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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Some new inequalities of Hermite-Hadamard type for s-convex functions with applications

Muhammad Adil Khan / Yuming Chu / Tahir Ullah Khan / Jamroz Khan
Published Online: 2017-12-09 | DOI: https://doi.org/10.1515/math-2017-0121

Abstract

In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.

Keywords: s-convex function; Hermite-Hadamard inequality; Hölder inequality; Trapezoidal formula

MSC 2010: 26D15; 26A51; 26D20

1 Introduction

Let I ⊆ ℝ be an interval. Then a real-valued function f : I → ℝ is said to be convex (concave) on I if the inequality f[λx+(1λ)y]()λf(x)+(1λ)f(y)

holds for all x, yI and λ ∈ [0, 1].

A large number of important properties and inequalities have been established for the class of convex (concave) functions since the convexity (concavity) was introduced more than a hundred years ago [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. But one of the most important inequalities for the convex (concave) function is the Hermite-Hadamard inequality [22], which can be stated as follows:

Theorem 1.1

Let I ⊆ ℝ be an interval and f : I → ℝ be a convex function on I. Then the inequality fa+b21baabf(x)dxf(a)+f(b)2(1)

holds for all a, bI with a < b. Both inequalities given in (1) hold in the reversed direction if f is concave on the interval I.

Recently, the improvements, generalizations, refinements and applications for the Hermite-Hadamard inequality have attracted the attention of many researchers [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

Hudzik and Maligranda [42] generalized the convex (concave) function to s-convex (concave) function.

Let s ∈ (0, 1]. Then the function f : [0, ∞)→ ℝ is said to be s-convex on the interval [0, ∞) if the inequality f[λx+(1λ)y]λsf(x)+(1λ)sf(y)(2)

takes place for all x, y ∈ [0, ∞) and λ∈ [0, 1]. f is said to be s-concave if inequality (2) is reversed.

We clearly see that the s-convexity (concavity) defined on [0, ∞) reduces to ordinary convexity (concavity) if s = 1.

In [43], the authors established the Hermite-Hadamard type inequality for the s-convex (concave) functions as follows.

Theorem 1.2

([43]). Let s ∈ (0, 1] and f : I ⊆ [0, ∞)→ ℝ be an s-convex function on I. Then the double inequality 2s1fa+b21baabf(x)dxf(a)+f(b)s+1(3)

holds for all a, bI with a < b. Both inequalities given in (3) hold in the reversed direction if f is s-concave on the interval I.

Both of the upper and lower bounds given in (3) for the s-convex (concave) functions were improved by Jagers in [44].

Hussian et al. [45] provided the Hermite-Hadamard type inequalities for the twice differentiable functions by using the following Lemma 1.3.

Lemma 1.3

([45]). Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I, and a, bI with a < b. Then the identity f(a)+f(b)21baabf(x)dx=(ba)2201t(1t)f[ta+(1t)b]dt

is valid if f″ ∈ L[a, b], where and in what follows I denotes the interior of the interval I.

Theorem 1.4

([45]). Let s ∈ (0, 1], q > 1, f : I ⊆ [0, ∞) → ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality f(a)+f(b)21baabf(x)dx(ab)22×6(q1)/q|f(a)|q+|f(b)|q(s+2)(s+3)1/q

holds if f″∈ L[a, b] and |f″|q is s-convex on [a, b].

Theorem 1.5

([45]). Let s ∈ (0, 1], p, q > 1 with 1/p + 1/q = 1, f : I ⊆ [0, ∞) → ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality f(a)+f(b)21baabf(x)dx2(s1q)/q(ba)2fa+b2Γ2(p+1)Γ(2p+2)p

holds if f″∈ L[a, b] and |f″|q is s-convex on [a, b], where Γ(x) = 0tx1etdt [46, 47, 48, 49, 50] is the classical gamma function.

In [51], Chu et al. discovered a new identity for the twice differentiable function.

Lemma 1.6

([51]). Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I, and a, bI with a < b. Then the identity [(xa)2(bx)2]f(x)+2(bx)f(b)+2(xa)f(a)2(ba)1baabf(t)dt=(xa)32(ba)01(1t2)f[ta+(1t)x]dt+(bx)32(ba)01(1t2)f[tb+(1t)x]dt(4)

holds for all x ∈ [a, b] if f″ ∈ L[a, b].

Next, we recall the definition of the fractional integrals [52].

Let 0 ≤ a < b, η > 0 and fL[a, b]. Then the left-sided and right-sided Riemann-Liouville fractional integrals Ja+ηf and Jbηf of order η are defined by Ja+ηf(x)=1Γ(η)ax(xt)η1f(t)dt,Jbηf(x)=1Γ(η)xb(tx)η1f(t)dt,

respectively.

We clearly see that Ja+0f(x)=Jb0f(x)=f(x). In particular, the fractional integral reduces to the classical integral if η = 1.

In [53], Set dealt with the fractional Ostrowski inequalities involving the Riemann-Liouville fractional integrals. More results and applications for the fractional derivatives and integrals can be found in the literature [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. Sarikaya et al. [67] and Özdemir et al. [68] established the Hermite-Hadamard type inequalities for the Riemann-Liouville fractional integrals as follows.

Theorem 1.7

([67]). Let η > 0, 0 ≤ a < b, f : [a, b] → (0, ∞) be a positive real-valued function with fL[a, b]. Then the double inequality fa+b2<Γ(η+1)2(ba)ηJa+ηf(b)+Jbηf(a)f(a)+f(b)2(5)

holds if f is convex on [a, b].

Theorem 1.8

([68]). Let η > 0, f : I ⊆ [0, ∞)→ ℝ be a differentiable mapping on I, and a, bI with a < b. Then the inequality (xa)ηf(a)+(bx)ηf(b)baΓ(η+1)baJxηf(a)+Jx+ηf(b)η(xa)η+1+(bx)η+1|f(x)|(s+1)(η+s+1)(ba)+1s+1Γ(η+1)Γ(s+1)Γ(η+s+3)(xa)η+1|f(a)|+(bx)η+1|f(b)|ba

is valid for all x ∈ [a, b] if f′ ∈ L(a, b) and |f′| is s-convex on [a, b].

Remark 1.9

We clearly see that inequality (5) reduces to inequality (1) if η = 1.

The following identity for the twice differentiable function, which was discovered by Chu et al. [51], will be used in the next section.

Lemma 1.10

([51]). Let η > 0, f : I ⊆ℝ→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the identity (xa)η+1(bx)η+1f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)ba=1baabf(t)dt+(xa)η+2(η+1)(ba)011tη+1f[ta+(1t)x]dt+(bx)η+2(η+1)(ba)011tη+1f[tb+(1t)x]dt(6)

holds for all x ∈ [a, b] if f″ ∈ L(a, b).

The main purpose of this paper is to establish several new Hermite-Hadamard type inequalities for s-convex (concave) functions via the classical and Riemann-Liouville fractional integrals, and provide the error estimatimations for the trapezoidal formula.

2 Hermite-Hadamard type inequalities for s-convex functions via classical integrals

Theorem 2.1

Let s ∈ (0, 1], f : I ⊆ [0, ∞) → ℝ be a twice differentiable function on I, and a, bI with a < b. Then the inequality [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dt(xa)32(ba)2(s+1)(s+3)|f(a)|+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|+(ba)32(ba)2(s+1)(s+3)|f(b)|+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|(7)

holds for all x ∈ [a, b] if |f″| is s-convex on [a, b] and f″∈ L[a, b].

Proof

It follows from (4) and the triangular inequality together with the s-convexity of |f″| that [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dt(xa)32(ba)01(1t2)f(ta+(1t)x)dt+(bx)32(ba)01(1t2)f(tb+(1t)x)dt(xa)32(ba)01(1t2)ts|f(a)|+(1t)s|f(x)|dt+(bx)32(ba)01(1t2)ts|f(b)|+(1t)s|f(x)|dt=(xa)32(ba)2(s+1)(s+3)|f(a)|+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|+(ba)32(ba)2(s+1)(s+3)|f(b)|+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|.

Corollary 2.2

Under the assumptions of Theorem 2.1, one has f(a)+f(b)21baabf(t)dt(ba)281(s+1)(s+3)(|f(a)|+|f(b)|)+(s+2)(s+3)2(s+1)(s+2)(s+3)fa+b2(ba)281(s+1)(s+3)+(s+2)(s+3)22s(s+1)(s+2)(s+3)(|f(a)|+|f(b)|).(8)

Proof

Let x = (a + b)/2, then the first inequality of (8) follows easily from (7). While the second inequality of (8) can be derived from the s-convexity of |f″|.□

Remark 2.3

Let s = 1, then the second inequality of (8) becomes f(a)+f(b)21baabf(t)dt(ba)224(|f(a)|+|f(b)|).

Theorem 2.4

Let s ∈ (0, 1], p, q > 1 with 1/p + 1/q = 1, f : I ⊆ [0, ∞) → ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dtΓ(1/2)Γ(p+1)2Γ(p+3/2)1/p(xa)3|f(a)|q+|f(x)|q1/q+(bx)3|f(b)|q+|f(x)|q1/q2(s+1)1/q(ba)(9)

holds for each x ∈ [a, b] if f″∈ L[a, b] and |f″|q is s-convex on [a, b].

Proof

From (4) together with the triangular and Hölder inequalities we clearly see that [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dt(xa)32(ba)01(1t2)pdt1/p01f(ta+(1t)x)qdt1/q+(bx)32(ba)01(1t2)pdt1/p01f(tb+(1t)x)qdt1/q.(10)

Making use of the s-convexity of |f″|q, we get 01f(ta+(1t)x)qdt01ts|f(a)|q+(1t)s|f(x)|qdt=|f(a)|q+|f(x)|qs+1,(11) 01f(tb+(1t)x)qdt01ts|f(b)|q+(1t)s|f(x)|qdt=|f(b)|q+|f(x)|qs+1.(12)

Note that 01(1t2)pdt1/p=Γ(1/2)Γ(p+1)2Γ(p+3/2)1/p.(13)

Therefore, inequality (9) follows easily from the (10)-(13).□

Corollary 2.5

Under the assumptions of Theorem 2.4, we have f(a)+f(b)21baabf(t)dt(ba)216(s+1)1/qΓ(1/2)Γ(p+1)2Γ(p+3/2)1/p×|f(a)|q+fa+b2q1/q+|f(b)|q+fa+b2q1/q(ba)216(s+1)1/qΓ(1/2)Γ(p+1)2Γ(p+3/2)1/p12s1/q+1+12s1/q|f(a)|+|f(b)|.(14)

Proof

Let x = (a + b)/2, then inequality (9) leads to the first inequality of (14) immediately. While the second inequality of (14) can be derived easily from the s-convexity of |f″|q and the elementary inequality k=1n(αk+βk)σk=1nαkσ+k=1nβkσ

for α1, β1, α2, β2, ⋯, αn, βn ≥ 0 and 0 ≤ σ ≤ 1.□

Remark 2.6

If s = 1, then the second inequality of (14) becomes f(a)+f(b)21baabf(t)dt(ba)232Γ(1/2)Γ(p+1)Γ(p+3/2)1/p121/q+321/q|f(a)|+|f(b)|.

Theorem 2.7

Let s ∈ (0, 1], p, q > 1 with 1/p + 1/q = 1, f : I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dtΓ(1/2)Γ(p+1)2Γ(p+3/2)1/p(xa)3fx+a2+(bx)3fx+b221+(1s)/q(ba)(15)

holds for any x ∈ [a, b] if f″ ∈ L[a, b] and |f″|q is s-concave on [a, b].

Proof

It follows from the s-concavity of |f″|q and (3) that 01|f(ta+(1t)x)|q2s1fx+a2q,(16) 01|f(tb+(1t)x)|q2s1fx+b2q.(17)

Therefore, inequality (15) follows from (4), (13), (16) and (17) together with the Hölder inequalities 01(1t2)f(ta+(1t)x)dt01(1t2)pdt1/p01f(ta+(1t)x)qdt1/q,01(1t2)f(tb+(1t)x)dt01(1t2)pdt1/p01f(tb+(1t)x)qdt1/q.

Corollary 2.8

Under the assumptions of Theorem 2.7, one has f(a)+f(b)21baabf(t)dt2s/q(ba)232Γ(1/2)Γ(p+1)Γ(p+3/2)1/pf3a+b4+fa+3b42(1+1/q)s(ba)232Γ(1/2)Γ(p+1)Γ(p+3/2)1/pfa+b2.(18)

Proof

Let x = (a + b)/2, then inequality (15) leads to the first inequality of (18) immediately. While the second inequality of (18) can be obtained by the s-concavity of |f″| due to the fact that |f″|q is s-concave, indeed, the s-concavity of |f″|q leads to the conclusion that ts|f(a)|+(1t)s|f(b)|qts|f(a)|q+(1t)s|f(b)|q|f(ta+(1t)b)|q.

Remark 2.9

Let s = 1, then from the second inequality of (18), we get f(a)+f(b)21baabf(t)dt21/q(ba)216Γ(1/2)Γ(p+1)Γ(p+3/2)1/pfa+b2.

Theorem 2.10

Let s ∈ (0, 1], q > 1, f : I ⊆ [0, ∞) → ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dt321/q(xa)32(s+1)(s+3)|f(a)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|q1/q3(ba)+321/q(bx)32(s+1)(s+3)|f(b)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|q1/q3(ba)(19)

holds for any x ∈ [a, b] if f″∈ L[a, b] and |f″|q is s-convex on [a, b].

Proof

It follows from (4) and the power-mean inequality that [(xa)2(bx)2]f(x)+2f(b)(bx)+2f(a)(xa)2(ba)1baabf(t)dt(xa)32(ba)01(1t2)|f(ta+(1t)x)|dt+(bx)32(ba)01(1t2)|f(tb+(1t)x)|dt(xa)32(ba)01(1t2)dt11/q01(1t2)|f(ta+(1t)x)|qdt1/q+(bx)32(ba)01(1t2)dt11/q01(1t2)|f(tb+(1t)x)|qdt1/q.(20)

From the s-convexity of |f″|q on [a, b] we get 01(1t2)|f(ta+(1t)x)|qdt01(1t2)ts|f(a)|q+(1t)s|f(x)|qdt=2(s+1)(s+2)|f(a)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|q(21)

and 01(1t2)|f(tb+(1t)x)|qdt01(1t2)ts|f(b)|q+(1t)s|f(x)|qdt=2(s+1)(s+2)|f(b)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)|f(x)|q.(22)

Note that 01(1t2)dt=23.(23)

Therefore, inequality (19) follows from (20)-(23).□

Corollary 2.11

Under the assumptions of Theorem 2.10, one has f(a)+f(b)21baabf(t)dt321/q(ba)2242(s+1)(s+3)|f(a)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)fa+b2q1/q+321/q(ba)2242(s+1)(s+3)|f(b)|q+(s+2)(s+3)2(s+1)(s+2)(s+3)fa+b2q1/q321/q(ba)2242(s+1)(s+3)+(s+2)(s+3)22s(s+1)(s+2)(s+3)1/qf(a)+f(b)+321/q(ba)224(s+2)(s+3)22s(s+1)(s+2)(s+3)1/qf(a)+f(b).(24)

Proof

Let x = (a + b)/2, then the first inequality of (24) can be obtained from inequality (19) immediately. While the second inequality of (24) follows from the s-convexity of |f″|q and the inequality k=1n(αk+βk)σk=1nαkσ+k=1nβkσ

for α1, β1, α2, β2, ⋯, αn, βn ≥ 0 and 0 ≤ σ ≤ 1.□

Remark 2.12

If s = 1, the inequality (24) leads to f(a)+f(b)21baabf(t)dt321/q(ba)22411241/q+5241/qf(a)+f(b).

3 Hermite-Hadamard type inequalities for fractional integrals

Theorem 3.1

Let s ∈ (0, 1], η > 0, f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2(η+1)(ba)η+1(s+1)(s+η+2)|f(a)|+1s+1Γ(η+2)Γ(s+1)Γ(η+s+3)|f(x)|+(bx)η+2(η+1)(ba)η+1(s+1)(s+η+2)|f(b)|+1s+1Γ(η+2)Γ(s+1)Γ(η+s+3)|f(x)|.(25)

holds for all x ∈ [a, b] if f″ ∈ L[a, b] and |f″| is s-convex on [a, b].

Proof

It follows from (6) and the triangle inequality together with the s-convexity of |f″| that [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2(η+1)(ba)011tη+1|f(ta+(1t)x)|dt+(bx)η+2(η+1)(ba)011tη+1|f(tb+(1t)x)|dt(xa)η+2(η+1)(ba)011tη+1[ts|f(a)|+(1t)s|f(x)|]dt+(bx)η+2(η+1)(ba)011tη+1[ts|f(b)|+(1t)s|f(x)|]dt=(xa)η+2(η+1)(ba)η+1(s+1)(s+η+2)|f(a)|+1s+1Γ(η+2)Γ(s+1)Γ(η+s+3)|f(x)|+(bx)η+2(η+1)(ba)η+1(s+1)(s+η+2)|f(b)|+1s+1Γ(η+2)Γ(s+1)Γ(η+s+3)|f(x)|.

Remark 3.2

Let η = 1 in Theorem 3.1, then we get inequality (7) given in Theorem 2.1.

Corollary 3.3

Under the assumptions of Theorem 3.1, we have ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+12η+2(η+1)(η+1)(|f(a)|+|f(b)|)(s+1)(s+η+1)+21s+1Γ(s+1)Γ(η+2)Γ(s+η+3)fa+b2(ba)η+1(|f(a)|+|f(b)|)2η+2(η+1)η+1(s+1)(s+η+1)+21s1s+1Γ(s+1)Γ(η+2)Γ(s+η+3).(26)

Proof

Let x = (a+b)/2, then inequality (25) leads to the first inequality of (26). While the second inequality of (26) can be derived from the s-convexity of |f″|. □

Remark 3.4

Let s = 1, then the second inequality of (26) leads to ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+1(|f(a)|+|f(b)|)2η+3(η+1)η+1η+22Γ(η+2)Γ(η+4)+1.

Theorem 3.5

Let η > 0, s ∈ (0, 1], p, q > 1 with 1/p+1/q = 1, M = Γ(1+p)Γ(1/(η+1))/[(η+1)Γ(1+p+1/(η+1))], f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2|f(a)|q+|f(x)|q1/q+(bx)η+2|f(b)|q+|f(x)|q1/q(η+1)(s+1)1/q(ba)M1/p(27)

holds for all x ∈ [a, b] if f″ ∈ L[a, b] and |f″|q is s-convex on [a, b].

Proof

It follows from (6) and the Hölder inequality together with the s-convexity of |f″|q that [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2(η+1)(ba)011tη+1pdt1/p01f(ta+(1t)x)qdt1/q+(bx)η+2(η+1)(ba)011tη+1pdt1/p01f(tb+(1t)x)qdt1/q,(28)

and inequalities (11) and (12) hold.

Note that 011tη+1pdt=1η+101u1/(η+1)1(1u)pdu=M.(29)

Therefore, inequality (27) follows from (11), (12), (28) and (29). □

Remark 3.6

Let η = 1, then Theorem 3.5 leads to Theorem 2.4.

Let x = (a+b)/2, then the following Corollary 3.7 can be obtained from (27) and the s-convexity of |f″|q together with the inequality k=1n(αk+βk)σk=1nαkσ+k=1nβkσ

for α1, β1, α2, β2, ⋯, αn, βn ≥ 0 and 0 ≤σ ≤ 1.

Corollary 3.7

Under the assumptions of Theorem 3.5, we have the inequality as follows: ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+1M1/p2η+2(η+1)(s+1)1/q|f(a)|q+fa+b2q1/q+|f(b)|q+fa+b2q1/q(ba)η+1(|f(a)|+|f(b)|)M1/p2η+2(η+1)(s+1)1/q12s1/q+1+12s1/q.(30)

Remark 3.8

Let s = 1, then the second inequality of (30) leads to ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+1(|f(a)|+|f(b)|)M1/p2η+2+1/q(η+1)121/q+321/q.

Theorem 3.9

Let η > 0, s ∈ (0, 1], p, q > 1 with 1/p+1/q = 1, M = Γ(1+p)Γ(1/(η+1))/[(η+1)Γ(1+p+1/(η+1))], f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2fx+a2+(bx)η+2fx+b2(η+1)2(1s)/q(ba)M1/p(31)

holds for all x ∈ [a, b] if f″ ∈ L[a, b] and |f″|q is s-concave on [a, b].

Proof

Theorem 3.9 follows easily from (16), (17), (28) and (29). □

Remark 3.10

Let η = 1, then Theorem 3.9 becomes Theorem 2.7.

Letting x = (a+b)/2 and making use of the s-concavity of |f″|, then inequality (31) leads to Corollary 3.11 immediately.

Corollary 3.11

Under the assumptions of Theorem 3.9, one has ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b22(s1)/q(ba)η+1M1/p2η+2(η+1)f3a+b4+fa+3b42[s(q+1)1]/q(ba)η+1M1/p2η+2(η+1)fa+b2.(32)

Remark 3.12

Let s = 1 in the second inequality of (32), then we get ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+1M1/p2η+1(η+1)fa+b2.

Theorem 3.13

Let η > 0, s ∈ (0, 1], q > 1, f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, and a, bI with a < b. Then the inequality [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2(η+1)|f(a)|q(s+1)(s+η+2)+1s+1Γ(s+1)Γ(η+2)Γ(s+η+3)|f(x)|q1/q(η+2)(ba)η+2η+11/q+(bx)η+2(η+1)|f(b)|q(s+1)(s+η+2)+1s+1Γ(s+1)Γ(η+2)Γ(s+η+3)|f(x)|q1/q(η+2)(ba)η+2η+11/q(33)

holds for all x ∈ [a, b] if f″ ∈ L[a, b] and |f″|q is s-convex on [a, b].

Proof

By use of (6) and the power-mean inequality, we have [(xa)η+1(bx)η+1]f(x)+(η+1)f(b)(bx)+(η+1)f(a)(xa)(η+1)(ba)1baabf(t)dt(xa)η+2(η+1)(ba)011tη+1|f(ta+(1t)x)|dt+(bx)η+2(η+1)(ba)011tη+1|f(tb+(1t)x)|dt(xa)η+2(η+1)(ba)011tη+1dt11/q011tη+1|f(ta+(1t)x)|qdt1/q+(bx)η+2(η+1)(ba)011tη+1dt11/q011tη+1|f(tb+(1t)x)|qdt1/q.(34)

It follows from the s-convexity of |f″|q on [a, b] that 011tη+1|f(ta+(1t)x)|qdt(η+1)|f(a)|q(s+1)(s+η+2)+1s+1Γ(s+1)Γ(η+2)Γ(s+η+3)|f(x)|q,(35) 011tη+1|f(tb+(1t)x)|qdt(η+1)|f(b)|q(s+1)(s+η+2)+1s+1Γ(s+1)Γ(η+2)Γ(s+η+3)|f(x)|q.(36)

Note that 011tη+1dt=η+1η+2.(37)

Therefore, Theorem 3.13 follows from (34)-(37). □

Remark 3.14

Let η = 1, then Theorem 3.13 becomes Theorem 2.10.

Let x = (a+b)/2, then from (33) and the s-convexity of |f″|q we get Corollary 3.15 immediately.

Corollary 3.15

Under the assumptions of Theorem 3.13, one has ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+12η+2(η+2)η+2η+11/qM1|f(a)|q+M2fa+b2q1/q+(ba)η+12η+2(η+2)η+2η+11/qM1|f(b)|q+M2fa+b2q1/q(ba)η+12η+2(η+2)η+2η+11/qM11/q+2M22s1/q(|f(a)|+|f(b)|),(38)

where M1 = (η+1)/[(s+1)(s+η+2)] and M2 = 1/(s+1)−Γ(s+1)Γ(η+2)/Γ(s+η+3).

Remark 3.16

Let s = 1, then inequality (38) leads to ba2η1f(a)+f(b)2Γ(η+1)baJa+ηfa+b2+Jbηfa+b2(ba)η+12η+2(η+2)η+2η+11/qη+12η+61/q+214Γ(η+2)2Γ(η+4)1/q(|f(a)|+|f(b)|).

4 Applications to trapezoidal formula

Let d be a division a = x0 < x1 < x2 < ⋯ < xn−1 < xn = b of the interval [a, b] and consider the quadrature formula abf(x)dx=T(f,d)+E(f,d),

where T(f,d)=i=0n1f(xi)+f(xi+1)2(xi+1xi)

is the trapezoidal version and E(f,d) denotes the associated approximation error.

Theorem 4.1

Let s ∈ (0, 1], f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, a, bI with a < b and d be a division a = x0 < x1 < x2 < ⋯ < xn−1 < xn = b of the interval [a, b]. Then the inequality |E(f,d)|1(s+1)(s+3)+(s+2)(s+3)22s(s+1)(s+2)(s+3)i=0n1(xi+1xi)3[|f(xi)|+|f(xi+1)|]8

holds if f″ ∈ L[a, b] and |f″| is s-convex on [a, b].

Proof

Let i ∈ {0, 1, 2, ⋯, n−1}, then applying Corollary 2.2 on the interval [xi, xi+1] we get f(xi)+f(xi+1)21xi+1xixixi+1f(x)dx(xi+1xi)281(s+1)(s+3)+(s+2)(s+3)22s(s+1)(s+2)(s+3)[|f(xi)|+|f(xi+1)|].

Therefore, |E(f,d)|=abf(x)dxT(f,d)=i=0n1xixi+1f(x)dxf(xi)+f(xi+1)2(xi+1xi)i=0n1xixi+1f(x)dxf(xi)+f(xi+1)2(xi+1xi)1(s+1)(s+3)+(s+2)(s+3)22s(s+1)(s+2)(s+3)×i=0n1(xi+1xi)3[|f(xi)|+|f(xi+1)|]8.

Making use of the similar arguments as in Theorem 4.1, we can get Theorems 4.2-4.4 from the Corollaries 2.5, 2.8 and 2.11 immediately.

Theorem 4.2

Let s ∈ (0, 1], p, q > 1 with 1/p+1/q = 1, f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, a, bI with a < b and d be a division a = x0 < x1 < x2 < ⋯ < xn−1 < xn = b of the interval [a, b]. Then the inequality |E(f,d)|Γ(1/2)Γ(p+1)2Γ(p+3/2)1/p12s1/q+1+12s1/q×i=0n1(xi+1xi)3[|f(xi)|+|f(xi+1)|]16(s+1)1/q

holds if f″ ∈ L[a, b] and |f″|q is s-convex on [a, b].

Theorem 4.3

Let s ∈ (0, 1], p, q > 1 with 1/p+1/q = 1, f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, a, bI with a < b and d be a division a = x0 < x1 < x2 < ⋯ < xn−1 < xn = b of the interval [a, b]. Then the inequality |E(f,d)|2(q+1)s/q32Γ(1/2)Γ(p+1)Γ(p+3/2)1/pi=0n1(xi+1xi)3fxi+1+xi2

holds if f″ ∈ L[a, b] and |f″|q is s-concave on [a, b].

Theorem 4.4

Let s ∈ (0, 1], q > 1, f: I ⊆ [0, ∞)→ ℝ be a twice differentiable mapping on I, a, bI with a < b and d be a division a = x0 < x1 < x2 < ⋯ < xn−1 < xn = b of the interval [a, b]. Then the inequality |E(f,d)|321/q2(s+1)(s+3)+Δ(s)1/q+Δ1/q(s)×i=0n1(xi+1xi)3[|f(xi)|+|f(xi+1)|]24

holds if f″ ∈ L[a, b] and |f″|q is s-convex on [a, b], where Δ(s)=(s+2)(s+3)22s(s+1)(s+2)(s+3).

5 Conclusion

In the article, we present several new Hermite-Hadamard type inequalities and error estimatimations for the trapezoidal formula involving the s-convex and s-concave functions for the classical and Riemann-Liouville fractional integrals.

Acknowledgement

The authors express their gratitude to the referees for very helpful and detailed comments and suggestions, which have significantly improved the presentation of this paper. The research was supported by the Natural Science Foundation of China under Grants 61673169, 61374086, 11371125, 11401191, 11701176, the Tianyuan Special Funds of the National Natural Science Foundation of China under Grant 11626101 and the Natural Science Foundation of the Department of Education of Zhejiang Province under GrantY201635325.

References

  • [1]

    Borwein J. M., Vanderwerff J. D., Convex Functions: Constructions, Characterizations and Counterexamples, Cambrige University Press, Cambridge, 2010 Google Scholar

  • [2]

    Zhang X. M., Chu Y. M., Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math., 2010, 40(3), 1061–1068 CrossrefGoogle Scholar

  • [3]

    Wang M. K., Qiu S. L., Chu Y. M., Jiang Y. P., Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221–229 CrossrefGoogle Scholar

  • [4]

    Chu Y. M., Xia W. F., Zhang X. H., The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications, J. Multivariate Anal., 2012, 105, 412–421 CrossrefGoogle Scholar

  • [5]

    Chu Y. M., Wang M. K., Qiu S. L., Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Adac. Sci. Math. Sci., 2012, 122(1), 41–51 CrossrefGoogle Scholar

  • [6]

    Qiu S. L., Qiu Y. F., Wang M. K., Chu Y. M., Hölder mean inequalities for the generalized Grötzsch ring and Hersch-Pfluger distortion functions, Math. Inequal. Appl., 2012, 15(1), 237–245 Google Scholar

  • [7]

    Wang M. K., Chu Y. M., Qiu S. L., Jiang Y. P., Bounds for the perimeter of an ellipse, J. Approx. Theory, 2012, 164(7), 928–937 CrossrefGoogle Scholar

  • [8]

    Chu Y. M., Wang M. K., Optimal Lehmer mean bounds for the Toader mean, Results Math., 2012, 61(3-4), 223–229 CrossrefGoogle Scholar

  • [9]

    Chu Y. M., Wang M. K., Jiang Y. P., Qiu S. L., Concavity of the complete elliptic integrals of the second kind with respect to Hölder means, J. Math. Anal. Appl., 2012, 395(2), 637–642 CrossrefGoogle Scholar

  • [10]

    Chu Y. M., Qiu Y. F., Wang M. K., Hölder mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 2012, 23(7), 521–527 CrossrefGoogle Scholar

  • [11]

    Xia W. F., Zhang X. H., Wang G. D., Chu Y. M., Some properties for a class of symmetric functions with applications, Indian J. Pure Appl. Math., 2012, 43(3), 227–249 CrossrefGoogle Scholar

  • [12]

    Chu Y. M., Wang M. K., Wang Z. K., Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means, Math. Inequal. Appl., 2012, 15(2), 415–422 Google Scholar

  • [13]

    Ma X. Y., Wang M. K., Zhong G. H., Qiu S. L., Chu Y. M., Some inequalities for the generalized distortion functions, Math. Inequal. Appl., 2012, 15(4), 941–954 Google Scholar

  • [14]

    Wang M. K., Chu Y. M., Asymptotical bounds for complete elliptic integrals of the second kind, J. Math. Anal. Appl., 2013, 402(1), 119–126 CrossrefGoogle Scholar

  • [15]

    Wang M. K., Chu Y. M., Song Y. Q., Asymptotical formulas for Gaussian and generalized hypergeometric functions, Appl. Math. Comput., 2016, 276, 44–60 CrossrefGoogle Scholar

  • [16]

    Wang M. K., Chu Y. M., Refinements of transformation inequalities for zero-balanced hypergeometric functions, Acta Math. Sci., 2017, 37B(3), 602–622 Google Scholar

  • [17]

    Yang Z. H., Chu Y. M., A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 2017, 20(3), 729–735 Google Scholar

  • [18]

    Yang Z. H., Qian W. M., Chu Y. M., Zhang W., Monotonicity rule for the quotient of two functions and its applications, J. Inequal. Appl., 2017, 2017, Article 106, 13 pages Google Scholar

  • [19]

    Yang Z. H., Qian W. M., Chu Y. M., On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages Google Scholar

  • [20]

    Wang M. K., Li Y. M., Chu Y. M., Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., DOI: 10.1007/s11139-017-9888-3 Google Scholar

  • [21]

    Yang Z. H., Qian W. M., Chu Y. M., Zhang W., On approximating the error function, Math. Inequal. Appl. (in press) Google Scholar

  • [22]

    Hadamard J., Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl., 1893, 58, 171–215 Google Scholar

  • [23]

    Wu S. H., On the weighted genealization of the Hermite-Hadamard inequality and its applications, Rocky Mountain J. Math., 2009, 39(5), 1741–1749 CrossrefGoogle Scholar

  • [24]

    Zhang X. M., Chu Y. M., Zhang X. H., The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl., 2010, 2010, Article ID 507560, 11 pages Google Scholar

  • [25]

    Wang M. K., Chu Y. M., Qiu S. L., Jiang Y. P., Convexity of the complete elliptic integrals of the first kind with respect to Hölder mean, J. Math. Anal. Appl., 2012, 388(2), 1141–1146 CrossrefGoogle Scholar

  • [26]

    Chu Y. M., Wang M. K., Qiu S. L., Jiang Y. P., Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 2012, 63(7), 1177–1184 CrossrefGoogle Scholar

  • [27]

    Wang M. K., Wang Z. K., Chu Y. M., An optimal double inequality between geometric and identric means, Appl. Math. Lett., 2012, 25(3), 471–475 CrossrefGoogle Scholar

  • [28]

    Chu Y. M., Wang M. K., Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal., 2012, 2012, Article ID 830585, 11 pages Google Scholar

  • [29]

    Chu Y. M., Hou S. W., Sharp bounds for Seiffert mean in terms of contraharmonic mean, Abstr. Appl. Anal., 2012, 2012, Article ID 425175, 6 pages Google Scholar

  • [30]

    Li Y. M., Long B. Y., Chu Y. M., Sharp bounds by the power mean for the generalized Heronian mean, J. Inequal. Appl., 2012, 2012, Article 129, 9 pages Google Scholar

  • [31]

    Gong W. M., Song Y. Q., Wang M. K., Chu Y. M., A sharp double inequality between Seiffert, arithmetic, and geometric means, Abstr. Appl. Anal., 2012, 2012, Article ID 684834, 7 pages Google Scholar

  • [32]

    Xia W. F., Janous W., Chu Y. M., The optimal convex combination bounds of arithmetic and harmonic means in terms of power mean, J. Math. Inequal., 2012, 6(2), 241–248 Google Scholar

  • [33]

    Li Y. M., Long B. Y., Chu Y. M., Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean, J. Math. Inequal., 2012, 6(4), 567–577 Google Scholar

  • [34]

    İşcan İ., Wu S. H., Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 2014, 238, 237–244 CrossrefGoogle Scholar

  • [35]

    Chen F. X., Wu S. H., Fejér and Hermite-Hadamard type inequalities for harmonically convex functions, J. Appl. Math., 2014, 2014, Article ID 386806, 6 pages Google Scholar

  • [36]

    Yang Z. H., Chu Y. M., Wang M. K., Monotonicity criterion for quotient of power series with applications, J. Math. Anal. Appl., 2015, 428(1), 586–604 Google Scholar

  • [37]

    Chen F. X., Wu S. H., Several completementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci Appl., 2016, 9(2), 705–716 CrossrefGoogle Scholar

  • [38]

    Yang Z. H., Chu Y. M., Zhang W., Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean, J. Inequal. Appl., 2016, 2016, Article 176, 10 pages Google Scholar

  • [39]

    Yang Z. H., Chu Y. M., Zhang W., Accurate approximations for the complete elliptic integral of the second kind, J. Math. Anal. Appl., 2016, 438(1), 875–888 CrossrefGoogle Scholar

  • [40]

    Chu Y. M., Adil Khan M., Ali T., Dragomir S. S., Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017, 2017, Article 93, 12 pages Google Scholar

  • [41]

    Yang Z. H., Chu Y. M., Zhang X. H., Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind, J. Nonlinear Sci. Appl., 2017, 10(3), 939–936 Google Scholar

  • [42]

    Hudzik H., Maligranda L., Some remarks on s-convex functions, Aequationes Math., 1994, 48(1), 100–111 CrossrefGoogle Scholar

  • [43]

    Dragomir S. S., Fitzpatrick S., The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math., 1999, 32(4), 687–696 Google Scholar

  • [44]

    Jagers B., On Hadamard-type inequality for s-convex functions, avaliable online at http://www3.cs.utwente.nl/~jagersaa/alphaframes/Alpha.pdf 

  • [45]

    Hussain S., Bhatti M. I., Iqbal M., Hadamard-type inequalities for s-convex function I, Punjab Univ. J. Math., 2009, 41, 51–60 Google Scholar

  • [46]

    Zhao T. H., Chu Y. M., Jiang Y. P., Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl., 2009, 2009, Article ID 728612, 13 pages Google Scholar

  • [47]

    Zhang X. M., Chu Y. M., A double inequalities for gamma function, J. Inequal. Appl., 2009, 2009, Article ID 503782, 7 pages Google Scholar

  • [48]

    Zhao T. H., Chu Y. M., A class of logarithmically completely monotonic functions associated with a gamma function, J. Inequal. Appl., 2010, 2010, Article ID 392431, 11 pages Google Scholar

  • [49]

    Zhao T. H., Chu Y. M., Wang H., Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011, 2011, Article ID 896483, 13 pages Google Scholar

  • [50]

    Yang Z. H., Chu Y. M., Zhang X. H., Sharp bounds for psi function, Appl. Math. Comput., 2015, 268,1055-1063 CrossrefGoogle Scholar

  • [51]

    Chu Y. M., Adil Khan M., Khan T. U., Ali T., Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 2016, 9(6), 4305–4316 CrossrefGoogle Scholar

  • [52]

    Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999 Google Scholar

  • [53]

    Set E., New inequalities of Ostrowski type for mappings whose derivaties are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 2012, 63(7), 1147–1154 CrossrefGoogle Scholar

  • [54]

    Cheng J. F., Chu Y. M., Solution to the linear fractional differential equation using Adomian decompoisition method, Math. Probl. Eng., 2011, 2011, Article ID 587086, 14 pages Google Scholar

  • [55]

    Cheng J. F., Chu Y. M., On the fractional difference equations of order (2, q), Abstr. Appl. Anal., 2011, 2011, Article ID 497259, 16 pages Google Scholar

  • [56]

    Cheng J. F., Chu Y. M., Fractional difference equations with real variable, Abstr. Appl. Anal., 2012, 2012, Article ID 918529, 24 pages Google Scholar

  • [57]

    Zhu C., Fečkan M., Wang J. R., Fractional integral inequalities for defferentiable convex mappings and applicaions to special means and a midpoint formula, J. Appl. Math. Stat. Inform., 2012, 8(2), 21–28 Google Scholar

  • [58]

    İşcan İ., New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., 2013, 2013, Article 491, 15 pages Google Scholar

  • [59]

    Srivastava H. M., Agarwal P., Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math., 2013, 8(2), 333–345 Google Scholar

  • [60]

    Wang J. R., Deng J. H., Fečkan M., Hermite-Hadamard-type inequalities for r-convex functions based on the use of Riemann-Liouville fractional integrals, Ukrainian Math., 2013, 65(2), 193–211 CrossrefGoogle Scholar

  • [61]

    Wang J. R., Li X. Z., Fečkan M., Zhou Y., Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 2013, 92(11), 2241–2253 CrossrefGoogle Scholar

  • [62]

    Yang X. J., Baleanu D., Srivastava H. M., Tenreiro Machado J. A., On local fractional continuous wavelet transform, Abstr. Appl. Anal., 2013, 2013, Article ID 725416, 5 pages Google Scholar

  • [63]

    Yang A. M., Chen Z. S., Srivastava H. M., Yang X. J., Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving logal fractional derivative operators, Abstr. Appl. Anal., 2013, 2013, Article ID 259125, 6 pages Google Scholar

  • [64]

    Yang X. J., Srivastava H. M., He J. H., Baleanu D., Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett. A, 2013, 377(28-30), 1696–1700 CrossrefGoogle Scholar

  • [65]

    Wang J. R., Li X. Z., Zhu C., Refinements of Hermite-Hadamard type inequalities involving fractional integrals, Bull. Belg. Math. Soc. Simon. Stevin, 2013, 20(4), 655–666 Google Scholar

  • [66]

    Adil Khan M., Khurshid Y., Ali T., Hermite-Hadamard inequality for fractional integrals via η-convex functions, Acta Math. Univ. Comenian, 2017, 86(1), 153–164 Google Scholar

  • [67]

    Sarikaya M. Z., Set E., Yaldiz H., Başak N., Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 2013, 57(9), 2403–2407 CrossrefGoogle Scholar

  • [68]

    Özdemir M. E., Avci M., Kavurmaci H., Hermite-Hadamard type inequalities for s-concave functions via fractional integrals, arXiv: 1202.0380v1 [math.FA] Google Scholar

About the article

Received: 2016-08-28

Accepted: 2017-10-11

Published Online: 2017-12-09


Competing interests: The authors declare that they have no competing interests.

Authors’ contributions: All authors contributed equally to the manuscript, and they read and approved the final manuscript.


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1414–1430, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0121.

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