Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 980438 | 14 pages | https://doi.org/10.1155/2012/980438

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

Academic Editor: Lars Diening
Received21 Feb 2012
Accepted21 May 2012
Published30 Jul 2012

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.

1. Introduction

In [1], the following Hermite-Hadamard type inequalities for differentiable convex functions were proved.

Theorem 1.1 (see [1, Theorem 2.2]). Let π‘“βˆΆπΌβˆ˜βŠ†β„β†’β„ be a differentiable mapping on 𝐼∘,   π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏. If |π‘“ξ…ž(π‘₯)| is convex on [π‘Ž,𝑏], then ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀||𝑓𝑓(π‘₯)𝑑π‘₯(π‘βˆ’π‘Ž)ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ(𝑏)8.(1.1)

Theorem 1.2 (see [1, Theorem 2.3]). Let π‘“βˆΆπΌβˆ˜βŠ†β„β†’β„ be a differentiable mapping on 𝐼∘,π‘Ž,π‘βˆˆπΌβˆ˜ with π‘Ž<𝑏, and let 𝑝>1. If the new mapping |π‘“ξ…ž(π‘₯)|𝑝/(π‘βˆ’1) is convex on [π‘Ž,𝑏], then ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž2(𝑝+1)1/𝑝||π‘“ξ…ž||(π‘Ž)𝑝/(π‘βˆ’1)+||π‘“ξ…ž||(𝑏)𝑝/(π‘βˆ’1)2ξƒ­(π‘βˆ’1)/𝑝.(1.2)

In [2], the above inequalities were generalized as follows.

Theorem 1.3 (see [2, Theorems 1 and 2]). Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and let π‘žβ‰₯1. If |𝑓′(π‘₯)|π‘ž is convex on [π‘Ž,𝑏], then ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4||π‘“ξ…ž(||π‘Ž)π‘ž+||π‘“ξ…ž(||𝑏)π‘ž2ξƒ­1/π‘ž,||||π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4||π‘“ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘ž2ξƒ­1/π‘ž.(1.3)

In [3], the above inequalities were further generalized as follows.

Theorem 1.4 (see [3, Theorems 2.3 and 2.4]). Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and let 𝑝>1. If |π‘“ξ…ž(π‘₯)|𝑝/(π‘βˆ’1) is convex on [π‘Ž,𝑏], then |||1βˆ«π‘βˆ’π‘Žπ‘π‘Žξ‚€π‘“(π‘₯)𝑑π‘₯βˆ’π‘“π‘Ž+𝑏2|||β‰€π‘βˆ’π‘Žξ‚΅416𝑝+11/𝑝×||π‘“ξ…ž||(π‘Ž)𝑝/(π‘βˆ’1)||𝑓+3ξ…ž||(𝑏)𝑝/(π‘βˆ’1)ξ‚„(π‘βˆ’1)/𝑝+3||π‘“ξ…ž||(π‘Ž)𝑝/(π‘βˆ’1)+||π‘“ξ…ž||(𝑏)𝑝/(π‘βˆ’1)ξ‚„(π‘βˆ’1)/𝑝,||||1ξ€œπ‘βˆ’π‘Žπ‘π‘Žξ‚€π‘“(π‘₯)𝑑π‘₯βˆ’π‘“π‘Ž+𝑏2||||β‰€π‘βˆ’π‘Ž4ξ‚΅4𝑝+11/𝑝||π‘“ξ…ž(||+||π‘“π‘Ž)ξ…ž(||ξ€Έ.𝑏)(1.4)

In [4], an inequality similar to the above ones was given as follows.

Theorem 1.5 (see [4, Theorem 3]). Let π‘“βˆΆ[π‘Ž,𝑏]→ℝ be an absolutely continuous mapping on [π‘Ž,𝑏] whose derivative belongs to 𝐿𝑝[π‘Ž,𝑏]. Then ||||13𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€+2π‘“π‘Ž+𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀1𝑓(π‘₯)𝑑π‘₯6ξ‚Έ2π‘ž+1+1ξ‚Ή3(π‘ž+1)1/π‘ž(π‘βˆ’π‘Ž)1/π‘žβ€–β€–π‘“ξ…žβ€–β€–π‘,(1.5) where (1/𝑝)+(1/π‘ž)=1 and 𝑝>1.

Recently, the following inequalities were obtained in [5].

Theorem 1.6. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž for π‘žβ‰₯1 is convex on [π‘Ž,𝑏], then ||||16𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ‚Έ212π‘ž+1+1ξ‚Ή3(π‘ž+1)1/π‘žβŽ‘βŽ’βŽ’βŽ£ξƒ©3||π‘“ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘ž4ξƒͺ1/π‘ž+||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+3ξ…ž||(𝑏)π‘ž4ξƒͺ1/π‘žβŽ€βŽ₯βŽ₯⎦,||||16𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯5(π‘βˆ’π‘Ž)βŽ‘βŽ’βŽ’βŽ£ξƒ©||𝑓7261ξ…ž||(π‘Ž)π‘ž||𝑓+29ξ…ž||(𝑏)π‘žξƒͺ901/π‘ž+||𝑓29ξ…ž||(π‘Ž)π‘ž||𝑓+61ξ…ž||(𝑏)π‘žξƒͺ901/π‘žβŽ€βŽ₯βŽ₯⎦.(1.6)

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. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏. If π‘“β€²βˆˆπΏ[π‘Ž,𝑏] and πœ†,πœ‡βˆˆβ„, then πœ†π‘“(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Ž=𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ€œ10(1βˆ’πœ†βˆ’π‘‘)π‘“ξ…žξ‚€π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏2+(πœ‡βˆ’π‘‘)π‘“ξ…žξ‚€π‘‘π‘Ž+𝑏2+(1βˆ’π‘‘)𝑏𝑑𝑑.(2.1)

Proof. Integrating by part and changing variable of definite integral yield ξ€œ10(1βˆ’πœ†βˆ’π‘‘)π‘“ξ…žξ‚€π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏22𝑑𝑑=βˆ’ξ‚Έξ‚€π‘βˆ’π‘Ž(1βˆ’πœ†βˆ’π‘‘)π‘“π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏2|||10+ξ€œ10π‘“ξ‚€π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏2=2π‘‘π‘‘ξ‚ƒξ‚€π‘βˆ’π‘Žπœ†π‘“(π‘Ž)+(1βˆ’πœ†)π‘“π‘Ž+𝑏2βˆ’4(π‘βˆ’π‘Ž)2ξ€œπ‘Ž(π‘Ž+𝑏)/2𝑓(π‘₯)𝑑π‘₯.(2.2)
Similarly, we have ξ€œ10(πœ‡βˆ’π‘‘)π‘“ξ…žξ‚€π‘‘π‘Ž+𝑏22+(1βˆ’π‘‘)𝑏𝑑𝑑=βˆ’ξ‚Έξ‚€π‘‘π‘βˆ’π‘Ž(πœ‡βˆ’π‘‘)π‘“π‘Ž+𝑏2+|||(1βˆ’π‘‘)𝑏10+ξ€œ10π‘“ξ‚€π‘‘π‘Ž+𝑏2+=2(1βˆ’π‘‘)π‘π‘‘π‘‘ξ‚ƒξ‚€π‘βˆ’π‘Ž(1βˆ’πœ‡)π‘“π‘Ž+𝑏2ξ‚ξ‚„βˆ’4+πœ‡π‘“(𝑏)(π‘βˆ’π‘Ž)2ξ€œπ‘(π‘Ž+𝑏)/2𝑓(π‘₯)𝑑π‘₯.(2.3) Adding these two equations leads to Lemma 2.1.

Lemma 2.2. For 𝑠>0 and 0β‰€πœ‰β‰€1, one has ξ€œ10||||πœ‰βˆ’π‘‘π‘ πœ‰π‘‘π‘‘=𝑠+1+(1βˆ’πœ‰)𝑠+1,ξ€œπ‘ +110𝑑||||πœ‰βˆ’π‘‘π‘ πœ‰π‘‘π‘‘=𝑠+2+(𝑠+1+πœ‰)(1βˆ’πœ‰)𝑠+1.(𝑠+1)(𝑠+2)(2.4)

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 Theorem 3.1.

Theorem 3.1. Let π‘“βˆΆπΌβŠ†β„β†’β„ be a differentiable function on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, 0β‰€πœ†,πœ‡β‰€1, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž for π‘žβ‰₯1 is convex on [π‘Ž,𝑏], then ||||πœ†f(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž8ξ‚€161/π‘žξ‚†ξ€·1βˆ’2πœ†+2πœ†2ξ€Έ1βˆ’1/π‘žΓ—ξ€Ίξ€·4βˆ’9πœ†+12πœ†2βˆ’2πœ†3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·2βˆ’3πœ†+2πœ†3ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€·1βˆ’2πœ‡+2πœ‡2ξ€Έ1βˆ’(1/π‘ž)Γ—ξ€Ίξ€·2βˆ’3πœ‡+2πœ‡3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·4βˆ’9πœ‡+12πœ‡2βˆ’2πœ‡3ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡.(3.1)

Proof. For π‘ž>1, by Lemma 2.1, the convexity of |π‘“ξ…ž(π‘₯)|π‘ž on [π‘Ž,𝑏], and the noted HΓΆlder's integral inequality, we have ||||πœ†π‘“(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ‚Έξ€œ10|||||||𝑓1βˆ’πœ†βˆ’π‘‘ξ…žξ‚€π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏2|||ξ€œπ‘‘π‘‘+10|||||||π‘“πœ‡βˆ’π‘‘ξ…žξ‚€π‘‘π‘Ž+𝑏2|||≀+(1βˆ’π‘‘)π‘π‘‘π‘‘π‘βˆ’π‘Ž4ξƒ―ξ‚΅ξ€œ10||||ξ‚Ά1βˆ’πœ†βˆ’π‘‘π‘‘π‘‘1βˆ’1/π‘žξ‚Έξ€œ10||||ξ‚€1βˆ’πœ†βˆ’π‘‘1+𝑑2||π‘“ξ…ž||(π‘Ž)π‘ž+1βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘ž+ξ‚΅ξ€œ10||||ξ‚Άπœ‡βˆ’π‘‘π‘‘π‘‘1βˆ’1/π‘žξ‚Έξ€œ10||||ξ‚€π‘‘πœ‡βˆ’π‘‘2||π‘“ξ…ž||(π‘Ž)π‘ž+2βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘žξƒ°.(3.2) In virtue of Lemma 2.2, a direct calculation yields ξ€œ10||||ξ‚€1βˆ’πœ†βˆ’π‘‘1+𝑑2||π‘“ξ…ž||(π‘Ž)π‘ž+1βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚=1𝑑𝑑2ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€œ10||||11βˆ’πœ†βˆ’π‘‘π‘‘π‘‘+2ξ€·||π‘“ξ…ž||(π‘Ž)π‘žβˆ’||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€œ10𝑑||||=11βˆ’πœ†βˆ’π‘‘π‘‘π‘‘2ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘žξ€Έξ‚€12βˆ’πœ†+πœ†2+1ξ€·||𝑓12ξ…ž||(π‘Ž)π‘žβˆ’||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€Ί(1βˆ’πœ†)3+(1βˆ’πœ†)2ξ€»=1(3βˆ’πœ†)ξ€·124βˆ’9πœ†+12πœ†2βˆ’2πœ†3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+1ξ€·122βˆ’3πœ†+2πœ†3ξ€Έ||π‘“ξ…ž||(𝑏)π‘ž,ξ€œ10||||ξ‚€π‘‘πœ‡βˆ’π‘‘2||π‘“ξ…ž(||π‘Ž)π‘ž+2βˆ’π‘‘2||π‘“ξ…ž(||𝑏)π‘žξ‚=1𝑑𝑑2ξ€·||π‘“ξ…ž||(π‘Ž)π‘žβˆ’||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€œ10𝑑||||||π‘“πœ‡βˆ’π‘‘π‘‘π‘‘+ξ…ž||(𝑏)π‘žξ€œ10||||=1πœ‡βˆ’π‘‘π‘‘π‘‘ξ€Ίπœ‡123+(1βˆ’πœ‡)2(||𝑓2+πœ‡)ξ€»ξ€·ξ…ž(||π‘Ž)π‘žβˆ’||π‘“ξ…ž(||𝑏)π‘žξ€Έ+ξ‚€12βˆ’πœ‡+πœ‡2||π‘“ξ…ž(||𝑏)π‘ž=1ξ€·122βˆ’3πœ‡+2πœ‡3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+1ξ€·124βˆ’9πœ‡+12πœ‡2βˆ’2πœ‡3ξ€Έ||π‘“ξ…ž||(𝑏)π‘ž.(3.3)
Substituting the above two equalities into the inequality (3.2) and utilizing Lemma 2.2 result in the inequality (3.1) for π‘ž>1.
For π‘ž=1, from Lemmas 2.1 and 2.2 it follows that ||||πœ†π‘“(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ‚Έξ€œ10||||ξ‚€1βˆ’πœ†βˆ’π‘‘1+𝑑2||π‘“ξ…ž||+(π‘Ž)1βˆ’π‘‘2||π‘“ξ…ž||+ξ€œ(𝑏)𝑑𝑑10||||ξ‚€π‘‘πœ‡βˆ’π‘‘2||π‘“ξ…ž||+(π‘Ž)2βˆ’π‘‘2||π‘“ξ…ž||=(𝑏)π‘‘π‘‘π‘βˆ’π‘Ž48ξ€Ίξ€·4βˆ’9πœ†+12πœ†2βˆ’2πœ†3ξ€Έ||π‘“ξ…ž||+ξ€·(π‘Ž)2βˆ’3πœ†+2πœ†2ξ€Έ||π‘“ξ…ž||+ξ€·(𝑏)2βˆ’3πœ‡+2πœ‡2ξ€Έ||π‘“ξ…ž||+ξ€·(π‘Ž)4βˆ’9πœ‡+12πœ‡2βˆ’2πœ‡3ξ€Έ||π‘“ξ…ž||ξ€»,(𝑏)(3.4) which is just equivalent to (3.1) for π‘ž=1. Theorem 3.1 is proved.

If taking πœ†=πœ‡ in Theorem 3.1, we derive the following corollary.

Corollary 3.2. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, 0β‰€πœ†β‰€1, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž is convex on [π‘Ž,𝑏] for π‘žβ‰₯1, then ||||πœ†2[𝑓]+ξ‚€(π‘Ž)+𝑓(𝑏)(1βˆ’πœ†)π‘“π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž8ξ‚€161/π‘žξ€·1βˆ’2πœ†+2πœ†2ξ€Έ1βˆ’(1/π‘ž)×4βˆ’9πœ†+12πœ†2βˆ’2πœ†3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·2βˆ’3πœ†+2πœ†3ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ίξ€·2βˆ’3πœ†+2πœ†3ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·4βˆ’9πœ†+12πœ†2βˆ’2πœ†3ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡.(3.5)

If letting πœ†=πœ‡=1/2,2/3,1/3, respectively, in Theorem 3.1, we can deduce the inequalities below.

Corollary 3.3. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |𝑓′(π‘₯)|π‘ž is convex on [π‘Ž,𝑏] for π‘žβ‰₯1, then ||||12𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€+π‘“π‘Ž+𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ‚€116121/π‘žξ‚»ξ€Ί9||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+3ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί3||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+9ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚Ό,||||13𝑓(π‘Ž)+𝑓(𝑏)+π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯5(π‘βˆ’π‘Ž)ξ‚€1721801/π‘žξ‚ƒξ€·||𝑓74ξ…ž||(π‘Ž)π‘ž||𝑓+16ξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·||𝑓16ξ…ž||(π‘Ž)π‘ž||𝑓+74ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚„,||||16𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯5(π‘βˆ’π‘Ž)ξ‚€172901/π‘žξ‚†ξ€Ί||𝑓61ξ…ž||(π‘Ž)π‘ž||𝑓+29ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί||𝑓29ξ…ž||(π‘Ž)π‘ž||𝑓+61ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡.(3.6)

If setting π‘ž=1 in Corollary 3.3, then one has the following.

Corollary 3.4. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)| is convex on [π‘Ž,𝑏], then ||||12𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€+π‘“π‘Ž+𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ€·||𝑓16ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ,||||1(𝑏)3𝑓(π‘Ž)+𝑓(𝑏)+π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯5(π‘βˆ’π‘Ž)ξ€·||𝑓144ξ…ž(||+||π‘“π‘Ž)ξ…ž(||ξ€Έ,||||1𝑏)6𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯5(π‘βˆ’π‘Ž)ξ€·||𝑓72ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ.(𝑏)(3.7)

The second main result is Theorem 3.5.

Theorem 3.5. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, 0 ≀ λ, ΞΌ  ≀ 1, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž is convex on [π‘Ž,𝑏] for π‘žβ‰₯1, then ||||πœ†π‘“(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ‚Έ12ξ‚Ή(π‘ž+1)(π‘ž+2)1/π‘žξ‚†ξ€·ξ€Ί(π‘ž+3βˆ’πœ†)(1βˆ’πœ†)π‘ž+1+(2π‘ž+4βˆ’πœ†)πœ†π‘ž+1ξ€»||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€Ί(π‘ž+1+πœ†)(1βˆ’πœ†)π‘ž+1+πœ†π‘ž+2ξ€»||π‘“ξ…ž(||𝑏)π‘žξ€Έ1/π‘ž+ξ€·ξ€Ί(π‘ž+1+πœ‡)(1βˆ’πœ‡)π‘ž+1+πœ‡π‘ž+2ξ€»||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€Ί(π‘ž+3βˆ’πœ‡)(1βˆ’πœ‡)π‘ž+1+(2π‘ž+4βˆ’πœ‡)πœ‡π‘ž+1ξ€»||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡.(3.8)

Proof. For π‘ž>1, by the convexity of |π‘“ξ…ž(π‘₯)|π‘ž on [π‘Ž,𝑏], Lemma 2.1, and HΓΆlder's integral inequality, it follows that ||||πœ†π‘“(π‘Ž)+πœ‡π‘“(𝑏)2+2βˆ’πœ†βˆ’πœ‡2π‘“ξ‚€π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ‚Έξ€œ10|||||||𝑓1βˆ’πœ†βˆ’π‘‘ξ…žξ‚€π‘‘π‘Ž+(1βˆ’π‘‘)π‘Ž+𝑏2|||ξ€œπ‘‘π‘‘+10|||||||π‘“πœ‡βˆ’π‘‘ξ…žξ‚€π‘‘π‘Ž+𝑏2|||≀+(1βˆ’π‘‘)π‘π‘‘π‘‘π‘βˆ’π‘Ž4ξƒ―ξ‚΅ξ€œ10𝑑𝑑1βˆ’1/π‘žξ‚Έξ€œ10||||1βˆ’πœ†βˆ’π‘‘π‘žξ‚€1+𝑑2||π‘“ξ…ž||(π‘Ž)π‘ž+1βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘ž+ξ‚΅ξ€œ10𝑑𝑑1βˆ’1/π‘žξ‚Έξ€œ10||||πœ‡βˆ’π‘‘π‘žξ‚€π‘‘2||π‘“ξ…ž||(π‘Ž)π‘ž+2βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘žξƒ°=π‘βˆ’π‘Ž4ξƒ―ξ‚Έξ€œ10||||1βˆ’πœ†βˆ’π‘‘π‘žξ‚€1+𝑑2||π‘“ξ…ž||(π‘Ž)π‘ž+1βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘ž+ξ‚Έξ€œ10||||πœ‡βˆ’π‘‘π‘žξ‚€π‘‘2||π‘“ξ…ž||(π‘Ž)π‘ž+2βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚ξ‚Ήπ‘‘π‘‘1/π‘žξƒ°.(3.9) By Lemma 2.2, we have ξ€œ10||||1βˆ’πœ†βˆ’π‘‘π‘žξ‚€1+𝑑2||π‘“ξ…ž||(π‘Ž)π‘ž+1βˆ’π‘‘2||π‘“ξ…ž||(𝑏)π‘žξ‚=1𝑑𝑑2ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€œ10||||1βˆ’πœ†βˆ’π‘‘π‘ž1𝑑𝑑+2ξ€·||π‘“ξ…ž||(π‘Ž)π‘žβˆ’||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€œ10𝑑||||1βˆ’πœ†βˆ’π‘‘π‘ž=1𝑑𝑑||𝑓2(π‘ž+1)ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘žπœ†ξ€Έξ€Ίπ‘ž+1+(1βˆ’πœ†)π‘ž+1ξ€»+1ξ€·||𝑓2(π‘ž+1)(π‘ž+2)ξ…ž||(π‘Ž)π‘žβˆ’||π‘“ξ…ž||(𝑏)π‘žξ€Έξ€Ί(1βˆ’πœ†)π‘ž+2+(π‘ž+2βˆ’πœ†)πœ†π‘ž+1ξ€»=12(π‘ž+1)(π‘ž+2)ξ€½ξ€Ί(π‘ž+3βˆ’πœ†)(1βˆ’πœ†)π‘ž+1+(2π‘ž+4βˆ’πœ†)πœ†π‘ž+1ξ€»||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€Ί(π‘ž+1+πœ†)(1βˆ’πœ†)π‘ž+1+πœ†π‘ž+2ξ€»||π‘“ξ…ž||(𝑏)π‘žξ€Ύ,ξ€œ10||||πœ‡βˆ’π‘‘π‘žξ‚€π‘‘2||π‘“ξ…ž(||π‘Ž)π‘ž+2βˆ’π‘‘2||π‘“ξ…ž(||𝑏)π‘žξ‚=1𝑑𝑑(2(π‘ž+1)(π‘ž+2)ξ€½ξ€Ίπ‘ž+1+πœ‡)(1βˆ’πœ‡)π‘ž+1+πœ‡π‘ž+2ξ€»||π‘“ξ…ž(||π‘Ž)π‘ž+ξ€Ί(π‘ž+3βˆ’πœ‡)(1βˆ’πœ‡)π‘ž+1+(2π‘ž+4βˆ’πœ‡)πœ‡π‘ž+1ξ€»||π‘“ξ…ž||(𝑏)π‘žξ€Ύ.(3.10) Substituting these two equalities into the inequality (3.9) yields (3.8) for π‘ž>1.
For π‘ž=1, the proof is the same as the deduction of (3.4). Thus, Theorem 3.5 is proved.

As the derivation of corollaries of Theorem 3.1, we can obtain the following corollaries of Theorem 3.5.

Corollary 3.6. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, 0β‰€πœ†β‰€1, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž is convex for π‘žβ‰₯1 on [π‘Ž,𝑏], then ||||πœ†π‘“(π‘Ž)+𝑓(𝑏)2+ξ‚€(1βˆ’πœ†)π‘“π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž4ξ‚Έ12ξ‚Ή(π‘ž+1)(π‘ž+2)1/π‘žΓ—ξ‚†ξ€·ξ€Ί(π‘ž+3βˆ’πœ†)(1βˆ’πœ†)π‘ž+1+(2π‘ž+4βˆ’πœ†)πœ†π‘ž+1ξ€»||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€Ί(π‘ž+1+πœ†)(1βˆ’πœ†)π‘ž+1+πœ†π‘ž+2ξ€»||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·ξ€Ί(π‘ž+1+πœ†)(1βˆ’πœ†)π‘ž+1+πœ†π‘ž+2ξ€»||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€Ί(π‘ž+3βˆ’πœ†)(1βˆ’πœ†)π‘ž+1+(2π‘ž+4βˆ’πœ†)πœ†π‘ž+1ξ€»||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡.(3.11)

Corollary 3.7. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž is convex for π‘žβ‰₯1 on [π‘Ž,𝑏], then ||||12𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€+π‘“π‘Ž+𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Ž8ξ‚Έ14ξ‚Ή(π‘ž+1)(π‘ž+2)1/π‘žξ‚†ξ€Ί||𝑓(3π‘ž+6)ξ…ž||(π‘Ž)π‘ž||𝑓+(π‘ž+2)ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί||𝑓(π‘ž+2)ξ…ž||(π‘Ž)π‘ž||𝑓+(3π‘ž+6)ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡,||||16𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ‚Έ112ξ‚Ή18(π‘ž+1)(π‘ž+2)1/π‘žΓ—ξ‚†ξ€Ίξ€·11+6π‘ž+(3π‘ž+8)2π‘ž+1ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·1+(3π‘ž+4)2π‘ž+1ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ίξ€·1+(3π‘ž+4)2π‘ž+1ξ€Έ||π‘“ξ…ž||(π‘Ž)π‘ž+ξ€·11+6π‘ž+(3π‘ž+8)2π‘ž+1ξ€Έ||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡.(3.12)

Corollary 3.8. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)| is convex on [π‘Ž,𝑏], then ||||12𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€+π‘“π‘Ž+𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ€·||𝑓16ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ,||||1(𝑏)6𝑓(π‘Ž)+𝑓(𝑏)+4π‘“π‘Ž+𝑏2βˆ’1ξ‚ξ‚„ξ€œπ‘βˆ’π‘Žπ‘π‘Ž||||≀𝑓(π‘₯)𝑑π‘₯2(π‘βˆ’π‘Ž)ξ€·||𝑓27ξ…ž(||||π‘“π‘Ž)+32ξ…ž(||ξ€Έ.𝑏)(3.13)

Corollary 3.9. Let π‘“βˆΆπΌβŠ†β„β†’β„ be differentiable on 𝐼∘, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and π‘“β€²βˆˆπΏ[π‘Ž,𝑏]. If |π‘“ξ…ž(π‘₯)|π‘ž is convex for π‘žβ‰₯1 on [π‘Ž,𝑏] and 𝑓(π‘Ž)+𝑓(𝑏)2ξ‚€=π‘“π‘Ž+𝑏2(3.14) then ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||=||||𝑓(π‘₯)𝑑π‘₯π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ‚Έ1164ξ‚Ή(π‘ž+1)(π‘ž+2)1/π‘žΓ—ξ‚†ξ€Ί||𝑓(2π‘ž+5)ξ…ž||(π‘Ž)π‘ž||𝑓+(2π‘ž+3)ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+(4π‘ž+7)ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί||𝑓(4π‘ž+7)ξ…ž||(π‘Ž)π‘ž+||π‘“ξ…ž||(𝑏)π‘žξ€»1/π‘ž+ξ€Ί||𝑓(2π‘ž+3)ξ…ž||(π‘Ž)π‘ž||𝑓+(2π‘ž+5)ξ…ž||(𝑏)π‘žξ€»1/π‘žξ‚‡.(3.15) In particular, when π‘ž=1, one has ||||𝑓(π‘Ž)+𝑓(𝑏)2βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||=||||𝑓(π‘₯)𝑑π‘₯π‘Ž+𝑏2ξ‚βˆ’1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“||||≀(π‘₯)𝑑π‘₯π‘βˆ’π‘Žξ€·||𝑓16ξ…ž||+||𝑓(π‘Ž)ξ…ž||ξ€Έ.(𝑏)(3.16)

4. Applications to Means

For two positive numbers π‘Ž>0 and 𝑏>0, define 𝐴(π‘Ž,𝑏)=π‘Ž+𝑏2√,𝐺(π‘Ž,𝑏)=π‘Žπ‘,𝐻(π‘Ž,𝑏)=2π‘Žπ‘,⎧βŽͺ⎨βŽͺβŽ©ξ‚Έπ‘π‘Ž+𝑏𝐿(π‘Ž,𝑏)=𝑠+1βˆ’π‘Žπ‘ +1ξ‚Ή(𝑠+1)(π‘βˆ’π‘Ž)1/π‘ βŽ§βŽͺ⎨βŽͺ⎩1,aβ‰ π‘π‘Ž,π‘Ž=𝑏,𝐼(π‘Ž,𝑏)=π‘’ξ‚΅π‘π‘π‘Žπ‘Žξ‚Ά1/(π‘βˆ’π‘Ž)𝐻,π‘Žβ‰ π‘π‘Ž,π‘Ž=𝑏,πœ”,π‘ βŽ§βŽͺ⎨βŽͺβŽ©ξ‚Έπ‘Ž(π‘Ž,𝑏)=𝑠+(π‘Žπ‘)𝑠/2πœ”+π‘π‘ ξ‚Ήπœ”+21/π‘ βˆš,𝑠≠0π‘Žπ‘,𝑠=0(4.1) for 0β‰€πœ”<∞. These means are, respectively, called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive number π‘Ž and 𝑏.

Applying Theorem 3.1 to 𝑓(π‘₯)=π‘₯𝑠 for 𝑠≠0 and π‘₯>0 leads to the following inequalities for means.

Theorem 4.1. Let π‘Ž,𝑏>0, π‘žβ‰₯1, either 𝑠>1 and (π‘ βˆ’1)π‘žβ‰₯1 or 𝑠<0, then ||||𝐴(π‘Žπ‘ ,𝑏𝑠)+𝐴𝑠(π‘Ž,𝑏)2βˆ’πΏπ‘ ||||≀(π‘Ž,𝑏)π‘βˆ’π‘Ž16|𝑠|π‘žξ‚Έ1ξ‚Ή4(π‘ž+1)(π‘ž+2)1/π‘žΓ—ξ‚†ξ€Ί(2π‘ž+5)π‘Ž(π‘ βˆ’1)π‘ž+(2π‘ž+3)𝑏(π‘ βˆ’1)π‘žξ€»1/π‘ž+ξ€Ίπ‘Ž(π‘ βˆ’1)π‘ž+(4π‘ž+7)𝑏(π‘ βˆ’1)π‘žξ€»1/π‘ž+ξ€Ί(4π‘ž+7)π‘Ž(π‘ βˆ’1)π‘ž+𝑏(π‘ βˆ’1)π‘žξ€»1/π‘ž+ξ€Ί(2π‘ž+3)π‘Ž(π‘ βˆ’1)π‘ž+(2π‘ž+5)𝑏(π‘ βˆ’1)π‘žξ€»1/π‘žξ‚‡.(4.2) In particular, if 𝑠β‰₯2 or 𝑠<0, then ||||𝐴(π‘Žπ‘ ,𝑏𝑠)+𝐴𝑠(π‘Ž,𝑏)2βˆ’πΏπ‘ ||||≀(π‘Ž,𝑏)π‘βˆ’π‘Ž8ξ€·π‘Ž|𝑠|π΄π‘ βˆ’1,π‘π‘ βˆ’1ξ€Έ.(4.3)

Taking 𝑓(π‘₯)=lnπ‘₯ for π‘₯>0 in Theorem 3.1 results in the following inequalities for means.

Theorem 4.2. For π‘Ž,𝑏>0 and π‘žβ‰₯1, one has |||ln𝐺(π‘Ž,𝑏)+ln𝐴(π‘Ž,𝑏)2|||β‰€βˆ’ln𝐼(π‘Ž,𝑏)π‘βˆ’π‘Žξ‚Έ116ξ‚Ή4(π‘ž+1)(π‘ž+2)1/π‘žξƒ¬ξ‚΅2π‘ž+5π‘Žπ‘ž+2π‘ž+3π‘π‘žξ‚Ά1/π‘ž+ξ‚΅1π‘Žπ‘ž+4π‘ž+7π‘π‘žξ‚Ά1/π‘ž+ξ‚΅4π‘ž+7π‘Žπ‘ž+1π‘π‘žξ‚Ά1/π‘ž+ξ‚΅2π‘ž+3π‘Žπ‘ž+2π‘ž+5π‘π‘žξ‚Ά1/π‘žξƒ­.(4.4) In particular, |||ln𝐴(π‘Ž,𝑏)+ln𝐺(π‘Ž,𝑏)2|||β‰€βˆ’ln𝐼(π‘Ž,𝑏)π‘βˆ’π‘Ž81.𝐻(π‘Ž,𝑏)(4.5)

Finally, we can establish an inequality for Heronian mean as follows.

Theorem 4.3. For 𝑏>π‘Ž>0, πœ”β‰₯0, and 𝑠β‰₯4 or 0≠𝑠<1, one has |||||π»π‘ πœ”,𝑠(π‘Ž,𝑏)𝐻(π‘Žπ‘ ,𝑏𝑠)+𝐻(𝑠/2)+1πœ”,(𝑠/2)+1ξ‚€π‘π‘Ž+π‘Žπ‘ξ‚,1βˆ’π»π‘ πœ”,π‘ ξƒ©πΏξ€·π‘Ž2,𝑏2𝐺2ξƒͺ|||||≀(π‘Ž,𝑏),1π‘βˆ’π‘Ž4ξƒ¬ξ€·π‘Ž|𝑠|𝐴(π‘Ž,𝑏)𝐴2(π‘ βˆ’1),𝑏2(π‘ βˆ’1)ξ€Έ(πœ”+2)𝐺2𝑠+ξ€·π‘Ž(π‘Ž,𝑏)πœ”π΄(π‘Ž,𝑏)π΄π‘ βˆ’2,π‘π‘ βˆ’2ξ€Έ2(πœ”+2)𝐺𝑠.(π‘Ž,𝑏)(4.6)

Proof. Let 𝑓(π‘₯)=(π‘₯𝑠+πœ”π‘₯𝑠/2+1)/(πœ”+2) for π‘₯>0 and π‘ βˆ‰(1,4). Then π‘“ξ…žπ‘ (π‘₯)=ξ‚€π‘₯πœ”+2π‘ βˆ’1+πœ”2π‘₯𝑠/2βˆ’1,||π‘“ξ…ž||(π‘₯)ξ…žξ…ž=π‘₯(𝑠/2)βˆ’3|𝑠|ξ‚ƒπœ”πœ”+28(π‘ βˆ’2)(π‘ βˆ’4)+(π‘ βˆ’1)(π‘ βˆ’2)π‘₯𝑠/2ξ‚„β‰₯0.(4.7) In virtue of Corollary 3.3, it follows that ||||12𝑓(𝑏/π‘Ž)+𝑓(π‘Ž/𝑏)2ξ‚€+𝑓𝑏/π‘Ž+π‘Ž/𝑏2ξ‚ξ‚Ήβˆ’1ξ€œπ‘/π‘Žβˆ’π‘Ž/𝑏𝑏/π‘Žπ‘Ž/𝑏𝑓||||=||||1(π‘₯)𝑑π‘₯2ξ‚»12𝑏𝑠+πœ”(π‘Žπ‘)𝑠/2+π‘Žπ‘ π‘Žπ‘ +π‘Ž(πœ”+2)𝑠+πœ”(π‘Žπ‘)𝑠/2+𝑏𝑠𝑏sξ‚Ή+(πœ”+2)(𝑏/π‘Ž+π‘Ž/𝑏)𝑠+πœ”(𝑏/π‘Ž+π‘Ž/𝑏)𝑠/2+1ξ‚Όβˆ’1πœ”+2ξ‚Έπœ”+2(𝑏/π‘Ž)𝑠+1βˆ’(π‘Ž/𝑏)𝑠+1(𝑠+1)(𝑏/π‘Žβˆ’π‘Ž/𝑏)+πœ”(𝑏/π‘Ž)𝑠/2+1βˆ’(π‘Ž/𝑏)𝑠/2+1(ξ‚Ή||||=|||||𝐻𝑠/2+1)(𝑏/π‘Žβˆ’π‘Ž/𝑏)+1π‘ πœ”,𝑠(π‘Ž,𝑏)𝐻(π‘Žπ‘ ,𝑏𝑠)+𝐻(𝑠/2)+1πœ”,(𝑠/2)+1ξ‚€π‘π‘Ž+π‘Žπ‘ξ‚,1βˆ’π»π‘ πœ”,π‘ ξƒ©πΏξ€·π‘Ž2,𝑏2𝐺2ξƒͺ|||||.(π‘Ž,𝑏),1(4.8) On the other hand, we have 𝑏/π‘Žβˆ’π‘Ž/𝑏|||𝑓16ξ…žξ‚€π‘Žπ‘ξ‚|||+|||π‘“ξ…žξ‚€π‘π‘Žξ‚|||ξ‚„=𝑏2βˆ’π‘Ž2𝑏16(πœ”+2)π‘Žπ‘|𝑠|π‘Žξ‚π‘ βˆ’1+ξ‚€π‘Žπ‘ξ‚π‘ βˆ’1ξ‚Ή+πœ”2𝑏|𝑠|π‘Žξ‚π‘ /2βˆ’1+ξ‚€π‘Žπ‘ξ‚π‘ /2βˆ’1=ξ‚Ήξ‚Ό(π‘βˆ’π‘Ž)|𝑠|4ξƒ¬ξ€·π‘Žπ΄(π‘Ž,𝑏)𝐴2(π‘ βˆ’1),𝑏2(π‘ βˆ’1)ξ€Έ(πœ”+2)𝐺2𝑠+ξ€·π‘Ž(π‘Ž,𝑏)πœ”π΄(π‘Ž,𝑏)π΄π‘ βˆ’2,π‘π‘ βˆ’2ξ€Έ2(πœ”+2)𝐺𝑠.(π‘Ž,𝑏)(4.9) The proof is complete.

Remark 4.4. Some inequalities of Hermite-Hadamard type were also obtained in [6–9] by the authors.

Acknowledgment

The first author was supported in part by the National Natural Science Foundation of China under Grant no. 10962004.

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Copyright © 2012 Bo-Yan Xi and Feng Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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