# Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means

Authors: Bo-Yan Xi; Feng Qi
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The authors establish some new inequalities for differentiable convex functions, which are similar to the celebrated Hermite-Hadamard's integral inequality for convex functions, and apply these inequalities to construct inequalities for special means of two positive numbers.


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## Body

## 1. Introduction

In [1], the following Hermite-Hadamard type inequalities for differentiable convex functions were proved.Theorem 1.1 (see [1, Theorem 2.2]).
Letf:I∘⊆ℝ→ℝ be a differentiable mapping on I∘,   a,b∈I∘ with a<b. If |f′(x)| is convex on [a,b], then
(1.1)|f(a)+f(b)2-1b-a∫ab‍f(x)dx|≤(b-a)(|f′(a)|+|f′(b)|)8.Theorem 1.2 (see [1, Theorem 2.3]).
Letf:I∘⊆ℝ→ℝ be a differentiable mapping on I∘,a,b∈I∘ with a<b, and let p>1. If the new mapping |f′(x)|p/(p-1) is convex on [a,b], then
(1.2)|f(a)+f(b)2-1b-a∫ab‍f(x)dx|≤b-a2(p+1)1/p[|f′(a)|p/(p-1)+|f′(b)|p/(p-1)2](p-1)/p.In [2], the above inequalities were generalized as follows.Theorem 1.3 (see [2, Theorems 1 and 2]).
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and let q≥1. If |f′(x)|q is convex on [a,b], then
(1.3)|f(a)+f(b)2-1b-a∫ab‍f(x)dx|≤b-a4[|f′(a)|q+|f′(b)|q2]1/q,|f(a+b2)-1b-a∫ab‍f(x)dx|≤b-a4[|f′(a)|q+|f′(b)|q2]1/q.In [3], the above inequalities were further generalized as follows.Theorem 1.4 (see [3, Theorems 2.3 and 2.4]).
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and let p>1. If |f′(x)|p/(p-1) is convex on [a,b], then
(1.4)|1b-a∫ab‍f(x)dx-f(a+b2)|≤b-a16(4p+1)1/p×{[|f′(a)|p/(p-1)+3|f′(b)|p/(p-1)](p-1)/p+[3|f′(a)|p/(p-1)+|f′(b)|p/(p-1)](p-1)/p},|1b-a∫ab‍f(x)dx-f(a+b2)|≤b-a4(4p+1)1/p(|f′(a)|+|f′(b)|).In [4], an inequality similar to the above ones was given as follows.Theorem 1.5 (see [4, Theorem 3]).
Letf:[a,b]→ℝ be an absolutely continuous mapping on [a,b] whose derivative belongs to Lp[a,b]. Then
(1.5)|13[f(a)+f(b)2+2f(a+b2)]-1b-a∫ab‍f(x)dx|≤16[2q+1+13(q+1)]1/q(b-a)1/q‖f′‖p,
where (1/p)+(1/q)=1 and p>1.Recently, the following inequalities were obtained in [5].Theorem 1.6.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)|q for q≥1 is convex on [a,b], then
(1.6)|16[f(a)+f(b)+4f(a+b2)]-1b-a∫ab‍f(x)dx|≤b-a12[2q+1+13(q+1)]1/q[(3|f′(a)|q+|f′(b)|q4)1/q+(|f′(a)|q+3|f′(b)|q4)1/q],|16[f(a)+f(b)+4f(a+b2)]-1b-a∫abf(x)dx|≤5(b-a)72[(61|f′(a)|q+29|f′(b)|q90)1/q+(29|f′(a)|q+61|f′(b)|q90)1/q].In this paper, we will establish some new Hermite-Hadamard type integral inequalities for differentiable functions and apply them to derive some inequalities of special means.


## 2. Lemmas

For establishing new integral inequalities of Hermite-Hadamard type, we need the lemmas below.Lemma 2.1.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b. If f′∈L[a,b] and λ,μ∈ℝ, then
(2.1)λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫ab‍f(x)dx=b-a4∫01‍[(1-λ-t)f′(ta+(1-t)a+b2)+(μ-t)f′(ta+b2+(1-t)b)]dt.Proof.
Integrating by part and changing variable of definite integral yield(2.2)∫01(1-λ-t)f′(ta+(1-t)a+b2)dt=-2b-a[(1-λ-t)f(ta+(1-t)a+b2)|01+∫01‍f(ta+(1-t)a+b2)dt]=2b-a[λf(a)+(1-λ)f(a+b2)]-4(b-a)2∫a(a+b)/2‍f(x)dx.
Similarly, we have(2.3)∫01(μ-t)f′(ta+b2+(1-t)b)dt=-2b-a[(μ-t)f(ta+b2+(1-t)b)|01+∫01‍f(ta+b2+(1-t)b)dt]=2b-a[(1-μ)f(a+b2)+μf(b)]-4(b-a)2∫(a+b)/2bf(x)dx.
Adding these two equations leads to Lemma 2.1.Lemma 2.2.
Fors>0 and 0≤ξ≤1, one has
(2.4)∫01‍|ξ-t|sdt=ξs+1+(1-ξ)s+1s+1,∫01‍t|ξ-t|sdt=ξs+2+(s+1+ξ)(1-ξ)s+1(s+1)(s+2).Proof.
This follows from a straightforward computation of definite integrals.


## 3. Some Integral Inequalities of Hermite-Hadamard Type

Now we are in a position to establish some new integral inequalities of Hermite-Hadamard type for differentiable convex functions.The first main result is Theorem3.1.Theorem 3.1.
Letf:I⊆ℝ→ℝ be a differentiable function on I∘,  a,b∈I with a<b, 0≤λ,μ≤1, and f′∈L[a,b]. If |f′(x)|q for q≥1 is convex on [a,b], then
(3.1)|λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫abf(x)dx|≤b-a8(16)1/q{(1-2λ+2λ2)1-1/q×[(4-9λ+12λ2-2λ3)|f′(a)|q+(2-3λ+2λ3)|f′(b)|q]1/q+(1-2μ+2μ2)1-(1/q)×[(2-3μ+2μ3)|f′(a)|q+(4-9μ+12μ2-2μ3)|f′(b)|q]1/q}.Proof.
Forq>1, by Lemma 2.1, the convexity of |f′(x)|q on [a,b], and the noted Hölder's integral inequality, we have
(3.2)|λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫abf(x)dx|≤b-a4[∫01‍|1-λ-t||f′(ta+(1-t)a+b2)|dt+∫01‍|μ-t||f′(ta+b2+(1-t)b)|dt]≤b-a4{(∫01‍|1-λ-t|dt)1-1/q[∫01|1-λ-t|(1+t2|f′(a)|q+1-t2|f′(b)|q)dt]1/q+(∫01‍|μ-t|dt)1-1/q[∫01|μ-t|(t2|f′(a)|q+2-t2|f′(b)|q)dt]1/q}.
In virtue of Lemma 2.2, a direct calculation yields
(3.3)∫01|1-λ-t|(1+t2|f′(a)|q+1-t2|f′(b)|q)dt=12(|f′(a)|q+|f′(b)|q)∫01|1-λ-t|dt+12(|f′(a)|q-|f′(b)|q)∫01t|1-λ-t|dt=12(|f′(a)|q+|f′(b)|q)(12-λ+λ2)+112(|f′(a)|q-|f′(b)|q)[(1-λ)3+(1-λ)2(3-λ)]=112(4-9λ+12λ2-2λ3)|f′(a)|q+112(2-3λ+2λ3)|f′(b)|q,∫01|μ-t|(t2|f′(a)|q+2-t2|f′(b)|q)dt=12(|f′(a)|q-|f′(b)|q)∫01t|μ-t|dt+|f′(b)|q∫01|μ-t|dt=112[μ3+(1-μ)2(2+μ)](|f′(a)|q-|f′(b)|q)+(12-μ+μ2)|f′(b)|q=112(2-3μ+2μ3)|f′(a)|q+112(4-9μ+12μ2-2μ3)|f′(b)|q.
Substituting the above two equalities into the inequality (3.2) and utilizing Lemma 2.2 result in the inequality (3.1) for q>1.
Forq=1, from Lemmas 2.1 and 2.2 it follows that
(3.4)|λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫ab‍f(x)dx|≤b-a4[∫01|1-λ-t|(1+t2|f′(a)|+1-t2|f′(b)|)dt+∫01|μ-t|(t2|f′(a)|+2-t2|f′(b)|)dt]=b-a48[(4-9λ+12λ2-2λ3)|f′(a)|+(2-3λ+2λ2)|f′(b)|+(2-3μ+2μ2)|f′(a)|+(4-9μ+12μ2-2μ3)|f′(b)|],
which is just equivalent to (3.1) for q=1. Theorem 3.1 is proved.If takingλ=μ in Theorem 3.1, we derive the following corollary.Corollary 3.2.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, 0≤λ≤1, and f′∈L[a,b]. If |f′(x)|q is convex on [a,b] for q≥1, then
(3.5)|λ2[f(a)+f(b)]+(1-λ)f(a+b2)-1b-a∫abf(x)dx|≤b-a8(16)1/q(1-2λ+2λ2)1-(1/q)×{{[(2-3λ+2λ3)|f′(a)|q+(4-9λ+12λ2-2λ3)|f′(b)|q]1/q}[(4-9λ+12λ2-2λ3)|f′(a)|q+(2-3λ+2λ3)|f′(b)|q]1/q+[(2-3λ+2λ3)|f′(a)|q+(4-9λ+12λ2-2λ3)|f′(b)|q]1/q}.If lettingλ=μ=1/2,2/3,1/3, respectively, in Theorem 3.1, we can deduce the inequalities below.Corollary 3.3.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)|q is convex on [a,b] for q≥1, then
(3.6)|12[f(a)+f(b)2+f(a+b2)]-1b-a∫abf(x)dx|≤b-a16(112)1/q{[9|f′(a)|q+3|f′(b)|q]1/q+[3|f′(a)|q+9|f′(b)|q]1/q},|13[f(a)+f(b)+f(a+b2)]-1b-a∫abf(x)dx|≤5(b-a)72(1180)1/q[(74|f′(a)|q+16|f′(b)|q)1/q+(16|f′(a)|q+74|f′(b)|q)1/q],|16[f(a)+f(b)+4f(a+b2)]-1b-a∫ab‍f(x)dx|≤5(b-a)72(190)1/q{[61|f′(a)|q+29|f′(b)|q]1/q+[29|f′(a)|q+61|f′(b)|q]1/q}.If settingq=1 in Corollary 3.3, then one has the following.Corollary 3.4.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)| is convex on [a,b], then
(3.7)|12[f(a)+f(b)2+f(a+b2)]-1b-a∫abf(x)dx|≤b-a16(|f′(a)|+|f′(b)|),|13[f(a)+f(b)+f(a+b2)]-1b-a∫abf(x)dx|≤5(b-a)144(|f′(a)|+|f′(b)|),|16[f(a)+f(b)+4f(a+b2)]-1b-a∫abf(x)dx|≤5(b-a)72(|f′(a)|+|f′(b)|).The second main result is Theorem3.5.Theorem 3.5.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, 0 ≤ λ, μ  ≤ 1, and f′∈L[a,b]. If |f′(x)|q is convex on [a,b] for q≥1, then
(3.8)|λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫abf(x)dx|≤b-a4[12(q+1)(q+2)]1/q{{([(q+3-μ)(1-μ)q+1+(2q+4-μ)μq+1]|f′(b)|q)1/q}([(q+3-λ)(1-λ)q+1+(2q+4-λ)λq+1]|f′(a)|q+[(q+1+λ)(1-λ)q+1+λq+2]|f′(b)|q)1/q+([(q+1+μ)(1-μ)q+1+μq+2]|f′(a)|q+[(q+3-μ)(1-μ)q+1+(2q+4-μ)μq+1]|f′(b)|q)1/q}.Proof.
Forq>1, by the convexity of |f′(x)|q on [a,b], Lemma 2.1, and Hölder's integral inequality, it follows that
(3.9)|λf(a)+μf(b)2+2-λ-μ2f(a+b2)-1b-a∫abf(x)dx|≤b-a4[∫01|1-λ-t||f′(ta+(1-t)a+b2)|dt+∫01|μ-t||f′(ta+b2+(1-t)b)|dt]≤b-a4{(∫01dt)1-1/q[∫01|1-λ-t|q(1+t2|f′(a)|q+1-t2|f′(b)|q)dt]1/q+(∫01dt)1-1/q[∫01|μ-t|q(t2|f′(a)|q+2-t2|f′(b)|q)dt]1/q}=b-a4{[∫01|1-λ-t|q(1+t2|f′(a)|q+1-t2|f′(b)|q)dt]1/q+[∫01|μ-t|q(t2|f′(a)|q+2-t2|f′(b)|q)dt]1/q}.
By Lemma 2.2, we have
(3.10)∫01|1-λ-t|q(1+t2|f′(a)|q+1-t2|f′(b)|q)dt=12(|f′(a)|q+|f′(b)|q)∫01|1-λ-t|qdt+12(|f′(a)|q-|f′(b)|q)∫01t|1-λ-t|qdt=12(q+1)(|f′(a)|q+|f′(b)|q)[λq+1+(1-λ)q+1]+12(q+1)(q+2)(|f′(a)|q-|f′(b)|q)[(1-λ)q+2+(q+2-λ)λq+1]=12(q+1)(q+2){[(q+3-λ)(1-λ)q+1+(2q+4-λ)λq+1]|f′(a)|q+[(q+1+λ)(1-λ)q+1+λq+2]|f′(b)|q},∫01|μ-t|q(t2|f′(a)|q+2-t2|f′(b)|q)dt=12(q+1)(q+2){[(q+1+μ)(1-μ)q+1+μq+2]|f′(a)|q+[(q+3-μ)(1-μ)q+1+(2q+4-μ)μq+1]|f′(b)|q}.
Substituting these two equalities into the inequality (3.9) yields (3.8) for q>1.
Forq=1, the proof is the same as the deduction of (3.4). Thus, Theorem 3.5 is proved.As the derivation of corollaries of Theorem3.1, we can obtain the following corollaries of Theorem 3.5.Corollary 3.6.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, 0≤λ≤1, and f′∈L[a,b]. If |f′(x)|q is convex for q≥1 on [a,b], then
(3.11)|λf(a)+f(b)2+(1-λ)f(a+b2)-1b-a∫abf(x)dx|≤b-a4[12(q+1)(q+2)]1/q×{{([(q+3-λ)(1-λ)q+1+(2q+4-λ)λq+1]|f′(b)|q)1/q}([(q+3-λ)(1-λ)q+1+(2q+4-λ)λq+1]|f′(a)|q+[(q+1+λ)(1-λ)q+1+λq+2]|f′(b)|q)1/q+([(q+1+λ)(1-λ)q+1+λq+2]|f′(a)|q+[(q+3-λ)(1-λ)q+1+(2q+4-λ)λq+1]|f′(b)|q)1/q}.Corollary 3.7.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)|q is convex for q≥1 on [a,b], then
(3.12)|12[f(a)+f(b)2+f(a+b2)]-1b-a∫abf(x)dx|≤b-a8[14(q+1)(q+2)]1/q{[(3q+6)|f′(a)|q+(q+2)|f′(b)|q]1/q+[(q+2)|f′(a)|q+(3q+6)|f′(b)|q]1/q},|16[f(a)+f(b)+4f(a+b2)]-1b-a∫abf(x)dx|≤b-a12[118(q+1)(q+2)]1/q×{{[(1+(3q+4)2q+1)|f′(a)|q+(11+6q+(3q+8)2q+1)|f′(b)|q]1/q}[(11+6q+(3q+8)2q+1)|f′(a)|q+(1+(3q+4)2q+1)|f′(b)|q]1/q+[(1+(3q+4)2q+1)|f′(a)|q+(11+6q+(3q+8)2q+1)|f′(b)|q]1/q}.Corollary 3.8.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)| is convex on [a,b], then
(3.13)|12[f(a)+f(b)2+f(a+b2)]-1b-a∫abf(x)dx|≤b-a16(|f′(a)|+|f′(b)|),|16[f(a)+f(b)+4f(a+b2)]-1b-a∫abf(x)dx|≤2(b-a)27(|f′(a)|+32|f′(b)|).Corollary 3.9.
Letf:I⊆ℝ→ℝ be differentiable on I∘,  a,b∈I with a<b, and f′∈L[a,b]. If |f′(x)|q is convex for q≥1 on [a,b] and
(3.14)f(a)+f(b)2=f(a+b2)
then
(3.15)|f(a)+f(b)2-1b-a∫abf(x)dx|=|f(a+b2)-1b-a∫abf(x)dx|≤b-a16[14(q+1)(q+2)]1/q×{[(2q+5)|f′(a)|q+(2q+3)|f′(b)|q]1/q+[|f′(a)|q+(4q+7)|f′(b)|q]1/q+[(4q+7)|f′(a)|q+|f′(b)|q]1/q+[(2q+3)|f′(a)|q+(2q+5)|f′(b)|q]1/q}.
In particular, when q=1, one has
(3.16)|f(a)+f(b)2-1b-a∫abf(x)dx|=|f(a+b2)-1b-a∫abf(x)dx|≤b-a16(|f′(a)|+|f′(b)|).


## 4. Applications to Means

For two positive numbersa>0 and b>0, define
(4.1)A(a,b)=a+b2,G(a,b)=ab,H(a,b)=2aba+b,L(a,b)={[bs+1-as+1(s+1)(b-a)]1/s,a≠ba,a=b,I(a,b)={1e(bbaa)1/(b-a),a≠ba,a=b,Hω,s(a,b)={[as+(ab)s/2ω+bsω+2]1/s,s≠0ab,s=0
for 0≤ω<∞. These means are, respectively, called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive number a and b.Applying Theorem3.1 to f(x)=xs for s≠0 and x>0 leads to the following inequalities for means.Theorem 4.1.
Leta,b>0, q≥1, either s>1 and (s-1)q≥1 or s<0, then
(4.2)|A(as,bs)+As(a,b)2-Ls(a,b)|≤b-a16|s|q[14(q+1)(q+2)]1/q×{[(2q+5)a(s-1)q+(2q+3)b(s-1)q]1/q+[a(s-1)q+(4q+7)b(s-1)q]1/q+[(4q+7)a(s-1)q+b(s-1)q]1/q+[(2q+3)a(s-1)q+(2q+5)b(s-1)q]1/q}.
In particular, if s≥2 or s<0, then
(4.3)|A(as,bs)+As(a,b)2-Ls(a,b)|≤b-a8|s|A(as-1,bs-1).Takingf(x)=lnx for x>0 in Theorem 3.1 results in the following inequalities for means.Theorem 4.2.
Fora,b>0 and q≥1, one has
(4.4)|ln⁡G(a,b)+ln⁡A(a,b)2-ln⁡I(a,b)|≤b-a16[14(q+1)(q+2)]1/q[(2q+5aq+2q+3bq)1/q+(1aq+4q+7bq)1/q+(4q+7aq+1bq)1/q+(2q+3aq+2q+5bq)1/q].
In particular,
(4.5)|ln⁡A(a,b)+ln⁡G(a,b)2-ln⁡I(a,b)|≤b-a81H(a,b).Finally, we can establish an inequality for Heronian mean as follows.Theorem 4.3.
Forb>a>0, ω≥0, and s≥4 or 0≠s<1, one has
(4.6)|Hω,ss(a,b)H(as,bs)+Hω,(s/2)+1(s/2)+1(ba+ab,1)-Hω,ss(L(a2,b2)G2(a,b),1)|≤b-a4|s|[A(a,b)A(a2(s-1),b2(s-1))(ω+2)G2s(a,b)+ωA(a,b)A(as-2,bs-2)2(ω+2)Gs(a,b)].Proof.
Letf(x)=(xs+ωxs/2+1)/(ω+2) for x>0 and s∉(1,4). Then
(4.7)f′(x)=sω+2(xs-1+ω2xs/2-1),|f′(x)|′′=x(s/2)-3|s|ω+2[ω8(s-2)(s-4)+(s-1)(s-2)xs/2]≥0.
In virtue of Corollary 3.3, it follows that
(4.8)|12[f(b/a)+f(a/b)2+f(b/a+a/b2)]-1b/a-a/b∫a/bb/af(x)dx|=|12{12[bs+ω(ab)s/2+asas(ω+2)+as+ω(ab)s/2+bsbs(ω+2)]+(b/a+a/b)s+ω(b/a+a/b)s/2+1ω+2}-1ω+2[(b/a)s+1-(a/b)s+1(s+1)(b/a-a/b)+ω(b/a)s/2+1-(a/b)s/2+1(s/2+1)(b/a-a/b)+1]|=|Hω,ss(a,b)H(as,bs)+Hω,(s/2)+1(s/2)+1(ba+ab,1)-Hω,ss(L(a2,b2)G2(a,b),1)|.
On the other hand, we have
(4.9)b/a-a/b16[|f′(ab)|+|f′(ba)|]=b2-a216(ω+2)ab{|s|[(ba)s-1+(ab)s-1]+ω2|s|[(ba)s/2-1+(ab)s/2-1]}=(b-a)|s|4[A(a,b)A(a2(s-1),b2(s-1))(ω+2)G2s(a,b)+ωA(a,b)A(as-2,bs-2)2(ω+2)Gs(a,b)].
The proof is complete.Remark 4.4.
Some inequalities of Hermite-Hadamard type were also obtained in [6–9] by the authors.



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