i (8, we have (a f gdμ T... T k T k+... f gdμ, T k = T k, ( f ;g; μ; p i = ( +k k (i,...,i S k ( i j p j g(x j i j p j f dμ (x, k N +. i j p j (b For each k N + ad all t [0,] f gdμ = H k (0 H k (t T k, H k (t =H k, (t; f ;g; μ; p i = ( +k k (i,...,i S k ( i j p j f t (c For each k N + ( T k, f ;g; μ; T k, ( f ;g; μ; p i, i j p j g(x j +( t gdμ dμ (x. i j p j

8 234 L. HORVÁTH AND J. PEČARIĆ T k, ( f ;g; μ; = ( +k 2 k f (i,...,i S k ( + k i j g(x j dμ (x (d If p i > 0 ( i, the for each k N + ad for each l N + f gdμ T k A l f gdμ, A l = A l, ( f ;g; μ; p i := ( +l l i +...+i =l i j N; j ( l i j p j f l i j p j g(x j l dμ (x. i j p j (e Suppose p i > 0 ( i. The lim T k = lim A l =! h(t,...,t,x,...,x dλ (t dμ (x, (0 k l E { E := (t,...,t R the fuctio h defied o E by h(t,...,t,x,...,x := with the otatio t := t j. ( t j p j } t j, t j 0, j =,...,, f t j p j ( t j p j g(x j Proof. (a ad (b come from Theorem 4. (c Theorem (b i [2] ca be applied. (d Accordig to Propositio 2 i [4], the sequece (A l l N+ is decreasig ad f gdμ A l f gdμ, l N +.

9 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 235 Defie for all (x,...,x the expressios G k, (x,...,x := i j p j f ad B l, (x,...,x := ( +k k ( +l l (i,...,i S k ( i +...+i =l i j N; j ( i j p j f i j p j i j p j i j p j g(x j, k N +, i j p j g(x j, l N +. Theorem (a i [2] shows that the sequece ( Gk, (x,...,x k N + ( is icreasig for all (x,...,x.byexample3i[0], the sequece ( Bl, (x,...,x l N + (2 is decreasig for all (x,...,x. It follows from Theorem 3 i [2]that lim G k, (x,...,x =lim B l, (x,...,x k l = h(t,...,t,x,...,x dλ (t, (x,...,x. (3 E Puttig all this together gives that for each k N + ad for each l N + G k, (x,...,x B l, (x,...,x, (x,...,x. By itegratig both sides over,wehavet k A l (k N +, l N +. (e Assumig that the fuctio h is λ μ -itegrable over E for the preset, the mootoicity properties of the sequeces ( ad(2, the limit formula (3, ad the Fubii s theorem imply (0. The measurability of h is obvious. To justify the supposed itegrability coditio, choose a fixed iterior poit a of I.Sice f is covex f (t f (a+ f + (a(t a, t I, f + (a meas the right-had derivative of f at a. By usig this ad the discrete Jese s iequality, we have ( t j p j f (a+ f + (a h(t,...,t,x,...,x ( t j p j g(x j a ( t j p j t j p j f (g(x j (4

10 236 L. HORVÁTH AND J. PEČARIĆ for all (t,...,t,x,...,x E. It is eough to prove that the fuctios o the left had side ad the right had side of the previous iequalities are λ μ - itegrable over E. We cosider oly the fuctio o the right had side of (4, the other case ca be hadled similarly. I provig this, we may suppose that f is oegative o I, ad therefore by the Fubii s theorem, ad the by Lemma 2. (a i [8] t j p j f (g(x j dλ μ (t,...,t,x,...,x E ( = E = f gdμ t j p j f (g(x j dμ (x dλ (t E ( t j p j dλ (t <. The proof is complete. REMARK 2. We stress that the sequeces (T k k N+ ad (A l l N+, compared i part (d of the previous result, are geerated from such refiemets of the discrete Jese s iequality which have bee obtaied by essetially differet methods. Now, the itegral variat of Theorem (a is obtaied. COROLLARY 2. Assume (H, (H 3, ad (Ĥ 4 Ĥ 6. By choosig α =, a= 0 ad b j = λ j ( j,λ j > ( j i (8, we have with the otatio d (λ := (a λ j for every k N + f gdμ T... T k T k+... f gdμ, T k = T k, ( f ;g; μ; p i = (k! (d (λ + k (i (i,...,i S!...(i! k λ i j j p j g(x j ( (λ j i j λ i j j p j f λ i j j p j dμ (x.

11 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 237 (b For each k N + ad all t [0,] f gdμ = H k (0 H k (t T k, H k (t =H k, (t; f ;g; μ; p i (c = (k! (d (λ + k (i (i,...,i S!...(i! k λ i j j p j g(x j ( λ i j j p j f λ i j lim T k = k (λ j i j +( t gdμ dμ (x. j p j f gdμ. Proof. We have oly to apply Theorem 4 to get (a ad (b. (c Let for all (x,...,x D k, (λ ;x,...,x = (k! (d (λ + k (i (i,...,i S!...(i! k λ i j j p j f ( (λ j i j By Theorem (a i [], the sequece ( Dk, (λ ;x,...,x k N + is icreasig, ad by (b of the same theorem lim D k, (λ ;x,...,x = k j p j g(x j. λ i j λ i j j p j p j f (g(x j, (x,...,x. It follows from these facts that lim T k = lim D k, (λ ;x,...,x dμ (x k k = p j f (g(x j μ (x= f gdμ. The proof is ow complete.

12 238 L. HORVÁTH AND J. PEČARIĆ 4. Meas geerated by the expressios i the ew refiemets I this sectio we itroduce some ew itegral meas (geeralized quasi-arithmetic meas ad study their properties. DEFINITION. Assume (H H 3 ad (H 4 Let( i,b i, μ i (i =,..., be probability spaces. Assume further (H 7 For each i =,...,,letg i be a measurable fuctio o i takig values i a iterval I R. (H 8 Letϕ, ψ : I R be cotiuous ad strictly mootoe fuctios. (a For each k N +,wedefie itegral meas with respect to (5by M ψ,ϕ (k = M ψ,ϕ (g i ; μ i ; p i ;a i ;k ( := ψ u k (i,...,i (i,...,i S k ( a j (i j p j (ψ ϕ T a j (i j p j ϕ (g j (x j dμt (x, (5 a j (i j p j if the itegrals exist ad fiite. (b For each k N + ad for all t [0,], itegral meas ca be defied with respect to (7 by M ψ,ϕ (t;k = M ψ,ϕ (t;g i ; μ i ; p i ;a i ;k := ψ t (i,...,i S k u k (i,...,i ( a j (i j p j ϕ (g j (x j if the itegrals exist ad fiite. a j (i j p j +( t a j (i j p j (ψ ϕ T a j (i j p j ϕ g j dμ j j dμ T (x, (6 a j (i j p j Ithasbeeshowi[3] that for ay j =,..., (i,...,i S k u k (i,...,i a j (i j =, (7

13 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 239 ad therefore by (H 3 This implies that u k (i,...,i ( a j (i j p j =, k N +. (8 (i,...,i S k M ψ,ϕ (k I ad M ψ,ϕ (t;k I, k N +, t [0,], that is they really defie meas. By Remark, ifϕ g i ad ψ g i are μ i -itegrable o i (i =,...,, ad ψ ϕ is either covex or cocave, the the itegrals i (5ad(6existadfiite. The followig itegral mea is also eeded: if (H 3 H 4 ad(h 7 are satisfied, ad χ : I R is a cotiuous ad strictly mootoe fuctio, the defie M χ = M χ (g i ; μ i ; p i := χ p i χ g i dμ i, (9 i= i if the itegrals exist ad fiite. Let q, g : [a,b] R be positive ad Lebesgue-itegrable fuctios, ad let χ : ]0, [ R be a cotiuous ad strictly mootoe fuctio. The so called geeralized weighted quasi-arithmetic mea of g with respect to the weight fuctio q b q(x χ (g(xdx M χ = M χ (g;q := χ a b (20 q(xdx is a special case of (9, ad it cotais differet remarkable meas (for example, weighted arithmetic, harmoic ad geometric meas. The properties of meas (20are studied itesively, we just metio two papers dealig with itegral meas: Haluška ad Hutik [6] ad Su, Log ad Chu [9]. We cotiue this sectio with a discussio o the mootoicity of the itroduced meas. THEOREM 5. Assume (H H 4, (H 7 H 8, ad assume that ϕ g i ad ψ g i are μ i -itegrable o i (i =,...,.The (a M ϕ M ψ,ϕ (... M ψ,ϕ (k... M ψ, k N +, (2 ad M ϕ M ψ,ϕ (0;k M ψ,ϕ (t;k M ψ,ϕ (k, k N +, t [0,], if either ψ ϕ is covex ad ψ is icreasig or ψ ϕ is cocave ad ψ is decreasig. a

14 240 L. HORVÁTH AND J. PEČARIĆ (b M ϕ M ψ,ϕ (... M ψ,ϕ (k... M ψ, k N +, (22 ad M ϕ M ψ,ϕ (0;k M ψ,ϕ (t;k M ψ,ϕ (k, k N +, t [0,], if either ψ ϕ is covex ad ψ is decreasig or ψ ϕ is cocave ad ψ is icreasig. Proof. (a ad (b ca be obtaied by applicatios of Theorem 2 to the fuctios ψ ϕ ad ϕ g i (i =,..., (ϕ (I is a iterval, if ψ ϕ is covex, ad to the fuctios ψ ϕ ad ϕ g i (i =,...,,ifψ ϕ is cocave, ad the upo takig ψ. Recetly, i [7] by Khuram Ali Kha ad Pečarić the iequalities (2 ad(22 have bee proved for the mea M ψ,ϕ (. It ca be see that our approach allows us to essetially geeralize ad exted some of the results from [7]. 5. Coectios to Hermite-Hadamard iequality Differet refiemets of the left had side of Hermite-Hadamard iequality ca be got from Theorem 4. THEOREM 6. Assume (H H 3, ad let f be a covex fuctio o [a,b]. The (a ( a + b f Tˆ... Tˆ k Tˆ k+... b f, 2 b a Tˆ k = Tˆ k, ( f ; p i ;a i = (b a u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j x j [a,b] f a j (i j p j dλ (x, k N +. (b For each k N + ad all t [0,] ( a + b f = Ĥ k (0 Ĥ k (t Ĥ k (= Tˆ k, 2 a

15 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 24 Ĥ k (t =Ĥ k, (t; f ; p i ;a i = (b a u k (i,...,i ( a j (i j p j (i,...,i S k [a,b] f t a j (i j p j x j a j (i j p j +( t a + b 2 dλ (x. (c Ĥ k is covex ad icreasig for each k N +. If f is cotiuous, the Ĥ k is also cotiuous. ( Proof. We ca apply Theorem 4 ad Theorem 3, whe the probability space is [a,b],b, b a λ (B ow meas the σ -algebra of Borel sets of [a,b], I :=[a,b], g is the idetity fuctio o [a, b],ad f is a covexfuctio o [a, b]. The ivestigatio of fuctios like Ĥ k, seems to be due to Dragomir, who has itroduced ad studied amog others the fuctio Ĥ, i [4]. May papers deal with similar fuctios, for example see Abdallah El Farissi [], Dragomir ad Agarwal [5], Yag ad Wag [20] ad Yag ad Tseg [2]. Our result gives a ew approach i treatig the problem. 6. Proofs of the mai iequalities We eed the followig well kow result: LEMMA. (see [2], 6. Lemma Let (Ω,A, μ be a measure space. Let E be a metric space, ad f : E Ω R a fuctio with the properties (i ω f (x,ω is μ -itegrable for each x E, (ii x f (x,ω is cotiuous at x 0 E for every ω Ω, (iii there is a oegative μ -itegrable fuctio h o Ω such that f (x,ω h(ω, (x,ω E Ω. The the fuctio ϕ defied o E by ϕ (x= f (x,ωdμ (ω is cotiuous at x 0. Ω Proof of Theorem 3. Fix k N +. (a Sice covexity is ivariat uder affie maps, the itegral is mootoic, ad the sum of covex fuctios is also covex, H k is covex o [0,].

16 242 L. HORVÁTH AND J. PEČARIĆ By applyig the classical Jese s iequality, we get for all t [0, ] that H k (t u k (i,...,i ( a j (i j p j (i,...,i S k a j (i j p j g j (x j f t +( t a j (i j p j T = u k (i,...,i ( a j (i j p j f (i,...,i S k = H k (0. a j (i j p j g j dμ j j dμ T (x a j (i j p j a j (i j p j g j dμ j j a j (i j p j Suppose 0 t < t 2. The covexity of H k,adh k (t H k (0 (t [0,] mea that H k (t 2 H k (t H k (t 2 H k (0 0, t 2 t t 2 (23 ad thus H k (t 2 H k (t. (b Whe (8 is combied with (23 ad with the discrete Jese s iequality, it follows that H k (0 f u k (i,...,i a j (i j p j g j dμ j (i,...,i S k j ( = f p j g j dμ j u k (i,...,i a j (i j = f (i j,...,i S k p j g j dμ j. j H k (=T k is obvious. (c It follows from (a. (d It remais oly to show that H k is cotiuous at. We check the coditios of Lemma. (i See Remark.

17 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 243 (ii Sice f is cotiuous, the fuctio a j (i j p j g j (x j t f t +( t a j (i j p j a j (i j p j g j dμ j j a j (i j p j is cotiuous at for every x T. (iii By applyig the discrete Jese s iequality, we have a j (i j p j g j (x j a j (i j p j g j dμ j f t j +( t a j (i j p j a j (i j p j tf a j (i j p j g j (x j +( t f a j (i j p j max f a j (i j p j g j (x j, f a j (i j p j a j (i j p j g j dμ j j a j (i j p j a j (i j p j g j dμ j j a j (i j p j for all t [0,] ad x T. Choose a fixed iterior poit a of I.Sice f is covex f (t f (a+ f + (a(z a, z I, f + (a meas the right-had derivative of f at a. It follows from this that f t a j (i j p j g j (x j f (a+ f + (a t a j (i j p j +( t a j (i j p j g j (x j a j (i j p j g j dμ j j a j (i j p j a j (i j p j +( t a j (i j p j g j dμ j j a a j (i j p j

18 244 L. HORVÁTH AND J. PEČARIĆ mi f (a+ f + (a f (a+ f + (a a j (i j p j g j (x j a a j (i j p j a j (i j p j g j dμ j j a a j (i j p j for all t [0,] ad x T. The result ow follows from Lemma. The proof is complete. Proof of Theorem 2. (a Sice S = {(,...,}, (3, (2ad(H 3 givethat T = dμ T (x. T f ( p j g j (x j From the classical Jese s iequality we therefore have T f p j g j (x j dμ T (x = f T Accordig to Theorem p i g i dμ i i= i. T k, (g(x,...,g(x ; p,...,p ;a,...,a T k+, (g(x,...,g(x ; p,...,p ;a,...,a, k N + for all fixed (x,...,x T l, ad hece T k T k+, k N +. Fially, it follows from the discrete Jese s iequality that T k u k (i,...,i (i,...,i S k T = u k (i,...,i (i,...,i S k ( = a j (i j p j f (g j (x j dμ T (x a j (i j p j (i,...,i S k u k (i,...,i a j (i j j p j f g j dμ j j f g j dμ j. (24

19 REFINEMENTS OF INTEGRAL FORM OF JENSEN S INEQUALITY 245 This ad (7 imply that T k p j j f g j dμ j, k N +. (b Apply Theorem 3 (b. The proof is complete. Ackowledgemets. The research of the st author has bee supported by Hugaria Natioal Foudatios for Scietific Research Grat No. K2086. The research of the 2d author has bee fully supported by Croatia Sciece Foudatio uder the project REFERENCES [] ABDALLAH EL FARISSI, Simple proof ad refiemet of Hermite Hadamard iequality, J. Math. Iequal. 4 (200, o. 3, [2] H. BAUER, Measure ad Itegratio Theory, Walter de Gruyter, Berli, New York, 200. [3] I. BRNETIĆ, C. E. M. PEARCE, J. PEČARIĆ, Refiemets of Jese s iequality, Tamkag J. Math. 3 (2000, o., [4] S.S.DRAGOMIR, Two mappigs i coectio to Hadamard s iequalities, J. Math. Aal. Appl. 67 (992, o., [5] S. S. DRAGOMIR, P. AGARWAL, Two ew mappigs associated with Hadamard s iequalities for covex fuctios, Appl. Math. Lett. (998, o. 3, [6] J. HALUŠKA, O. HUTNIK, Some iequalities ivolvig itegral meas, Tatra Mt. Math. Publ. 35 (2007, o., [7] G. H. HARDY, J. E. LITTLEWOOD, G. PÓLYA, Iequalities, Cambridge Uiversity Press, Cambridge, 978. [8] L. HORVÁTH, Iequalities correspodig to the classical Jese s iequality, J. Math. Iequal. 3 (2009, o. 2, [9] L. HORVÁTH, A refiemet of the itegral form of Jese s iequality, J. Iequal. Appl. ( :78, 9 pp. [0] L. HORVÁTH, J. PEČARIĆ, A refiemet of the discrete Jese s iequality, Math. Iequal. Appl. 4 (20, o. 4, [] L. HORVÁTH, A ew refiemet of the discrete Jese s iequality depedig o parameters, J. Iequal. Appl. ( :55, 6 pp. [2] L. HORVÁTH, Weighted form of a recet refiemet of the discrete Jese s iequality, Math. Iequal. Appl. 7 (204, o. 3, [3] L. HORVÁTH, Ifiite refiemets of the discrete Jese s iequality defied by recursio, J. Math. Iequal. 9 (205 o. 4, [4] L. HORVÁTH, J. PEČARIĆ, Refiemets of the classical Jese s iequality comig from refiemets of the discrete Jese s iequality, Adv. Iequal. Appl., Vol. 205 (205, Art ID 8, 7 pp. [5] L. HORVÁTH, KHURAM ALI KHAN, J. PEČARIĆ, Combiatorial Improvemets of Jese s Iequality, Moographs i Iequalities 8, Elemet, Zagreb, 204. [6] J. L. W. V. JENSEN, Sur les foctios covexes et les iegalités etre les valeurs moyees, Acta Math. 30 (906, [7] KHURAM ALI KHAN, J. PEČARIĆ, Mixed symmetric meas related to the classical Jese s iequality, J. Math. Iequal. 7 (203 o., [8] J. ROOIN, A refiemet of Jese s iequality, JIPAM. J. Iequal. Pure Appl. Math. 6 (2, Article 38 (2005. [9] H. SUN, B. LONG, Y. CHU, O the geeralized weighted quasi-arithmetic itegral mea, It. Joural of Math. Aalysis 7 (203 o. 4,

20 246 L. HORVÁTH AND J. PEČARIĆ [20] G. S. YANG,C.S.WANG, Some refiemets of Hadamard s iequality, Tamkag J. Math. 28 (997, [2] G. S. YANG, K. L. TSENG, O certai itegral iequalities related to Hermite-Hadamard iequalities, J. Math. Aal. Appl. 239 (999, o., (Received February 6, 206 László Horváth Departmet of Mathematics, Uiversity of Paoia Egyetem u. 0, 8200 Veszprém, Hugary lhorvath@almos.ui-pao.hu Josip Pečarić Faculty of Textile Techology, Uiversity of Zagreb Pierottijeva 6, 0000 Zagreb, Croatia pecaric@mahazu.hazu.hr Mathematical Iequalities & Applicatios mia@ele-math.com