We improve the classical Jensen inequality for convex functions by extending it to a wider class of functions. We also consider some weaker conditions for the weights occurring in this inequality

Josip Pečarić

Sanja Varošanec

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 32 (2003) no. 1, pp. 79–84ictp.acad.ro/jnaatON CONVEXITY–LIKE INEQUALITIES (II)JOSIP PEČARIĆ∗ and SANJA VAROŠANEC†Abstract. We improve the classical Jensen inequality for convex functions byextending it to a wider class of functions. We also consider some weaker condi-tions for the weights occurring in this inequality.MSC 2000. 26D15.Keywords. Jensen’s inequality, convexity-like inequalities, generalized Steffen-sen’s inequality.1. INTRODUCTIONOne of the most important inequality in the theory of convex functions andmany fields of mathematics connected with this theory is Jensen’s inequalitywhich states:Theorem 1. If f : I → R (I is an open interval in R) is a convex function,x1, x2, . . . , xn ∈ I, n ≥ 2, p1, p2, . . . , pn > 0, then(1) f(1Pnn∑i=1pixi)≤ 1Pnn∑i=1pif(xi),where Pn =n∑i=1pi.This inequality has an important role in mathematics and statistics andthere exist a lot of its generalization, improvements and refinements. For ex-ample, it is reasonable to ask whether the condition pi > 0 can be relaxed. Or,on the other hand, can we obtain inequality (8) for a wider class of functions?Some answers on the second question were given in [1] by Dragomirescu andIvan, and in [3] by Pečarić and Pearce. Answer on the first question was givenby Steffensen [8]. Namely, he obtained that the following theorem holds.Theorem 2. If f : I → R (I is an open interval in R) is a convex function,(x1, x2, . . . , xn) ∈ In, n ≥ 2, is a monotonic n-tuple, and if pi are real numberssuch that∗Faculty of Textile Technology, University of Zagreb, Pierrotijeva 6, 10000 Zagreb, Croa-tia, e-mail: pecaric@hazu.hr.†Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia,e-mail: varosans@math.hr.80 Josip Pečarić and Sanja Varošanec 2(2) x̄ = 1Pnn∑i=1pixi ∈ I, 0 ≤ Pk ≤ Pn, k = 1, 2, . . . , n− 1, Pn > 0,then (1) holds.The previous theorem is known in literature under the name: Jensen–Steffensen’s inequality. In this paper we show that the condition (2) canbe relaxed. In Section 2 we suppose that f is a differentiable function andmain result is a consequence of the generalized Steffensen’s inequality [7]. InSection 3 we will pointed out that differentiability is not a necessary conditionand we will be in position to prove a theorem about Jensen’s type inequalitydirectly, without using some other results. The proof is based on the identitywhich is first used in the proof of the Jensen-Steffensen inequality by J. Pečarićin [4].2. CONSEQUENCE OF THE GENERALIZED STEFFENSEN INEQUALITYWe need the following Steffensen type inequality which is given in [7].Here, we shall use these notation:t = (t1, . . . , tn), [a,b] ={t : ai ≤ ti ≤ bi, 1 ≤ i ≤ n}=n∏i=1[ai, bi].If t and x are two n-tuples then by t + x we will denote the n-tuple (t1 +x1, . . . , tn + xn).Theorem 3. Let µ be a measure such that [a,b] is a finite µ-measurableset, f , g and fg be µ-integrable functions on [a,b]. Let Λ = (λ1, . . . λn) be apositive n-tuple, [a,a + λ] ⊂ [a,b] and A be a real number such thatAµ([a,a + Λ])=∫[a,b]gdµ,and let one of the following cases be satisfied µ-almost everywhere:1. g(t) ≤ A and f(t) ≥ f(a + Λ), for t ∈ [a,a + Λ];g(t) ≥ 0 and f(t) ≤ f(a + Λ), for t ∈ [a,b]\[a,a + Λ];2. g(t) ≥ A and f(t) ≤ f(a + Λ), for t ∈ [a,a + Λ];g(t) ≤ 0 and f(t) ≥ f(a + Λ), for t ∈ [a,b]\[a,a + Λ].ThenA∫[a,a+Λ]fdµ ≥∫[a,b]fgdµ.This is a direct multidimensional generalization of the right-hand side ofSteffensen’s inequality which states that if f is a nonincreasing function on[a, b] and g is integrable with g(t) ∈ [0, 1], then the following inequalities hold3 On convexity–like inequalities (II) 81(3)∫ bb−λf(t)dt ≤∫ baf(t)g(t)dt ≤∫ a+λaf(t)dt, λ =∫ bag(t)dt.Generalization of the left-hand side of the inequality is given in the followingtheorem.Theorem 4. Let µ be a measure such that [a,b] is a finite µ-measurableset, f, g and fg be µ-integrable functions on [a,b]. Let Λ = (λ1, . . . , λn) be apositive n-tuple, [b− Λ,b] ⊂ [a,b] and A be a real number such thatAµ([b− Λ,b])=∫[a,b]gdµ,and let one of the following three cases be satisfied µ-almost everywhere:1. g(t) ≤ A and f(t) ≤ f(b− Λ), for t ∈ [b− Λ,b];g(t) ≥ 0 and f(t) ≥ f(b− Λ), for t ∈ [a,b]\[b− Λ,b];2. g(t) ≥ A and f(t) ≥ f(b− Λ), for t ∈ [b− Λ,b];g(t) ≤ 0 and f(t) ≤ f(b− Λ), for t ∈ [a,b]\[b− Λ,b].ThenA∫[b−Λ,b]fdµ ≤∫[a,b]fgdµ.Proofs of the above mentioned theorems are given in [7]. Now, we are in thesituation to pointed out the following result which improve classical Jensen’sinequality.Theorem 5. Let f : (a, b)→ R be a differentiable function, (x1, x2, . . . , xn)∈ (a, b)n be a nonincreasing n-tuple, and p1, . . . , pn be a real numbers suchthat there exists integer k ∈ {1, 2, . . . , n− 1} with properties1Pnn∑i=1pixi ∈ (xk+1, xk](4)Pj ≥ 0, j = 1, 2, . . . , k;P̄j ≥ 0 j = k + 1, . . . , n, Pn > 0,(5)where Pj =j∑i=1pi and P̄j =n∑i=jpi.If f has property thatf ′(x) ≤ f ′(1Pnn∑i=1pixi), for x ≤ 1Pnn∑i=1pixi andf ′(x) ≥ f ′(1Pnn∑i=1pixi), for x ≥ 1Pnn∑i=1pixi,(6)then (1) holds.Proof. We use idea from [5]. Setting g(t) = gj on (xj+1, xj ], j = 1, . . . , n−1,wheregj = PjPn , and g(xn) = gn = 1,82 Josip Pečarić and Sanja Varošanec 4we easily check that the following holds:gj ≤ 1 for j = n, . . . , k,gj ≥ 0 for j = 1, . . . , kand after some simple calculations we getλ =∫ x1xng(t)dt = 1Pnn∑i=1pixi − xn.Functions g and −f ′ satisfy assumptions from Theorem 3, case 1, for n = 1,so, the following inequality holds∫ x1xnf ′(t)g(t)dt ≥∫ xn+λxnf ′(t)dtand we getn−1∑k=1(f(xk)− f(xk+1))gk ≥ f(xn + λ)− f(xn)wherefrom it can be obtained (1). Furthermore, Theorem 5 is also true if (x1, . . . , xn) is a nondecreasing se-quence; the only change needed in the hypothesis is to require that1Pnn∑i=1pixi ∈ (xk, xk+1].In this case, the proof is based on the inequality described in Theorem 4.Once more, we stress that under the conditions of the previous Theorem 5,the Jensen inequality holds not only for convex functions but also for the widerclass of functions.3. FURTHER RESULTLet c ∈ (a, b) and λ be a real number. We will say that f : (a, b) → R hasproperty K1(c,λ), if for any z, y ∈ (a, b), z, y ≥ c,(7) f(z)− f(y)z − y≥ λholds.If for any z, y ∈ (a, b), z, y ≤ c inequality (7) holds then we will say that fhas property K2(c,λ).If in (7) the reversed inequality holds, then we will say that f has propertyRK1(c,λ) or RK2(c,λ) respectively.If f is differentiable function on (a, b) and if f has a property K1(c,λ) forsome c ∈ (a, b), then for f ′ the following holdsf ′(x) ≥ λ, ∀x ≥ c.Obviously, if f is a nondecreasing function on (a, b) then f has propertiesK1(c,0) and K2(c,0) for any c ∈ (a, b). Also, it is known that a convex function5 On convexity–like inequalities (II) 83f defined on an interval (a, b) has properties K1(c, f ′+(c)) and RK2(c, f ′−(c))for any c ∈ (a, b).Theorem 6. Let (x1, . . . , xn) ∈ (a, b)n be a nonincreasing sequence andp1, . . . , pn be real numbers such that Pn =n∑i=1pi > 0. Let us suppose that thereexists an integer m ∈ {1, 2, . . . , n− 1} such that x̄ = 1Pnn∑i=1pixi ∈ (xm+1, xm].1. If Pi ≥ 0, for i = 1, . . . ,m, P̄i ≥ 0, for i = m, . . . , n and if f : (a, b)→R has properties K1(x̄, λ) and RK2(x̄, λ) then(8) f(1Pnn∑i=1pixi)≤ 1Pnn∑i=1pif(xi)holds, where Pi =i∑j=1pj and P̄i =n∑j=ipj.2. If Pi ≥ 0, for i = 1, . . . ,m, P̄i ≤ 0, for i = m, . . . , n and if f : (a, b)→R has properties K1(x̄, λ) and K2(x̄, λ) then (8) holds.3. Similarly, if Pi ≤ 0, for i = 1, . . . ,m, P̄i ≥ 0, for i = m, . . . , n and iff : (a, b)→ R has properties RK1(x̄, λ) and RK2(x̄, λ) then (8) holds.Proof. The proof is based on the following identity [4]:f(x̄)− 1Pnn∑i=1pif(xi) =m−1∑i=1[λ(xi − xi+1)−(f(xi)− f(xi+1))] PiPn+[λ(xm − x̄)− (f(xm)− f(x̄))]PmPn+[f(x̄)− f(xm+1)− λ(x̄− xm+1)] P̄m+1Pn+n−1∑i=m+1[f(xi)− f(xi+1)− λ(xi − xi+1)] P̄i+1Pn,where Pi =i∑j=1pj and P̄i =n∑j=ipj .Let us suppose that assumptions from case (1) hold. Then the right-handside of the previous identity is non-positive, so the left-hand side is also non-positive, and inequality (8) is proved. Other cases can be proved on the sameway. REFERENCES[1] Dragomirescu, M. and Ivan, C., Convexity–like inequalities for averages in a convexset, Aequationes Math., 45, pp. 179–199, 1993.[2] Mitrinović, D. S., Pečarić, J. and Fink, A. M., Classical and New Inequalities inAnalysis, Kluwer Acad. Publ., Dordrecht, 1993.[3] Pearce, C. E. M. and Pečarić, J., On convexity–like inequalites, Rev. RoumaineMath. Pure Appl., 42, nos. 1–2, pp. 133–138, 1997.84 Josip Pečarić and Sanja Varošanec 6[4] Pečarić, J., A simple proof of the Jensen–Steffensen inequality, Amer. Math. Monthly,91, pp. 195–196, 1984.[5] Pečarić, J., On the Jensen–Steffensen inequality, Univ. Beograd, Publ. ETF,634–677, pp. 101–107, 1979.[6] Pečarić, J., Proschan, F. and Tong, Y. L., Convex functions, Partial Orderingsand Statistical Applications, Academic Press, 1992.[7] Pečarić, J. and Varošanec, S., Multidimensional and discrete Steffensen’s inequality,Southeast Asian Bulletin of Mathematics, 23, pp. 277–284, 1999.[8] Steffensen, J. F., On certain inequalities and methods of approximation, J. InstituteActuaries, 51, pp. 274–297, 1919.Received by the editors: March 9, 1998.

On convexity-like inequalities (II)

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On convexity-like inequalities (II)

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