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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Generalizations of Steffensen’s inequality via the extension of Montgomery identity

Andrea Aglić Aljinović
  • Corresponding author
  • University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Josip Pečarić / Anamarija Perušić Pribanić
Published Online: 2018-04-26 | DOI: https://doi.org/10.1515/math-2018-0039

Abstract

In this paper, we obtained new generalizations of Steffensen’s inequality for n-convex functions by using extension of Montgomery identity via Taylor’s formula. Since 1-convex functions are nondecreasing functions, new inequalities generalize Stefensen’s inequality. Related Ostrowski type inequalities are also provided. Bounds for the reminders in new identities are given by using the Chebyshev and Grüss type inequalities.

Keywords: Steffensen’s inequality; n-convex functions; Montgomery identity; Ostrowski-type inequality; Grüss-type inequality

MSC 2010: 26D15; 26A51

1 Introduction

In [1], the authors obtain the following extension of Montgomery identity using Taylor’s formula:

Theorem 1.1

Let f : I → ℝ be suct that f(n − 1) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b. Then the following identity holds f(x)=1baabftdti=0n2fi+1xbxi+2axi+2i+2!ba+1(n1)!abTnx,sfnsds(1)

where Tn(x,s)=1nbaasn,asx;1nbabsn,x<sb.

Remark 1.2

The last identity holds also for n = 1. In this special case, we assume that i=0n2 is an empty sum. Thus (1) reduces to well-known Montgomery identity (e.g. [2]) f(x)=1baabf(t)dt+abT1(x,s)f(s)ds

where the Peano kernel is T1(x,s)=saba,asx;sbba,x<sb.

The aim of this paper is to obtain some new generalizations of Steffensen’s inequality using above extension of Montgomery identity. The Steffensen’s inequality was first given and proved by Steffensen in 1918 ([3]):

Theorem 1.3

Suppose that f is nonincreasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and λ=abg(t)dt. Then we have bλbf(t)dtabf(t)g(t)dtaa+λf(t)dt.(2)

The inequalities are reversed for f nondecreasing.

In [4] Jakšetić and Pečarić generalized Steffensen’s inequality for positive measures.

Mitrinović stated in [5] that the inequalities in (2) follow from the identities which will be the starting point for our generalizations of Steffensen’s inequality.

First, let us recall the definition of n-convex functions.

Let f be a real-valued function defined on the segment [a, b]. The divided difference of order n of the function f at distinct points x0, …, xn ∈ [a, b], is defined recursively (see [6, 7]) by f[xi]=f(xi),(i=0,,n)

and f[x0,,xn]=f[x1,,xn]f[x0,,xn1]xnx0.

The value f[x0, …, xn] is independent of the order of the points x0, …, xn.

The definition may be extended to include the case when some (or all) of the points coincide. Assuming that f(j − 1)(x) exists, we define f[x,,xjtimes]=f(j1)(x)(j1)!.(3)

The notion of n-convexity goes back to Popoviciu ([8]). We follow the definition given by Karlin ([9]):

Definition 1.4

A function f: [a, b] → ℝ is said to be n-convex on [a, b], n ≥ 0, if for all choices of (n + 1) distinct points in [a, b], nth order divided difference of f satisfies f[x0,...,xn]0.

Note that, 1–convex functions are nondecreasing functions. If f(n) exists, then f is n–convex iff f(n) ≥ 0.

The paper is organized as follows. After this Introduction, in Section 2 we obtain new identities related to Steffensen’s inequality. Using these new identities we generalize Steffensen’s inequality for n–convex functions. Further, in Section 3 we give Ostrowski-type inequalities related to our new generalizations. We conclude this paper with some new bounds for our identities, using the Chebyshev and Grüss type inequalities.

Throughout the paper, it is assumed that all integrals under consideration exist and that they are finite.

2 Generalizations of Steffensen’s inequality via the extension of Montgomery identity

In this section we obtain generalizations of Steffensen’s inequality for n-convex functons using identity (1).

Theorem 2.1

Let f : I → ℝ be suct that f(n − 1) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤ 1. Let aa+λp(t)dt=abg(t)p(t)dt and let the function G1 be defined by G1(x)=ax(1g(t))p(t)dt,x[a,a+λ],xbg(t)p(t)dt,x[a+λ,b].(4)

Then aa+λf(t)p(t)dtabf(t)g(t)p(t)dt+abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx=1(n2)!ababG1(x)Tn1(x,s)dxf(n)(s)ds.(5)

Proof

Using identity [a,a+λ]f(t)p(t)dt[a,b]f(t)g(t)p(t)dt=[a,a+λ]f(t)(1g(t))p(t)dt(a+λ,b]f(t)g(t)p(t)dt

and integration by parts we have aa+λf(t)p(t)dtabf(t)g(t)p(t)dt=aa+λ[f(t)f(a+λ)][1g(t)]p(t)dt+a+λb[f(a+λ)f(t)]g(t)p(t)dt=aa+λax(1g(t))p(t)dtdf(x)a+λbxbg(t)p(t)dtdf(x)=abG1(x)df(x)=abG1(x)f(x)dx.

Applying identity (1) to f and replacing n with n − 1 we have f(x)=f(b)f(a)bai=0n3fi+1xbxi+2axi+2i+2!ba+1(n2)!abTn1(x,s)f(n)(s)ds.

Now we obtain abG1(x)f(x)dx=abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx+1(n2)!abG1(x)abTn1(x,s)f(n)(s)dsdx.(6)

After applying Fubini’s theorem on the last term in (6) we obtain (5). □

Theorem 2.2

Let f : I → ℝ be suct that f(n − 1) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤1. Let bλbp(t)dt=abg(t)p(t)dt and let the function G2 be defined by G2(x)=axg(t)p(t)dt,x[a,bλ],xb(1g(t))p(t)dt,x[bλ,b].(7)

Then abf(t)g(t)p(t)dtbλbf(t)p(t)dt+abG2(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx=1(n2)!ababG2(x)Tn1(x,s)dxf(n)(s)ds.(8)

Proof

Similarly as in the proof of Theorem 2.1, we use the identity [a,b]f(t)g(t)p(t)dt(bλ,b]f(t)p(t)dt=[a,bλ]f(t)g(t)p(t)dt(bλ,b]f(t)(1g(t))p(t)dt.

Now, using the above obtained identites we give generalization of Steffensen’s inequality for n-convex functions.

Theorem 2.3

Let f : I → ℝ be suct that f(n − 1) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤1. Let aa+λp(t)dt=abg(t)p(t)dt and let the function G1 be defined by (4). If f is nconvex and abG1(x)Tn1(x,s)dx0,s[a,b],(9)

then abf(t)g(t)p(t)dtaa+λf(t)p(t)dt+abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx.(10)

Proof

If the function f is n-convex, without loss of generality we can assume that f is n–times differentiable and f(n) ≥ 0 see [7, p. 16 and p. 293]. Now we can apply Theorem 2.1 to obtain (10). □

Theorem 2.4

Let f : I → ℝ be suct that f(n − 1) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤1. Let bλbp(t)dt=abg(t)p(t)dt and let the function G2 be defined by (7). If f is nconvex and abG2(x)Tn1(x,s)dx0,s[a,b],(11)

then abf(t)g(t)p(t)dtbλbf(t)p(t)dtabG2(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx.(12)

Proof

Similar to the proof of Theorem 2.3. □

Remark 2.5

If the integrals in (9) and (11) are nonpositive, then the reverse inequalities in (10) and (12) hold. Note that in this case for some odd n ≥ 3, functions Gi, i = 1, 2 are nonnegative so integrals in (9) and (11) are nonpositive. Hence, inequalities (10) and (12) are reversed.

3 Ostrowski-type inequalities

In this section we give the Ostrowski-type inequalities related to generalizations obtained in the previous section.

Here, the symbol Lp [a, b] (1 ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a, b] equipped with the norm fp=abftpdt1p

and L[a, b] denotes the space of essentially bounded functions on [a, b] with the norm f=esssupta,bft.

Theorem 3.1

Suppose that all assumptions of Theorem 2.1 hold. Assume also that (p, q) is a pair of conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p+1/q = 1 and f(n)Lp[a, b] for some n ≥ 2. Then we have aa+λf(t)p(t)dtabf(t)g(t)p(t)dt+abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx1(n2)!fnpabG1(x)Tn1(x,)dxq.(13)

The constant on the right-hand side of (13) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.

Proof

Let’s denote C(s)=1(n2)!abG1(x)Tn1(x,s)dx.

By taking the modulus of (5) and applying Hölder’s inequality we obtain aa+λf(t)p(t)dtabf(t)g(t)p(t)dt+abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx=abC(s)f(n)(s)dsfnpCq.

For the proof of the sharpness of the constant ∥Cq let us find a function f for which the equality in (13) is obtained.

For 1 < p < ∞ take f to be such that f(n)(s)=sgn C(s)C(s)1p1.

For p = ∞ take f(n)(s) = sgn C(s).

For p = 1 we prove that abC(s)f(n)(s)dtmaxt[a,b]C(t)abf(n)(s)ds(14)

is the best possible inequality. C(⋅) is a continuous function on [a, b] and so is |C(⋅)| . Suppose that |C(⋅)| attains its maximum at s0 ∈ [a, b]. First we assume that C(s0) > 0. For ε > 0 small enough we define fε(s) by fε(s)=0,ass0,1εn!(ss0)n,s0ss0+ε,1n!(ss0)n1,s0+εsb.

Then abC(s)fε(n)(s)ds=s0s0+εC(s)1εds=1εs0s0+εC(s)ds.

Now from the inequality (14) we have 1εs0s0+εC(s)ds1εC(s0)s0s0+εds=C(s0).

Since, limε01εs0s0+εC(s)ds=C(s0)

the statement follows. In the case C(s0) < 0, we define fε(s) by fε(s)=1n!(ss0ε)n1,ass0,1εn!(ss0ε)n,s0ss0+ε,0,s0+εsb,

and the rest of the proof is the same as above. □

Theorem 3.2

Suppose that all assumptions of Theorem 2.2 hold. Assume also (p, q) is a pair of conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p+1/q = 1. Let f(n)Lp[a, b] for some n ≥ 2. Then we have abf(t)g(t)p(t)dtbλbf(t)p(t)dt+abG2(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx1(n2)!fnpabG2(x)Tn1(x,)dxq.(15)

The constant on the right-hand side of (15) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.

Proof

Similar to the proof of Theorem 3.1. □

4 Generalizations related to the bounds for the Chebyshev functional

Let f, h : [a, b] → ℝ be Lebesgue integrable functions. We define the Chebyshev functional T(f, h) by T(f,h):=1baabf(t)h(t)dt1baabf(t)dt1baabh(t)dt.

In 1882 Chebyshev proved that Tf,h112fhba2,

provided that f, h exist and are continuous on [a, b] and ∥f = supt ∈ [a, b] | f(t) |. It also holds if f, h : [a, b] → are absolutely continuous and f, gL[a, b] while ∥f = ess supt ∈ [a, b]| f(t) |.

In 1934, Grüss in his paper [10] proved that Tf,h14MmNn,

provided that there exist the real numbers m, M, n, N such that mftM,nhtN

for a.e. t ∈ [a, b]. The constant 1/4 is the best possible.

In [11] Cerone and Dragomir proved the following theorems:

Theorem 4.1

Let f : [a, b] → ℝ be a Lebesgue integrable function and h : [a, b] → ℝ be an absolutely continuous function with (⋅ −a)(b – ⋅)[h]2L1[a, b]. Then we have the inequality |T(f,h)|12[T(f,f)]121baab(xa)(bx)[h(x)]2dx12.(16)

The constant 12 in (16) is the best possible.

Theorem 4.2

Assume that h : [a, b] → ℝ is monotonic nondecreasing on [a, b] and f : [a, b] → ℝ is absolutely continuous with fL[a, b]. Then we have the inequality |T(f,h)|12(ba)fab(xa)(bx)dh(x).1(17)

The constant 12 in (17) is the best possible.

In the sequel we use the above theorems to obtain some new bounds for integrals on the left hand side in the perturbed version of identities (5) and (8).

Firstly, let us denote Ωi(s)=abGi(x)Tn1(x,s)dx,i=1,2.(18)

Theorem 4.3

Let f : I → ℝ be suct that f(n) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b and (⋅ –a)(b – ⋅)[f(n+1)]2L1[a, b]. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤ 1. Let aa+λp(t)dt=abg(t)p(t)dt and let the functions G1 and Ω1 be defined by (4) and (18). Then aa+λf(t)p(t)dtabf(t)g(t)p(t)dt+abG1(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx+f(n1)(b)f(n1)(a)(ba)(n2)!abΩ1(s)ds=Sn1(f;a,b),(19)

where the remainder Sn1(f;a,b) satisfies the estimation Sn1(f;a,b)ba2(n2)!T(Ω1,Ω1)12ab(sa)(bs)[f(n+1)(s)]2ds12.(20)

Proof

Applying Theorem 4.1 for fΩ1 and hf(n) we obtain 1baabΩ1(s)f(n)(s)ds1baabΩ1(s)ds1baabf(n)(s)ds12T(Ω1,Ω1)121baab(sa)(bs)[f(n+1)(s)]2dt12.(21)

Now if we add 1(ba)(n2)!abΩ1(s)dsabf(n)(s)ds=f(n1)(b)f(n1)(a)(ba)(n2)!abΩ1(s)ds

to both sides of identity (5) and use inequality (21), we obtain representation (19) and bound (20). □

Similarly, using identity (8) we obtain the following result:

Theorem 4.4

Let f : I → ℝ be suct that f(n) is absolutely continuous for some n ≥ 2, I ⊂ ℝ an open interval, a, bI, a < b and (⋅ –a)(b – ⋅)[f(n+1)]2L1[a, b]. Let g, p : [a, b] → ℝ be integrable functions such that p is positive and 0 ≤ g ≤ 1. Let bλbp(t)dt=abg(t)p(t)dt and let the functions G2 and Ω2 be defined by (7) and (18). Then abf(t)g(t)p(t)dtbλbf(t)p(t)dt+abG2(x)f(b)f(a)bai=0n3f(i+2)(x)bxi+2axi+2i+2!badx+f(n1)(b)f(n1)(a)(ba)(n2)!abΩ2(s)ds=Sn2(f;a,b),(22)

where the remainder Sn2(f;a,b) satisfies the estimation Sn2(f;a,b)ba2(n2)!T(Ω2,Ω2)12ab(sa)(bs)[f(n+1)(s)]2ds12.

Proof

Similar to the proof of Theorem 4.3. □

The following Grüss-type inequalities also hold.

Theorem 4.5

Let f : I → ℝ be suct that f(n) is absolutely continuous for some n ≥ 2 and f(n+1) ≥ 0 on [a, b]. Let functions Ωi, i = 1, 2 be defined by (18).

  1. Let aa+λp(t)dt=abg(t)p(t)dt. Then we have representation (19) and the remainder Sn1(f;a,b) satisfies the bound Sn1(f;a,b)ba(n2)!Ω1f(n1)(b)+f(n1)(a)2f(n2)(b)f(n2)(a)ba.(23)

  2. Let bλbp(t)dt=abg(t)p(t)dt.. Then we have representation (22) and the remainder Sn2(f;a,b) satisfies the bound Sn2(f;a,b)ba(n2)!Ω2f(n1)(b)+f(n1)(a)2f(n2)(b)f(n2)(a)ba.

Proof

  1. Applying Theorem 4.2 for fΩ1, hf(n) we obtain 1baabΩ1(s)f(n)(s)ds1baabΩ1(s)ds1baabf(n)(s)ds12(ba)Ω1ab(sa)(bs)f(n+1)(s)ds.(24)

    Since ab(sa)(bs)f(n+1)(s)ds=ab[2s(a+b)]f(n)(s)ds=(ba)f(n1)(b)+f(n1)(a)2f(n2)(b)f(n2)(a)

    Using representation (5) and inequality (24) we deduce (23).

  2. Similar to the (a)-part.

References

  • [1]

    Aglić Aljinović A., Pečarić J., On some Ostrowski type inequalities via Montgomery identity and Taylors’s formula, TamkangJ. Math., 2005, 36 (3), 199–218. CrossrefGoogle Scholar

  • [2]

    Aglić Aljinović A., A note on generalization of weighted Čebyšev and Ostrowski inequalities, J. Math. Inequal., 2009, 3 (3), 409–416. Google Scholar

  • [3]

    Steffensen J. F., On certain inequalities between mean values and their application to actuarial problems, Skand. Aktuarietids. 1918, 82–97. Google Scholar

  • [4]

    Jakšetić J., Pečarić J., Steffensen’s inequality for positive measures, Math. Inequal. Appl., 2015, 18 (3), 1159–1170. Web of ScienceGoogle Scholar

  • [5]

    Mitrinović D. S., The Steffensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 1969, 247–273, 1–14. Google Scholar

  • [6]

    Atkinson K. E., An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989. Google Scholar

  • [7]

    Pečarić J. E., Proschan F., Tong Y. L., Convex functions, partial orderings, and statistical applications, Mathematics in science and engineering, 1992, 187, Academic Press. Google Scholar

  • [8]

    Popoviciu T., Sur l’approximation des fonctions convexes d’ordre superieur, Mathematica, 1934, 10, 49–54. Google Scholar

  • [9]

    Karlin S., Total Positivity, Stanford Univ. Press, Stanford, 1968. Google Scholar

  • [10]

    Grüss G., Über das Maximum des absoluten Betrages von 1baabfxgxdx1ba2abfxdxabgxdx, Math. Z.,1935, 39, 215-226. Google Scholar

  • [11]

    Cerone P., Dragomir S. S., Some new bounds for the Čebyšev functional in terms of the first derivative and applications, J. Math. Inequal., 2014, 8(1), 159–170. Google Scholar

About the article

Received: 2016-10-13

Accepted: 2018-02-07

Published Online: 2018-04-26


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 420–428, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0039.

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