Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure

Journal of Inequalities and Applications, Jan 2017

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results. MSC: 26D15, 26A51, 26A24.

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Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure

Mikic´ et al. Journal of Inequalities and Applications Levinson's type generalization of the Jensen inequality and its converse for real Stieltjes measure Rozarija Mikic´ 0 Josip Pecˇaric´ 0 Mirna Rodic´ 0 0 Faculty of Textile Technology, University of Zagreb , Prilaz baruna Filipovic ́a 28a, Zagreb, 10000 , Croatia We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi's inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results. Jensen's inequality; converse Jensen's inequality; Hermite-Hadamard's inequality; Giaccardi's inequality; Levinson's inequality; Green function; mean-value theorems 1 Introduction and preliminary results © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The following theorem states the well-known Levinson inequality. ≥  and let pi, xi, yi, i = , . . . , n be such Theorem . ([]) Let f : , c → R satisfy f that pi > , in= pi = ,  ≤ xi ≤ c and x + y = x + y = · · · = xn + yn. Then the following inequality is valid: pif (xi) – f (x¯) ≤ pif (yi) – f (y¯), where x¯ = in= pixi and y¯ = in= piyi denote the weighted arithmetic means. Numerous papers have been devoted to extensions and generalizations of this result, as well as to weakening the assumptions under which inequality (.) is valid (see for instance [–], and []). A function f : I → R is called k-convex if [x, . . . , xk]f ≥  for all choices of k +  distinct points x, x, . . . , xk ∈ I. If the kth derivative of a convex function exists, then f (k) ≥ , but f (k) may not exist (for properties of divided differences and k-convex functions see []). max{x, . . . , xn} ≤ max{y, . . . , yn}, then (.) holds. (ii) Pečarić [] proved that inequality (.) is valid when one weakens the previous assumption (.) to xi + xn–i+ ≤ c and (iii) Mercer [] made a significant improvement by replacing condition (.) with a weaker one, i.e. he proved that inequality (.) holds under the following conditions: ≥ , pi = , a ≤ xi, yi ≤ b, max{x, . . . , xn} ≤ max{y, . . . , yn}, pi(xi – x¯) = (iv) Witkowski [] showed that it is enough to assume that f is -convex in Mercer’s assumptions. Furthermore, Witkowski weakened the assumption (.) and showed that equality can be replaced by inequality in a certain direction. Furthermore, Baloch, Pečarić, and Praljak in their paper [] introduced a new class of functions Kc(a, b) that extends -convex functions and can be interpreted as functions that are ‘-convex at point c ∈ a, b ’. They showed that Kc(a, b) is the largest class of functions for which Levinson’s inequality (.) holds under Mercer’s assumptions, i.e. that f ∈ Kc(a, b) if and only if inequality (.) holds for arbitrary weights pi > , in= pi =  and sequences xi and yi that satisfy xi ≤ c ≤ yi for i = , , . . . , n. We give the definition of the class Kc(a, b) extended to an arbitrary interval I. Definition . Let f : I → R and c ∈ I◦, where I◦ is the interior of I. We say that f ∈ Kc(I) (f ∈ Kc(I)) if there exists a constant D such that the function F(x) = f (x) – D x is concave (convex) on –∞, c] ∩ I and convex (concave) on [c, +∞ ∩ I. Remark . For the class Kc(a, b) the following useful results hold (see []): Jakšetić, Pečarić, and Praljak in [] gave the following Levinson type generalization of the Jensen-Boas inequality. Theorem . ([]) Let c ∈ I◦ and let f : [a, b] → R and g : [a, b] → R be continuous monotonic functions (either increasing or decreasing) with ranges I ∩ –∞, c] and I ∩ [c, +∞ , respectively. Let λ : [a, b] → R and μ : [a, b] → R be continuous or of bounded variation satisfying On the other hand, in [] Pečarić, Perić, and Rodić Lipanović generalized the Jensen inequality (.) for a real Stieltjes measure. They considered the Green function G defined on [α, β] × [α, β] by which is convex and continuous with respect to both s and t. The function G is continuous under s and continuous under t, and it can easily be shown by integrating by parts that any function ϕ : [α, β] → R, ϕ ∈ C([α, β]), can be represented by ϕ(x) = ββ –– αx ϕ(α) + βx –– αα ϕ(β) + G(x, s)ϕ (s) ds. Using that fact, the authors in [] gave the conditions under which inequality (.) holds for a real Stieltjes measure, which is not necessarily positive nor increasing. This result is stated in the following theorem. Theorem . ([]) Let g : [a, b] → R be continuous function and [α, β] interval such that the image of g is a subset of [α, β]. Let λ : [a, b] → R be continuous function or the function of bounded variation, such that λ(a) = λ(b) and Then the following two statements are equivalent: , s ≤ Furthermore, the statements () and () are also equivalent if we change the sign of inequality in both (.) and (.). Corollary . ([]) Let the conditions from the previous theorem hold. Then the following two statements are equivalent: Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality in both statements ( ) and ( ). The main aim of our paper is to give a Levinson type generalization of the result from Theorem .. In that way, a generalization of Theorem . for real Stieltjes measure, not necessarily positive nor increasing, will also be obtained. 2 Main results In order to simplify the notation, throughout this paper we use the following notation: The following theorem states our main result. Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆ R an interval and c ∈ α, β such that f ([a, b]) ⊆ [α, c] and g([a, b]) ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded variation such that λ(a) = λ(b) and μ(a) = μ(b) and such that C := ab f (x) dλ(x) – f¯ = ab g(x) dμ(x) – g¯ holds. If for all s ∈ [α, c] and for all s ∈ [c, β] we have G(f¯, s) ≤ G(g¯, s) ≤ where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β]) we have The statement also holds if we reverse all signs of inequalities in (.) and (.). When we rearrange the previous inequality, we get and after rearranging we get – g¯ ≤ Since the function φ is continuous and convex on [c, β] and for all s ∈ [c, β] (.) holds, from Theorem . it follows that Inequality (.) follows directly by combining inequalities (.) and (.), and taking into account the condition (.). Remark . It is obvious from the proof of the previous theorem that if we replace the equality (.) by a weaker condition then (.) becomes – ϕ(f¯) ≤ C ≤ C ≤ Since the function ϕ belongs to class Kc([α, β]), we have ϕ–(c) ≤ D ≤ ϕ+(c) (see []), so if additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to – f¯ ≤ Remark . It is easy to see that Theorem . further generalizes the Levinson type generalization of the Jensen-Boas inequality given in Theorem .. Namely, if in Theorem . we set the functions f and g to be monotonic, and the functions λ and μ to satisfy for all uk ∈ vk–, vk , v = a, vn = b, and μ(a) < μ(b), then since the function G is continuous and convex in both variables, we can apply the Jensen inequality and see that for all s ∈ [α, c] and s ∈ [c, β] inequalities (.) hold, so we get exactly Theorem .. 3 Discrete case In this section we give the results for the discrete case. The proofs are similar to those in the integral case given in the previous section, so we will state these results without the proofs. In Levinson’s inequality (.) and its generalizations (see []) we see that pi (i = , . . . , n) are positive real numbers. Here, we will give a generalization of that result, allowing pi to also be negative, with the sum not equal to zero, but with a supplementary demand on pi and xi given by using the Green function G defined in (.). Here we use the common notation: for real n-tuples (x, . . . , xn) and (p, . . . , pn) we set Pk = ik= pi, P¯k = Pn – Pk– (k = , . . . , n) and x¯ = Pn in= pixi. Analogously, for real mtuples (y, . . . , ym) and (q, . . . , qm) we define Qk , Q¯k (k = , . . . , m) and y¯. We already know from the first section that we can represent any function ϕ : [α, β] → R, ϕ ∈ C([α, β]), in the form (.), where the function G is defined in (.), and by some calculation it is easy to show that the following holds: Pn i= piϕ(xi) = Pn i= piG(xi, s) ϕ (s) ds. Using that fact the authors in [] derived the analogous results of Theorem . and Corollary . for discrete case, and here, similarly as in the previous section, we get the following results. Theorem . Let [α, β] ⊆ R be an interval and c ∈ α, β . Let xi ∈ [a, b] ⊆ [α, c], pi ∈ R (i = , . . . , n) be such that Pn =  and x¯ ∈ [α, c], and let yj ∈ [a, b] ⊆ [c, β], qj ∈ R (j = , . . . , m) be such that Qm =  and y¯ ∈ [c, β] and let C :=  Pn i= pixi – x¯ = Qm j= where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β]) we have Pn i= qjϕ(yj) – ϕ(y¯), where D is the constant from Definition .. Inequality (.) is reversed if we change the signs of inequalities in (.). Remark . Theorem . is the generalization of Levinson’s type inequality given in []. Namely, since the function G is convex in both variables, in the case when all pi >  and qj >  we can apply the Jensen inequality and we see that for all s ∈ [α, c] and s ∈ [c, β] inequalities (.) hold. Now from Theorem . and Corollary . we get the result from []. Remark . We can replace the equality from the condition (.) by a weaker condition in the analogous way as in Remark . from the previous section. 4 Converses of the Jensen inequality The Jensen inequality for convex functions implies a whole series of other classical inequalities. One of the most famous ones amongst them is the so-called Edmundson-LahRibarič inequality which states that, for a positive measure μ on [, ] and a convex function φ on [m, M] (–∞ < m < M < +∞), if f is any μ-measurable function on [, ] such that m ≤ f (x) ≤ M for x ∈ [, ], one has ≤ MM–– mf¯ φ(m) + Mf¯ ––mm φ(M), where f¯ =  f dμ/  dμ. It was obtained in . by Lah and Ribarič in their paper []. Since then, there have been many papers written on the subject of its generalizations and converses (for instance, see [] and []). In [] the authors gave a Levinson type generalization of inequality (.) for positive measures. In this section we will obtain a similar result involving signed measures, with a supplementary demand by using the Green function G defined in (.). In order to do so, we first need to state a result from [], which gives us a version of the Edmundson-LahRibarič inequality for signed measures. Theorem . ([]) Let g : [a, b] → R be continuous function and [α, β] be an interval such that the image of g is a subset of [α, β]. Let m, M ∈ [α, β] (m = M) be such that m ≤ g(t) ≤ M for all t ∈ [a, b]. Let λ : [a, b] → R be continuous function or the function of bounded variation, and λ(a) = λ(b). Then the following two statements are equivalent: M – g¯ ϕ(m) + g¯ – m ϕ(M) ≤ M – m M – m ≤ MM–– mg¯ G(m, s) + Mg¯ ––mm G(M, s) Furthermore, the statements () and () are also equivalent if we change the sign of inequality in both (.) and (.). Corollary . ([]) Let the conditions from the previous theorem hold. Then the following two statements are equivalent: Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality in both statements ( ) and ( ). In the following theorem we give the Levinson type generalization of the upper result, and we use a similar method to Section  of this paper. Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆ R an interval and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) = [m, M] ⊆ [c, β], where m = M and m = M. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded variation such that λ(a) = λ(b) and μ(a) = μ(b) and C := ≤ MM––mf¯ G(m, s) + Mf¯ ––mm  G(M, s) M – g¯ G(m, s) + g¯ – m G(M, s), ≤ M – m M – m where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β]) we have M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x) M – m M – m ab dλ(x) g – m ϕ(M) – ab ϕ(g(x)) dμ(x) . M – m ab dμ(x) The statement also holds if we reverse all signs of inequalities in (.), (.) and (.). Proof Let ϕ ∈ Kc([α, β]) be continuous function on [α, β] and let φ(x) = ϕ(x) – D x, where D is the constant from Definition .. M – f¯ φ(m) + f¯ – m φ(M). ≥ M – m M – m When we rearrange the previous inequality, we get M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x) M – m M – m ab dλ(x) and after rearranging we get M – g¯ φ(m) + g¯ – m φ(M), ≤ M – m M – m Since the function φ is continuous and convex on [c, β] and for all s ∈ [c, β] (.) holds, from Theorem . it follows that Inequality (.) follows directly by combining inequalities (.) and (.), and taking into account the condition (.). Remark . It is obvious from the proof of the previous theorem that if we replace the equality (.) by a weaker condition then (.) becomes M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x) M – m M – m ab dλ(x) Since ϕ–(c) ≤ D ≤ ϕ+(c) (see []), if additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to 5 Discrete form of the converses of the Jensen inequality In this section we give the Levinson type generalization for converses of Jensen’s inequality in discrete case. The proofs are similar to those in the integral case given in the previous section, so we give these results with the proofs omitted. As we can represent any function ϕ : [α, β] → R, ϕ ∈ C([α, β]), in the form (.), where the function G is defined in (.), by some calculation it is easy to show that the following holds: Pn i= piϕ(xi) – bb –– ax¯ ϕ(a) – bx¯ –– aa ϕ(b) Pn i= piG(xi, s) – bb –– ax¯ G(a, s) – bx¯ –– aa G(b, s) ϕ (s) ds. Theorem . ([]) Let –∞ < a ≤ A ≤ c ≤ b ≤ B < +∞. If xi ∈ [a, A], yj ∈ [b, B], pi > , qj >  for i = , . . . , n and j = , . . . , m are such that in= pi = jm= qj =  and pixi = BB –– by¯ b + By¯ –– bb B – where x¯ = in= pixi and y¯ = jm= qjyj, then for every f ∈ Kc(a, B) we have pif (xi) ≤ BB –– by¯ f (b) + By¯ –– bb f (B) – Our first result is a generalization of the result from [] stated above, in which it is allowed for pi, qj to also be negative, with the sum not equal to zero, but with supplementary demands on pi, qj and xi, yj given by using the Green function G defined in (.). C := bb––ax¯ a + bx¯––aa b –  = bb––ay¯ a + by¯––aa b – Qm j= Pn i= Qm j= b – y¯ G(a, s) + y¯ – a G(b, s), qjG(yj, s) ≤ b – a b – a Pn i= D b – y¯ ϕ(a) + y¯ – a ϕ(b) –  ≤  C ≤ b – a b – a Qm j= The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.). Remark . If we set all pi, qj to be positive, then Theorem . becomes the result from [] which is stated above in Theorem .. Remark . We can replace the equality from the condition (.) by a weaker condition in the analogous way as in Remark . from the previous chapter. 6 The Hermite-Hadamard inequality The classical Hermite-Hadamard inequality states that for a convex function ϕ : [a, b] → R the following estimation holds: Its weighted form is proved by Fejér in []: If ϕ : [a, b] → R is a convex function and p : [a, b] → R nonnegative integrable function, symmetric with respect to the middle point (a + b)/, then the following estimation holds: p(x) dx ≤ ϕ(x)p(x) dx ≤ ϕ(a) + ϕ(b) Fink in [] discussed the generalization of (.) by separately looking the left and right side of the inequality and considering certain signed measures. In their paper [], the authors gave a complete characterization of the right side of the Hermite-Hadamard inequality. Rodić Lipanović, Pečarić, and Perić in [] obtained the complete characterization for the left and the right side of the generalized Hermite-Hadamard inequality for the real Stieltjes measure. In this section a Levinson type generalization of the Hermite-Hadamard inequality for signed measures will be given as a consequence of the results given in Sections  and . Here we use the following notation: The statement also holds if we reverse all signs of the inequalities in (.) and (.). Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β and let [a, b] ⊆ [α, c] and [a, b] ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded variation such that λ(a) = λ(b), μ(a) = μ(b) and x˜ ∈ [α, c], y˜ ∈ [c, β], and such that C := ab x dλ(x) – x˜ = ab y dμ(y) – y˜. ab dλ(x) ab dμ(y) G(x˜, s) ≤ G(y˜, s) ≤ Remark . Let the conditions from the previous corollary hold. (i) If for all s ∈ [α, c] and s ∈ [c, β] inequalities (.) hold, then for every continuous function ϕ ∈ Kc([α, β]) the reverse inequalities in (.) hold. (ii) If for all s ∈ [α, c] and s ∈ [c, β] the reversed inequalities in (.) hold, then for every continuous function ϕ ∈ Kc([α, β]) (.) holds. Note that for the Levinson type generalization of the left-side inequality of the generalized Hermite-Hadamard inequality it is necessary to demand that x˜ ∈ [α, c] and y˜ ∈ [c, β]. Remark . If in Remark . we put f (x) = x and g(x) = x, we can obtain weaker conditions instead of equality (.) under which inequality (.) holds. Similarly, from the results given in the fourth section we get the Levinson type generalization of the right-side inequality of the generalized Hermite-Hadamard inequality. Here we allow that the mean value x˜ goes outside of the interval [α, c] and y˜ outside of the interval [c, β]. Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β , and let [a, b] ⊆ [α, c] and [a, b] ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded variation such that λ(a) = λ(b) and μ(a) = μ(b) and such that C := b – x˜ a + x˜ – a b – ab x dλ(x) b – a b – a ab dλ(x) ≤ bb––ax˜ G(a, s) + bx˜––aa G(b, s), b – y˜ G(a, s) + y˜ – a G(b, s), ≤ b – a b – a where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β]) we have b – x˜ ϕ(a) + x˜ – a ϕ(b) – ab ϕ(x) dλ(x) b – a b – a ab dλ(x) ≤ D C ≤ bb––ay˜ ϕ(a) + by˜––aa ϕ(b) – abaϕb(dy)μd(μy)(y) . The statement also holds if we reverse all signs of the inequalities in (.), (.) and (.). Remark . If in Remark . we put f (x) = x and g(x) = x, we can obtain analogous weaker conditions instead of equality (.) under which inequality (.) holds. It is easy to see that for λ(x) = x and μ(x) = x the conditions (.), (.) and (.) are always fulfilled. In that way we can obtain a Levinson type generalization of both sides in the classical weighted Hermite-Hadamard inequality. Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β , and let [a, b] ⊆ [α, c] and [a, b] ⊆ [c, β]. (i) If C :=  (b – a) =  (b – a) holds, then for every continuous function ϕ ∈ Kc([α, β]) ϕ(x) dx – ϕ ϕ(x) dx – ϕ (ii) If C :=  (b – a) =  (b – a) holds, then for every continuous function ϕ ∈ Kc([α, β]) 7 The inequalities of Giaccardi and Petrovic´ The well-known Petrović inequality [] for convex function f : [, a] → R is given by f (xi) ≤ f xi + (n – )f (), where xi (i = , . . . , n) are nonnegative numbers such that x, . . . , xn, in= xi ∈ [, a]. The following generalization of (.) is given by Giaccardi (see [] and []). Theorem . (Giaccardi, []) Let p = (p, . . . , pn) be a nonnegative n-tuple and x = (x, . . . , xn) be a real n-tuple such that pjxj – xi ≥ , i = , . . . , n, pixi ∈ [a, b] and pixi = x. If f : [a, b] → R is a convex function, then A = pif (xi) ≤ Af pi –  f (x), B = In this section we will use an analogous technique as in the previous sections to obtain a Levinson type generalization of the Giaccardi inequality for n-tuples p of real numbers which are not necessarily nonnegative. As a simple consequence, we will obtain a Levinson type generalization of the original Giaccardi inequality (.). In order to do so, we first need to state two results from []. Theorem . ([]) Let xi ∈ [a, b] ⊆ [α, β], a = b, pi ∈ R (i = , . . . , n) be such that Pn = . Then the following two statements are equivalent: Pn i= Pn i= pif (xi) ≤ bb –– ax¯ f (a) + bx¯ –– aa f (b) piG(xi, s) ≤ bb –– ax¯ G(a, s) + bx¯ –– aa G(b, s) Moreover, the statements () and () are also equivalent if we change the sign of the inequality in both inequalities, in (.) and in (.). Corollary . ([]) Under the conditions from the previous theorem, the following two statements are also equivalent: ( ) For every continuous concave function ϕ : [α, β] → R the reverse inequality in (.) holds. Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality in both statements ( ) and ( ). Our first result is a Levinson type generalization of the Giaccardi inequality for n-tuples p and m-tuples q of arbitrary real numbers instead of nonnegative real numbers. Theorem . Let [α, β] ⊆ R be an interval and c ∈ α, β such that [a, b] ⊆ [α, c] and [a, b] ⊆ [c, β]. Let p and x be n-tuples of real numbers, and let q and y be m-tuples of real numbers such that Pn = in= pi = , Qm = im= qi = , and pixi – xj ≥  (j = , . . . , n); pixi ∈ [a, b]; pixi = x; qiyi – yj ≥  (j = , . . . , m); qiyi ∈ [a, b]; qiyi = y. C := A = A A = A = i= m i= m B = B = piG(xi, s) ≤ AG pixi, s + B pi –  G(x, s), qjG(yj, s) ≤ AG qj –  G(y, s), pi –  ϕ(x) – qj –  ϕ(y) – The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.). Remark . One needs to notice that if we set pi (i = , . . . , n) and qj (j = , . . . , m) to be positive, Theorem . becomes the Levinson type generalization of the original Giaccardi inequality (.). Remark . As in the previous sections, we can replace the equality (.) by a weaker condition and then (.) becomes pi –  ϕ(x) – piϕ(xi) ≤ C ≤ C ≤ Aϕ qj –  ϕ(y) – Since ϕ–(c) ≤ D ≤ ϕ+(c) (see []), if additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to ≤ A 8 Mean-value theorems Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆ R an interval and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) = [m, M] ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded variation such that λ(a) = λ(b) and μ(a) = μ(b), and let ϕ ∈ Kc([α, β]) be a continuous function. Motivated by the results obtained in previous sections, we define the following linear functionals which, respectively, represent the difference between the right and the left side of inequalities (.) and (.): J (ϕ) = ab ϕ(g(x)) dμ(x) – ϕ(g¯) – ab ϕ(f (x)) dλ(x) + ϕ(f¯), ELR(ϕ) = M – g¯ ϕ(m) + g – m ϕ(M) – ab ϕ(g(x)) dμ(x) M – m M – m – M – f¯ ϕ(m) – f¯ – m ϕ(M) + ab ϕ(f (x)) dλ(x) , M – m M – m where m = M and m = M. We have: (i) J (ϕ) ≥ , when (.) holds and for all s ∈ [α, c], s ∈ [c, β] (.) holds; (ii) ELR(ϕ) ≥ , when (.) holds, and for all s ∈ [α, c] (.) holds and for all s ∈ [c, β] (.) holds. In the following two theorems we give the mean-value theorems of the Lagrange and Cauchy type, respectively. Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆ R an interval and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) = Proof Since ϕ (x) is continuous on [α, β], it attains its minimum and maximum value on [α, β], i.e. there exist m = minx∈[α,β] ϕ (x) and M = maxx∈[α,β] ϕ (x). The functions ϕ, ϕ : [α, β] → R defined by J (ϕ) = ELR(ϕ) = ϕ(x) = M x – ϕ(x) are -convex because ϕ (x) ≥  and ϕ (x) ≥ , so from Remark . it follows that they belong to the class Kc([α, β]). From Theorem . it follows that J (ϕ) ≥  and J (ϕ) ≥ , and from Theorem . it follows that ELR(ϕ) ≥  and ELR(ϕ) ≥  and so we get where ϕ˜(x) = x. Since the function ϕ˜ is -convex, we have ϕ˜ ∈ Kc([α, β]), so by applying Theorem . (resp. Theorem .) we get J (ϕ˜) ≥  (resp. ELR(ϕ˜) ≥ ). If J (ϕ˜) =  (resp. ELR(ϕ˜) = ), then (.) implies J (ϕ) =  (resp. (.) implies ELR(ϕ) = ), so (.) (resp. (.)) holds for every ξ ∈ [α, β]. Otherwise, dividing (.) by J (ϕ˜) >  (resp. (.) by ELR(ϕ˜) > ) we get resp. m ≤ and continuity of ϕ ensures the existence of ξ ∈ [α, β] satisfying (.) (resp. ξ ∈ [α, β] satisfying (.)). Theorem . Let the conditions from Theorem . hold. Let ϕ, ψ ∈ C([α, β]). If J (ψ ) =  and ELR(ψ ) = , then there exist ξ, ξ ∈ [α, β] such that where ϕ˜(x) = x. Now we have J (ϕ˜) = , because otherwise we would have J (ψ ) = , which is a contradiction with the assumption J (ψ ) = . So we have χ (ξ) = J (ψ )ϕ (ξ) – J (ϕ)ψ (ξ) = , and this gives us the first claim of the theorem. The second claim is proved in an analogous manner, by observing the linear functional ELR instead of J . Remark . Note that if we set the functions f , g, λ, and μ from our theorems to fulfill the conditions from Jensen’s integral inequality or Jensen-Steffensen’s, or Jensen-Brunk’s, or Jensen-Boas’ inequality, then - applying that inequality on the function G which is continuous and convex in both variables - we see that in these cases for all s ∈ [α, c], s ∈ [c, β] inequalities in (.) hold, and so from our results we directly get the results from the paper []. Remark . If in the definition of the functional J (resp. ELR) we set f (x) = x and g(x) = x, then we get a functional that represents the difference between the right and the left side of the left-hand part (resp. right-hand part) of the generalized Hermite-Hadamard inequality. In the same manner, adequate results of Lagrange and Cauchy type for those functionals can be derived directly from Theorem . and Theorem .. 8.1 Discrete case Qm j= Pn i= JD (ϕ) = piϕ(xi) + ϕ(x¯) – ϕ(y¯), – bb––ax¯ ϕ(a) – bx¯––aa ϕ(b) +  where a = b and a = b; G(ϕ) = Aϕ qj –  ϕ(y) – pi –  ϕ(x) + D(ϕ) = Qm j= Pn i= ELRD (ϕ) = Pn i= Qm j= G(ϕ) = j=  Theorem . Let the conditions of Theorem . hold and let ϕ, ψ ∈ C([α, β]). If JD (ψ ) = , ELRD (ψ ) = , and G(ψ ) = , then there exist ξ, ξ, ξ ∈ [α, β] such that all of the following statements hold: As a consequence of the previous two theorems, we now give some further results in which we give explicit conditions on pi, xi (i = , . . . , n) and qj, yj (j = , . . . , m) for (.) and (.) to hold, where using the properties of the function G we can skip the supplementary conditions on that function. Proof Note that pi, qj >  implies that x¯ ∈ [α, c] and y¯ ∈ [c, β], so we can set the interval [a, b] to be [α, c] and [a, b] to be [c, β]. The function G is convex, so by Jensen’s inequality we see that the inequalities in (.) hold for all s ∈ [α, c], s ∈ [c, β]. Now we can apply Theorem . and Theorem . to get the statements of this corollary. Corollary . Let (x, . . . , xn) be monotonic n-tuple, xi ∈ [α, c] (i = , . . . , n) and (y, . . . , ym) be monotonic m-tuple, yj ∈ [c, β] (j = , . . . , m). Let (p, . . . , pn) be a real n-tuple such that  ≤ Pk ≤ Pn (k = , . . . , n), Pn > , and (q, . . . , qm) be a real m-tuple such that  ≤ Qk ≤ Qm (k = , . . . , m), Qm > . Proof Suppose that x ≥ x ≥ · · · ≥ xn. We have Pn(x – x¯) = pi(x – xi) = (xj– – xj)(Pn – Pj–) ≥  so it follows that x ≥ x¯ . Furthermore, Pn(x¯ – xn) = pi(xi – xn) = (xj – xj+)Pj ≥ , and Theorem .. Competing interests The authors declare that they have no competing interests. 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Rozarija Mikić, Josip Pečarić, Mirna Rodić. Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure, Journal of Inequalities and Applications, 2017, 4, DOI: 10.1186/s13660-016-1274-y