ON JENSEN'S INEQUALITY INVOLVING AVERAGES OF ... - anubih

54 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIAfor r, t, u, v, s ∈ R such that r ≤ u, t ≤ v. All limiting cases of (2) and theproof of (3) can be found in [2]. Moreover, some related results of this typeare previously obtained in [8] and [12]. It is interesting that one year laterin the paper [16] the case s = 0 of this mean as an extension of Stolarskymeans is considered without mentioning reference [2] ( see (2.2) in [16] ).Also, in [16] there is no mention of any results from [8, 11, 12] (see Theorem4.1 in [16] and compare it with results from [8, 11, 12] ).D. E. Wulbert in, [18] proved that for a convex function ψ : I → R,⎧⎪⎨F (a, b) =⎪⎩∫1bb−aψ(x)dx, a ≠ b;aψ(a), a = b.is convex on I × I.In [4], D. E. Wulbert’s result is given in the following form:Theorem 1.1. Let ψ be a continuous function on an interval I with anonempty interior. If ψ is convex on I, then the integral arithmetic meanF defined in (4) is convex on I 2 .Furthermore, for a i , b i ∈ I, i = 1, . . . , n and non-negative real weightsw i , i = 1, . . . , n such that ∑ ni=1 w i = 1, the following holds(4)1b − a∫ baψ(x)dx ≤n∑i=1w i1b i − a i∫ bia iψ(x)dx (5)and(ā ) + ¯bψ ≤ 12 ¯b − ā∫ā¯bψ(t)dt≤n∑i=1where ā = ∑ ni=1 w ia i , ¯b = ∑ ni=1 w ib i .w i1b i − a i∫b ia iψ(t)dt≤n∑i=1w iψ(a i ) + ψ(b i )2(6)Using Theorem 1.1, we gave a discrete extension of means (2) and we getnew means of Cauchy type defined by :⎡Mr,t(w s i , a s i , b s i ; n)= ⎢t 3 −ts 2⎣r 3 − rs 2n∑aw r+si −b r+si i a s ii=1−bs in∑aw t+si −b t+si i a s ii=1−bs i⎤− (M w s i(a i )) r+s −(Mw s i(b i )) r+s(Mw s i(a i )) s −(Mw s i(b i )) s⎥− (M w s i(a i )) t+s −(Mw s i(b i )) t+s ⎦(Mw s i(a i )) s −(Mw s i(b i )) s1r−t

ON JENSEN’S INEQUALITY INVOLVING AVERAGES OF CONVEX FUNCTIONS 55where⎧( n∑⎪⎨ w i x SMw S ii(x i ) = i=1n∏ ⎪⎩i=1) 1S, S ≠ 0,x w ii, S = 0,r ≠ t ≠ s, r ≠ s, −s, s ≠ 0.Limiting cases are also derived and we have shown that these means aremonotonic, that is for r, t, u, v, s ∈ R such that r ≤ u, t ≤ vM s r,t(w i ; a i , b i ; n) ≤ M s u,v(w i ; a i , b i ; n). (7)If we let a i → b i , i = 1, . . . , n, we have discrete version of means in (2). Letus note that the case s = 0 is also considered in [16] (see (1.14) and (1.15)in [16]).In this paper, we will give further extension of results from [10]. To beginwith, let us note that the following integral version of (5) is valid:Theorem 1.2. Let α, β : Ω → I (I, an interval with non-empty interior)be functions from L 1 (µ) and let ψ : I → R be a continuous convex function.Then the following inequality is valid:1β − α∫ βα∫ψ(x)dx ≤Ω(1β(u) − α(u)where α = ∫ α(u)dµ(u), β = ∫ β(u)dµ(u).ΩΩ∫ β(u)α(u)ψ(x)dx)dµ(u), (8)Proof. Follows from integral version of convexity on I × I (see [9], p. 51),and the fact that the function F : I × I → R defined with (4) is convex. □In the later text J stands for an open interval in R; with R + we denotepositive reals.Definition 1. A function f : J → R + is a log-convex in Jensen’s sense if( ) x + yf 2 ≤ f(x)f(y),2for all x, y ∈ J.The following simple result gives us necessary and sufficient condition torecognize log-convex function in Jensen’s sense.Lemma 1.3. A function f : J → R + is a log-convex in Jensen’s sense ifand only if( ) x + yu 2 1f(x) + 2u 1 u 2 f + u 222f(y)) ≥ 0, (9)for all u 1 , u 2 ∈ R and x, y ∈ J.

56 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIANow we define exponentially convex functions (see [1]).Definition 2. A function f : J → R is exponentially convex on J if it iscontinuous andn∑ξ i ξ j f (x i + x j ) ≥ 0i,j=1for all n ∈ N and all choices ξ i ∈ R, i = 1, . . . , n such that x i + x j ∈ J, 1 ≤i, j ≤ n.For our purposes we will need the following simple characterization ofexponential convexity.Proposition 1.4. Let f : J → R. The following propositions are equivalent.(i) fis exponentially convex.(ii) f is continuous andn∑ξ i ξ j fi,j=1( )xi + x j≥ 0,2for all n ∈ N, ξ i ∈ R and x i ∈ J, i = 1, . . . , n.The following corollary gives the inclusion relation between two very importantsubclasses of the class of convex functions.Corollary 1.5. A positive exponentially convex function f : J → R + islog-convex function i.e. log f is convex function on J.Exponentially convex functions have strong analytical properties: theyare differentiable of all orders, and even more, they are analytical (see [3,5]). Previous results follow from the integral representation of exponentiallyconvex functions:Theorem 1.6. The function f : J → R is exponentially convex on J if andonly iff(x) =∫ ∞−∞for some non-decreasing function σ : R → R.Proof. See [1], p. 211.e tx dσ(t), x ∈ J (10)Remark 1.7. Theorem 1.6 enables us, in particular, to conclude that anyreal function that is a Laplace transform of some positive function is anexponentially convex function on its open domain.The next theorem from [7] will be very helpful in the sequel.□

ON JENSEN’S INEQUALITY INVOLVING AVERAGES OF CONVEX FUNCTIONS 57Theorem 1.8. Let f : J → R + be log-convex, derivable function.Let M : J × J → R + be defined by⎧( ) 1⎨ f(x) x−yM(x, y) =f(y), x ≠ y;)⎩exp , x = y.If x 1 ≤ x 2 , y 1 ≤ y 2 then(f ′ (x)f(x)(11)M(x 1 , y 1 ) ≤ M(x 2 , y 2 ). (12)It is well known that a real valued convex function ψ defined on I ischaracterized by the second order divided difference at mutually distinctpoints x 0 , x 1 , x 2 ∈ I:ψ(x 0 )[x 0 , x 1 , x 2 ; ψ] =(x 0 − x 1 )(x 0 − x 2 )ψ(x 1 )+(x 1 − x 0 )(x 1 − x 2 ) + ψ(x 2 )(x 2 − x 0 )(x 2 − x 1 ) ≥ 0,If ψ is a differentiable function, we have the support line convexity criteriafor x 0 , x 1 ∈ I[x 0 , x 0 , x 1 ; ψ] = ψ(x 1) − ψ(x 0 ) − (x 1 − x 0 )ψ ′ (x 0 )(x 1 − x 0 ) 2 ≥ 0.Also, if ψ is twice differentiable on I, we have a very useful result aboutconvexity for x 0 ∈ I[x 0 , x 0 , x 0 ; ψ] = ψ′′ (x 0 )22. Main results≥ 0.Let I be an interval with nonempty interior and J stand for an open intervalin R. Let α, β : Ω → I be functions from L 1 (µ) as in introduction and letψ : I → R be a continuous function.Denote∫F (α, β; ψ; µ) =Ω(1β(u) − α(u)∫ β(u)α(u)ψ(x)dx)dµ(u) − 1 ∫ βψ(x)dx,β − α α(13)where α = ∫ α(u)dµ(u), β = ∫ β(u)dµ(u). From Theorem 1.2 it follows thatΩΩif ψ is convex function, then F (α, β; ψ; µ) ≥ 0.Further, we introduce the following families of functions:

58 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIA• X 1 = {ψ t : I → R, t ∈ J}, a family of functions from C(I) suchthat t → [x 0 , x 1 , x 2 ; ψ t ] is log-convex in Jensen’s sense on J for everychoice of distinct points x 0 , x 1 , x 2 ∈ I;• X 2 = {ψ t : I → R, t ∈ J}, a family of differentiable functions on Isuch that t → [x 0 , x 0 , x 1 ; ψ t ] is log-convex in Jensen’s sense on J forevery choice of distinct points x 0 , x 1 ∈ I;• X 3 = {ψ t : I → R, t ∈ J}, a family of twice differentiable functionsfrom I such that t → [x 0 , x 0 , x 0 ; ψ t ] is log-convex in Jensen’s senseon J for every choice of x 0 ∈ I.Theorem 2.1. Let (Ω, A, µ) be a probability space, let α, β : Ω → I befunctions from L 1 (µ). Let F : J → R be function defined byF (t) = F (α, β; ψ t ; µ) (14)ψ t ∈ X k , t ∈ J; k = 1, 2, 3. Then the following is valid(i) the function F is log-convex in Jensen’s sense on J;(ii) if the function F is continuous, it is log-convex and for r, s, t ∈ Jsuch that r < s < t we have,(F (s)) t−r ≤ (F (r)) t−s (F (t)) s−r ; (15)(iii) if the function F is derivable, then for every t, r, u, v ∈ J such thatr ≤ u, t ≤ v, we haveM r,t (α, β, µ) ≤ M u,v (α, β, µ), (16)where⎧⎨( ) 1F (r) r−tM r,t (α, β, µ) = F (t), t ≠ r;⎩exp ( ddr log F (r)) , t = r.(17)Proof.(i) Let u 1 , u 2 ∈ R and t, r ∈ J be arbitrary. Define function h on I byh(x) = u 2 1ψ t (x) + 2u 1 u 2 ψ t+r (x) + u 2 2ψ r (x).2Then][x 0 , x 1 , x 2 ; h] = u 2 1 [x 0 , x 1 , x 2 ; ψ t ] + 2u 1 u 2[x 0 , x 1 , x 2 ; ψ t+r2+ u 2 2 [x 0 , x 1 , x 2 ; ψ r ]Since t → [x 0 , x 1 , x 2 ; ψ t ] is log-convex in Jensen’s sense on J[x 0 , x 1 , x 2 ; h] ≥ 0

ON JENSEN’S INEQUALITY INVOLVING AVERAGES OF CONVEX FUNCTIONS 59we conclude that h is convex function on I. Hence (8) gives( ) t + ru 2 1F (t) + 2u 1 u 2 F + u 222F (r) ≥ 0F is log-convex in the Jensen’s sense.(ii) Since F is continuous on J it is a log-convex function.(iii) This is a simple consequence of Theorem 1.8.Remark 2.2. The case X 3 of the Theorem 2.1 is a generalization of Theorem4.2 in [16]. It is interesting to note here that the author S. Simić hasproved exactly the same theorem twice: once as Theorem 4.2 in [16] and thesecond time as Theorem 1 in [17], but there is no reference to either.We now go a step further as we prove analogue of Theorem 2.1 for exponentiallyconvex functions. We first introduce the following families offunctions:• X 1 = {ψ t : I → R, t ∈ J}, a family of functions from C(I) suchthat t → [x 0 , x 1 , x 2 ; ψ t ] is an exponentially convex function on J forevery choice of 3 distinct points x 0 , x 1 , x 2 ∈ I;• X 2 = {ψ t : I → R, t ∈ J}, a family of differentiable functions on Isuch that t → [x 0 , x 0 , x 1 ; ψ t ] is an exponentially convex function onJ for every choice of 2 distinct points x 0 , x 1 ∈ I;• X 3 = {ψ t : I → R, t ∈ J}, a family of twice differentiable functionsfrom I such that t → [x 0 , x 0 , x 0 ; ψ t ] is an exponentially convexfunction on J for every choice of x 0 ∈ I.Theorem 2.3. Let (Ω, A, µ) be a probability space, let I be an interval inR with a nonempty interior and let α, β : Ω → I be functions from L 1 (µ).Let F : J → R be function defined byF (t) = F (α, β; ψ t ; µ), (18)ψ t ∈ X k , t ∈ J; k = 1, 2, 3. Then the following hold(i) the function F is exponentially convex on J;(ii) for each n ∈ N and t 1 , . . . , t n ∈ J the matrixpositive semi-definite. Particularly,[ (ti + t jdet F2)] ni,j=1≥ 0;[F(iii) for every t, r, u, v ∈ J such that r ≤ u, t ≤ v we have□( )]ti +t n j2isi,j=1M r,t (α, β, µ) ≤ M u,v (α, β, µ), (19)

60 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIAwhere⎧⎨( ) 1F (r) r−tM r,t (α, β, µ) =, r ≠ t;F (t)⎩exp ( ddr log F (r)) , t = r.(20)Proof. (i) Let u i ∈ R, t i ∈ J, i = 1, . . . , n, for arbitrary fixed n ∈ N. Definea function ν on I byn∑ν(x) = u i u j ψ t i +t j (x). (21)2By assumption[x 0 , x 1 , x 2 ; ν] =i,j=1n∑i,j=1[]x 0 , x 1 , x 2 ; ψ t i +t j ≥ 0,2hence ν is convex function on I. Now we can substitute ψ = ν in (8), andwe getn∑( )ti + t ju i u j F ≥ 02i,j=1concluding exponential convexity of F on J.(ii) and (iii)-parts are simple consequences of (i).Now we need to built mean value theorems in order to generate Cauchytype means.Theorem 2.4. Let (Ω, A, µ) be a probability space, let I be a compact intervalin R, and let α, β : Ω → I be functions from L 1 (µ) and let f ∈ C 2 (I).Then there exists ξ ∈ I such thatF (α, β; f; µ) = f ′′ (ξ)F (α, β; e 2 ; µ) (22)2where F is defined with (13) and e 2 (x) = x 2 .Proof. Analogous to the proof of Theorem 2.4 in [10].Following the steps of the proof of Theorem 2.5 in [10] , we can prove thefollowing mean value theorem.Theorem 2.5. Let (Ω, A, µ) be a probability space, I be a compact intervalin R, let α, β : Ω → I be functions from L 1 (µ) and let f 1 , f 2 ∈ C 2 (I). Thenthere exits ξ ∈ I such thatf ′′1 (ξ)f ′′2 (ξ) = F (α, β; f 1, µ)F (α, β; f 2 ; µ) , (23)□□

ON JENSEN’S INEQUALITY INVOLVING AVERAGES OF CONVEX FUNCTIONS 61assuming both denominators are non-zero.Remark 2.6. The previous Theorem is a generalization of results in [8, 11,12].Remark 2.7. Theorem (2.5) enables us to define various types of means,because if the function f 1′′has inverse from (23) we havef 2′′( ) f′′ −1 ( )1 F (α, β; f1 , µ)ξ =f 2′′.F (α, β; f 2 ; µ)The number ξ ∈ I we call Cauchy mean on I.3. ExamplesExample 3.1. Suppose that I is any compact interval in R + , J = R + andthat {ψ t : I → R, t ∈ J} is the family of functions from C 2 (I) defined byψ t (x) = e−x√ t.tSince t ↦→ d2 ψ t(x) = e −x√t , is exponentially convex on J by Example 3 indx 2[7] (see [15] p. 214), by Theorem 2.3 we conclude the exponential convexityof function F on J, whereF (t) =⎛−1 ∫( √ ⎝t) 3Ωe −√ tβ(u) − e −√ tα(u)β(u) − α(u)If we apply (20) on F we get expressions Mr,t(α, 1 β; µ) :⎧⎪⎨Mr,t(α, 1 β; µ) =⎪⎩⎞dµ(u) − e−β√t − e −α√ t⎠ . (24)β − α( ) 1F (α,β;ψr;µ) r−tF (α,β;ψ t ;µ)), r ≠ t;()exp − 1 r − F (α,β;id·ψ r;µ)2 √ rF (α,β;ψ r;µ), r = t,and we have the monotonicity property (19): for (r, t), (u, v) ∈ J × J suchthat r ≤ u, t ≤ vM 1 r,t(α, β; µ) ≤ M 1 u,v(α, β; µ). (25)Example 3.2. Suppose that I is a compact interval in R + and J = R + .Define ψ t ∈ X 3 by{t −x, t ≠ 1;ψ t (x) =(log t) 2x 22 , t = 1.

62 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIAExponential convexity of t ↦→ d2dx 2 ψ t (x) = t −x on J for x ∈ R + is given byExample 2 in [7] (see also [15], p. 210). For this family the exponentiallyconvex function is:⎧⎪⎨F (t) =⎪⎩[−1 ∫log 3 tΩ[1 ∫6Ω]t −β(u) −t −α(u)β(u)−α(u)dµ(u) − t−β −t −α, t ≠ 1;β−α]β 3 (u)−α 3 (u)β(u)−α(u) dµ(u) − β3 −α 3, t = 1.β−α(26)Then, as corresponding to (20), we define⎧⎪⎨Mr,t(α, 2 β; µ) =⎪⎩[ ] 1F (α,β;ψr ;µ) r−tF (α,β;ψ t ;µ), r ≠ t;[]exp − F (α,β;idψ r;µ)r F (α,β;ψ r ;µ) − 2r log r, r = t, r ≠ 1;[exp− F (α,β;id·ψ 1;µ)3F (α,β;ψ 1 ;µ)], r = t, t = 1.Again, we have the monotonicity property (19):If (r, t), (u, v) ∈ J × J such that r ≤ u, t ≤ v, thenM 2 r,t(α, β; µ) ≤ M 2 u,v(α, β; µ). (27)Example 3.3. Define a family of functions {ψ t : t ∈ J}, on I ⊆ R, I =[a, b], J = R by{ 1eψ t (x) =tx , t ≠ 0;t12 2 x2 , t = 0,(28)Then t ↦→ d2dx 2 (ψ t (x)) = e tx which is a basic example of exponentially convexfunction on R for each x ∈ [a, b]. Using Theorem 2.3, for this family we canconstruct a new exponentially convex function defined by⎧⎪⎨F (t) =⎪⎩1t 3 [ ∫Ω[1 ∫6Ω]e tβ(u) −e tα(u)β(u)−α(u)dµ(u) − etβ −e tα, t ≠ 0;β−α]β 3 (u)−α 3 (u)β(u)−α(u) dµ(u) − β3 −α 3, t = 0.β−α(29)

ON JENSEN’S INEQUALITY INVOLVING AVERAGES OF CONVEX FUNCTIONS 63Using (20) we now construct M 3 r,t(α, β; µ) on R × R⎧⎪⎨Mr,t(α, 3 β; µ) =⎪⎩[ ] 1F (α,β;ψr ;µ) r−tF (α,β;ψ t ;µ), r ≠ t;[ ]exp F (α,β;id·ψr;µ)F (α,β;ψ r;µ)− 2 r, r = t ≠ 0;exp[ ]F (α,β;id·ψ0 ;µ)3F (α,β;ψ 0 );µ, r = t = 0.(30)Again we have monotonicityM 3 r,t(α, β; µ) ≤ M 3 u,v(α, β; µ), r ≤ u, t ≤ v. (31)Observe here, that, according Theorem 2.5, we have here Cauchy means:e ξ =[ ] 1F (α, β; ψr ; µ) r−t, (32)F (α, β; ψ t ; µ)for some(unique) ξ ∈ I = [a, b]. So we conclude that M 3 r,t(α, β; µ) representsa Cauchy mean on [e a , e b ].Remark 3.4.(i) If we replace α → log α and β → log β in (30) we get the continuousversion of the 3-parameters mean for s = 0 obtained in [4].(ii) The case α = β on Ω, gives us the limiting cases of the new Cauchymeans for s = 0 in [2], that are means from [16].Example 3.5. Define a family of functions Φ = {ϕ t : t ∈ J}, on I ⊆ R + ,I = [a, b], J = R, by⎧x⎨tt(t−1), t ≠ 1, 0;ϕ t (x) =⎩− log x, t = 0;x log x, t = 1.(33)Then t ↦→ d2dx 2 (ϕ t (x)) = x t−2 is again an exponentially convex function on Jsince x t−2 = e (t−2) log x . Using Theorem 2.3, for this family construct a new

64 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIAexponentially convex function defined by⎧ [1 ∫t 3 −tΩ⎪⎨F (t) =[1 ∫2Ωβ t+1 (u)−α t+1 (u)β(u)−α(u)dµ(u) − βt+1 −α t+1β−α], t ≠ −1, 0, 1;]log(β(u))−log(α(u))β(u)−α(u)dµ(u) − log(β)−log(α) , t = −1;β−αβ log β−α log αβ−α− ∫ Ωβ(u) log β(u)−α(u) log α(u)β(u)−α(u)dµ(u), t = 0;⎪⎩∫Ωβ(u) 2 log β(u)−α(u) 2 log α(u)β(u)−α(u)dµ(u) − β2 log β−α 2 log α, t = 1β−αUsing (20), for family Φ we now construct M 4 r,t(α, β; µ; Φ) on R × R :⎧⎪⎨Mr,t(α, 4 β; µ; Φ) =⎪⎩[ ] 1F (α,β;ϕr;µ) r−tF (α,β;ϕ[ t;µ)exp 1−2r[exp 1 − F (α,β;ϕ2 0 ;µ)[expr 2 −r − F (α,β;ϕ 0ϕ r ;µ)F (α,β;ϕ r ;µ)2F (α,β;ϕ 0 ;µ)−1 − F (α,β;ϕ 0ϕ 1 ;µ)2F (α,β;ϕ 1 ;µ)M 4 r,t(α, β; µ; Φ) are Cauchy means since, for r ≠ t,, r ≠ t, t ≠ 0, 1;], r = t ≠ 0, 1;], r = t = 0;], r = t = 1.(34)(35)d 2(ϕdx 2 r (x))d 2(ϕdx 2 t (x)) = xr−tis an invertible function. Particularly,[ ] 1F (α, β; ϕr ; µ) r−ta ≤≤ b. (36)F (α, β; ϕ t ; µ)We now impose one additional parameter s ∈ R; after substitutions α →α s , β → β s , r → r s and t → t sin (36), we have[F (α smin{a s , b s , β s ; ψ rs} ≤; µ)] sr−tF (α s , β s ≤ max{a s , b s }, r ≠ t. (37); ψ t ; µ)sHence, we get the (generalized) Cauchy means on [a, b] since[F (α s , β s ; ψ rsa ≤; µ)F (α s , β s ; ψ t ; µ)s] 1r−t≤ b.

68 J. JAKŠETIĆ, J. PEČARIĆ AND G. ROQIA[5] W. Ehm, M. G. Genton and T. Gneiting, Stationary covariances associated with exponentiallyconvex functions, Bernoulli, 9 (4) (2003), 607–615.[6] E. Issacson and H. B. Keller, Analysis of Numerical methods, Dover Publications, Inc.New York, 1966.[7] J. Jakšetić and J. Pečarić, Exponential Convexity Method, (submitted).[8] A. McD. Mercer, Some new inequalities involving elementary mean values, J. Math.Anal. Appl., 229 1999), 677–681.[9] J. E. Pečarić, F. Proschan, and Y. L.Tong, Convex functions, partial orderings, andstatistical applications, Academic Press, Inc, 1992.[10] J. Pečarić and G. Roqia, Generalization of Stolarsky type means, J. Inequal. Appl.,Art. ID 720615, 15 pages, (2010).[11] J. Pečarić, M. R. Lipanović and H. N. Srivastava, Some mean value theorems ofCauchy type, Fract. Calc. Appl. Anal., 9 (2) (2006), 143–158.[12] J. Pečarić, I. Perić and H. N. Srivastava, A family of Cauchy type mean value theorems,J. Math. Anal. Appl.,36 (2005), 730–739.[13] J. Pečarić, I. Perić and M. R. Lipanović, Integral representations of generalized Whiteleymeans and related inequalities, Math. Inequal. Appl.,12 (2) (2009), 295–309.[14] J. Pečarić and V. Šimić, Stolarsky-Tobey mean in n variables, Math. Inequal. Appl.,2 (3) (1999), 325–341.[15] J. L. Schiff, The Laplace transform. Theory and applications. Undergraduate Textsin Mathematics. Springer-Verlag, New York, 1999.[16] S. Simić, An extension of Storalsky means to the multivariable case, Int. J. Math.Mat. Sci., Art. ID 432857, 14 pages, (2009).[17] S. Simić, On certain new inequalities in information theory, Acta Math. Hungar., 124(4) (2009), 353–361.[18] D. E. Wulbert, Favard’s Inequality on average values of convex functions, Math.Comp. Mod., 30 (3) (2000), 853–856.(Received: May 31, 2011)J. JakšetićFaculty of Mechanical Engineering andNaval ArchitectureUniversity of Zagreb10000 Zagreb, CrotiaE–mail: julije@math.hrJ. PečarićFaculty of Textile TechnologyUniversity of ZagrebPierottijeva 6, 10 000 Zagreb, CroatiaE–mail: pecaric@hazu.hrG. RoqiaAbdus Salam School ofMathematical Sciences68-B, New Muslim TownLahore 54000, PakistanE–mai: rukiyya@gmail.com