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 𝑓∶𝐼∘⊆ℝ→ℝ be a differentiable mapping on 𝐼∘,   ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏. If |ğ‘“î…ž(𝑥)| is convex on [ğ‘Ž,𝑏], then ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤||𝑓𝑓(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)||+||𝑓(ğ‘Ž)||(𝑏)8.(1.1)

Theorem 1.2 (see [1, Theorem 2.3]). Let 𝑓∶𝐼∘⊆ℝ→ℝ be a differentiable mapping on 𝐼∘,ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏, and let 𝑝>1. If the new mapping |ğ‘“î…ž(𝑥)|𝑝/(𝑝−1) is convex on [ğ‘Ž,𝑏], then ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž2(𝑝+1)1/𝑝||ğ‘“î…ž||(ğ‘Ž)𝑝/(𝑝−1)+||ğ‘“î…ž||(𝑏)𝑝/(𝑝−1)2(𝑝−1)/𝑝.(1.2)

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

Theorem 1.3 (see [2, Theorems 1 and 2]). Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and let ğ‘žâ‰¥1. If |𝑓′(𝑥)|ğ‘ž is convex on [ğ‘Ž,𝑏], then ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž4||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+||ğ‘“î…ž(||𝑏)ğ‘ž21/ğ‘ž,||||ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž4||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘ž21/ğ‘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 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and let 𝑝>1. If |ğ‘“î…ž(𝑥)|𝑝/(𝑝−1) is convex on [ğ‘Ž,𝑏], then |||1âˆ«ğ‘âˆ’ğ‘Žğ‘ğ‘Žî‚€ğ‘“(𝑥)ğ‘‘ğ‘¥âˆ’ğ‘“ğ‘Ž+𝑏2|||â‰¤ğ‘âˆ’ğ‘Žî‚µ416𝑝+11/𝑝×||ğ‘“î…ž||(ğ‘Ž)𝑝/(𝑝−1)||𝑓+3||(𝑏)𝑝/(𝑝−1)(𝑝−1)/𝑝+3||ğ‘“î…ž||(ğ‘Ž)𝑝/(𝑝−1)+||ğ‘“î…ž||(𝑏)𝑝/(𝑝−1)(𝑝−1)/𝑝,||||1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žî‚€ğ‘“(𝑥)ğ‘‘ğ‘¥âˆ’ğ‘“ğ‘Ž+𝑏2||||â‰¤ğ‘âˆ’ğ‘Ž44𝑝+11/𝑝||ğ‘“î…ž(||+||ğ‘“ğ‘Ž)(||.𝑏)(1.4)

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

Theorem 1.5 (see [4, Theorem 3]). Let 𝑓∶[ğ‘Ž,𝑏]→ℝ be an absolutely continuous mapping on [ğ‘Ž,𝑏] whose derivative belongs to 𝐿𝑝[ğ‘Ž,𝑏]. Then ||||13𝑓(ğ‘Ž)+𝑓(𝑏)2+2ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤1𝑓(𝑥)𝑑𝑥62ğ‘ž+1+13(ğ‘ž+1)1/ğ‘ž(ğ‘âˆ’ğ‘Ž)1/ğ‘žâ€–â€–ğ‘“î…žâ€–â€–ğ‘,(1.5) where (1/𝑝)+(1/ğ‘ž)=1 and 𝑝>1.

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

Theorem 1.6. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž for ğ‘žâ‰¥1 is convex on [ğ‘Ž,𝑏], then ||||16𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî‚¸212ğ‘ž+1+13(ğ‘ž+1)1/ğ‘žâŽ¡âŽ¢âŽ¢âŽ£îƒ©3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘ž41/ğ‘ž+||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+3||(𝑏)ğ‘ž41/ğ‘žâŽ¤âŽ¥âŽ¥âŽ¦,||||16𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥5(ğ‘âˆ’ğ‘Ž)⎡⎢⎢⎣||𝑓7261||(ğ‘Ž)ğ‘ž||𝑓+29||(𝑏)ğ‘žîƒª901/ğ‘ž+||𝑓29||(ğ‘Ž)ğ‘ž||𝑓+61||(𝑏)ğ‘žîƒª901/ğ‘žâŽ¤âŽ¥âŽ¥âŽ¦.(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 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏. If 𝑓′∈𝐿[ğ‘Ž,𝑏] and 𝜆,𝜇∈ℝ, then 𝜆𝑓(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž=𝑓(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž410(1−𝜆−𝑡)ğ‘“î…žî‚€ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏2+(𝜇−𝑡)ğ‘“î…žî‚€ğ‘¡ğ‘Ž+𝑏2+(1−𝑡)𝑏𝑑𝑡.(2.1)

Proof. Integrating by part and changing variable of definite integral yield 10(1−𝜆−𝑡)ğ‘“î…žî‚€ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏22𝑑𝑡=âˆ’î‚¸î‚€ğ‘âˆ’ğ‘Ž(1−𝜆−𝑡)ğ‘“ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏2|||10+10ğ‘“î‚€ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏2=2ğ‘‘ğ‘¡î‚ƒî‚€ğ‘âˆ’ğ‘Žğœ†ğ‘“(ğ‘Ž)+(1−𝜆)ğ‘“ğ‘Ž+𝑏2−4(ğ‘âˆ’ğ‘Ž)2î€œğ‘Ž(ğ‘Ž+𝑏)/2𝑓(𝑥)𝑑𝑥.(2.2)
Similarly, we have 10(𝜇−𝑡)ğ‘“î…žî‚€ğ‘¡ğ‘Ž+𝑏22+(1−𝑡)𝑏𝑑𝑡=âˆ’î‚¸î‚€ğ‘¡ğ‘âˆ’ğ‘Ž(𝜇−𝑡)ğ‘“ğ‘Ž+𝑏2+|||(1−𝑡)𝑏10+10ğ‘“î‚€ğ‘¡ğ‘Ž+𝑏2+=2(1−𝑡)ğ‘ğ‘‘ğ‘¡î‚ƒî‚€ğ‘âˆ’ğ‘Ž(1−𝜇)ğ‘“ğ‘Ž+𝑏2−4+𝜇𝑓(𝑏)(ğ‘âˆ’ğ‘Ž)2𝑏(ğ‘Ž+𝑏)/2𝑓(𝑥)𝑑𝑥.(2.3) Adding these two equations leads to Lemma 2.1.

Lemma 2.2. For 𝑠>0 and 0≤𝜉≤1, one has 10||||𝜉−𝑡𝑠𝜉𝑑𝑡=𝑠+1+(1−𝜉)𝑠+1,𝑠+110𝑡||||𝜉−𝑡𝑠𝜉𝑑𝑡=𝑠+2+(𝑠+1+𝜉)(1−𝜉)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 𝑓∶𝐼⊆ℝ→ℝ be a differentiable function on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, 0≤𝜆,𝜇≤1, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž for ğ‘žâ‰¥1 is convex on [ğ‘Ž,𝑏], then ||||𝜆f(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž8161/ğ‘žî‚†î€·1−2𝜆+2𝜆21−1/ğ‘žÃ—î€ºî€·4−9𝜆+12𝜆2−2𝜆3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+2−3𝜆+2𝜆3||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘ž+1−2𝜇+2𝜇21−(1/ğ‘ž)×2−3𝜇+2𝜇3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+4−9𝜇+12𝜇2−2𝜇3||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘žî‚‡.(3.1)

Proof. For ğ‘ž>1, by Lemma 2.1, the convexity of |ğ‘“î…ž(𝑥)|ğ‘ž on [ğ‘Ž,𝑏], and the noted Hölder's integral inequality, we have ||||𝜆𝑓(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž410|||||||𝑓1âˆ’ğœ†âˆ’ğ‘¡î…žî‚€ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏2|||𝑑𝑡+10|||||||ğ‘“ğœ‡âˆ’ğ‘¡î…žî‚€ğ‘¡ğ‘Ž+𝑏2|||≤+(1−𝑡)ğ‘ğ‘‘ğ‘¡ğ‘âˆ’ğ‘Ž410||||1−𝜆−𝑡𝑑𝑡1−1/ğ‘žî‚¸î€œ10||||1−𝜆−𝑡1+𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘ž+10||||𝜇−𝑡𝑑𝑡1−1/ğ‘žî‚¸î€œ10||||𝑡𝜇−𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+2−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘žîƒ°.(3.2) In virtue of Lemma 2.2, a direct calculation yields 10||||1−𝜆−𝑡1+𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚=1𝑑𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€œ10||||11−𝜆−𝑡𝑑𝑡+2||ğ‘“î…ž||(ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€œ10𝑡||||=11−𝜆−𝑡𝑑𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘žî€¸î‚€12−𝜆+𝜆2+1||𝑓12||(ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€º(1−𝜆)3+(1−𝜆)2=1(3−𝜆)124−9𝜆+12𝜆2−2𝜆3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1122−3𝜆+2𝜆3||ğ‘“î…ž||(𝑏)ğ‘ž,10||||𝑡𝜇−𝑡2||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+2−𝑡2||ğ‘“î…ž(||𝑏)ğ‘žî‚=1𝑑𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€œ10𝑡||||||𝑓𝜇−𝑡𝑑𝑡+||(𝑏)ğ‘žî€œ10||||=1𝜇−𝑡𝑑𝑡𝜇123+(1−𝜇)2(||𝑓2+𝜇)(||ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž(||𝑏)ğ‘žî€¸+12−𝜇+𝜇2||ğ‘“î…ž(||𝑏)ğ‘ž=1122−3𝜇+2𝜇3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1124−9𝜇+12𝜇2−2𝜇3||ğ‘“î…ž||(𝑏)ğ‘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 ğ‘ž>1.
For ğ‘ž=1, from Lemmas 2.1 and 2.2 it follows that ||||𝜆𝑓(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž410||||1−𝜆−𝑡1+𝑡2||ğ‘“î…ž||+(ğ‘Ž)1−𝑡2||ğ‘“î…ž||+(𝑏)𝑑𝑡10||||𝑡𝜇−𝑡2||ğ‘“î…ž||+(ğ‘Ž)2−𝑡2||ğ‘“î…ž||=(𝑏)ğ‘‘ğ‘¡ğ‘âˆ’ğ‘Ž484−9𝜆+12𝜆2−2𝜆3||ğ‘“î…ž||+(ğ‘Ž)2−3𝜆+2𝜆2||ğ‘“î…ž||+(𝑏)2−3𝜇+2𝜇2||ğ‘“î…ž||+(ğ‘Ž)4−9𝜇+12𝜇2−2𝜇3||ğ‘“î…ž||,(𝑏)(3.4) which is just equivalent to (3.1) for ğ‘ž=1. Theorem 3.1 is proved.

If taking 𝜆=𝜇 in Theorem 3.1, we derive the following corollary.

Corollary 3.2. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, 0≤𝜆≤1, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž is convex on [ğ‘Ž,𝑏] for ğ‘žâ‰¥1, then ||||𝜆2[𝑓]+(ğ‘Ž)+𝑓(𝑏)(1−𝜆)ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž8161/ğ‘žî€·1−2𝜆+2𝜆21−(1/ğ‘ž)×4−9𝜆+12𝜆2−2𝜆3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+2−3𝜆+2𝜆3||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘ž+2−3𝜆+2𝜆3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+4−9𝜆+12𝜆2−2𝜆3||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘žî‚‡.(3.5)

If letting 𝜆=𝜇=1/2,2/3,1/3, respectively, in Theorem 3.1, we can deduce the inequalities below.

Corollary 3.3. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |𝑓′(𝑥)|ğ‘ž is convex on [ğ‘Ž,𝑏] for ğ‘žâ‰¥1, then ||||12𝑓(ğ‘Ž)+𝑓(𝑏)2+ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî‚€116121/ğ‘žî‚»î€º9||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+3||(𝑏)ğ‘žî€»1/ğ‘ž+3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+9||(𝑏)ğ‘žî€»1/ğ‘žî‚¼,||||13𝑓(ğ‘Ž)+𝑓(𝑏)+ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥5(ğ‘âˆ’ğ‘Ž)1721801/ğ‘žî‚ƒî€·||𝑓74||(ğ‘Ž)ğ‘ž||𝑓+16||(𝑏)ğ‘žî€¸1/ğ‘ž+||𝑓16||(ğ‘Ž)ğ‘ž||𝑓+74||(𝑏)ğ‘žî€¸1/ğ‘žî‚„,||||16𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥5(ğ‘âˆ’ğ‘Ž)172901/ğ‘žî‚†î€º||𝑓61||(ğ‘Ž)ğ‘ž||𝑓+29||(𝑏)ğ‘žî€»1/ğ‘ž+||𝑓29||(ğ‘Ž)ğ‘ž||𝑓+61||(𝑏)ğ‘žî€»1/ğ‘žî‚‡.(3.6)

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

Corollary 3.4. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)| is convex on [ğ‘Ž,𝑏], then ||||12𝑓(ğ‘Ž)+𝑓(𝑏)2+ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî€·||𝑓16||+||𝑓(ğ‘Ž)||,||||1(𝑏)3𝑓(ğ‘Ž)+𝑓(𝑏)+ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥5(ğ‘âˆ’ğ‘Ž)||𝑓144(||+||ğ‘“ğ‘Ž)(||,||||1𝑏)6𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥5(ğ‘âˆ’ğ‘Ž)||𝑓72||+||𝑓(ğ‘Ž)||.(𝑏)(3.7)

The second main result is Theorem 3.5.

Theorem 3.5. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, 0 ≤ λ, μ  ≤ 1, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž is convex on [ğ‘Ž,𝑏] for ğ‘žâ‰¥1, then ||||𝜆𝑓(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž412(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žî‚†î€·î€º(ğ‘ž+3−𝜆)(1−𝜆)ğ‘ž+1+(2ğ‘ž+4−𝜆)ğœ†ğ‘ž+1||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+(ğ‘ž+1+𝜆)(1−𝜆)ğ‘ž+1+ğœ†ğ‘ž+2||ğ‘“î…ž(||𝑏)ğ‘žî€¸1/ğ‘ž+(ğ‘ž+1+𝜇)(1−𝜇)ğ‘ž+1+ğœ‡ğ‘ž+2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+(ğ‘ž+3−𝜇)(1−𝜇)ğ‘ž+1+(2ğ‘ž+4−𝜇)ğœ‡ğ‘ž+1||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡.(3.8)

Proof. For ğ‘ž>1, by the convexity of |ğ‘“î…ž(𝑥)|ğ‘ž on [ğ‘Ž,𝑏], Lemma 2.1, and Hölder's integral inequality, it follows that ||||𝜆𝑓(ğ‘Ž)+𝜇𝑓(𝑏)2+2−𝜆−𝜇2ğ‘“î‚€ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž410|||||||𝑓1âˆ’ğœ†âˆ’ğ‘¡î…žî‚€ğ‘¡ğ‘Ž+(1−𝑡)ğ‘Ž+𝑏2|||𝑑𝑡+10|||||||ğ‘“ğœ‡âˆ’ğ‘¡î…žî‚€ğ‘¡ğ‘Ž+𝑏2|||≤+(1−𝑡)ğ‘ğ‘‘ğ‘¡ğ‘âˆ’ğ‘Ž410𝑑𝑡1−1/ğ‘žî‚¸î€œ10||||1âˆ’ğœ†âˆ’ğ‘¡ğ‘žî‚€1+𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘ž+10𝑑𝑡1−1/ğ‘žî‚¸î€œ10||||ğœ‡âˆ’ğ‘¡ğ‘žî‚€ğ‘¡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+2−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘žîƒ°=ğ‘âˆ’ğ‘Ž410||||1âˆ’ğœ†âˆ’ğ‘¡ğ‘žî‚€1+𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘ž+10||||ğœ‡âˆ’ğ‘¡ğ‘žî‚€ğ‘¡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+2−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚î‚¹ğ‘‘ğ‘¡1/ğ‘žîƒ°.(3.9) By Lemma 2.2, we have 10||||1âˆ’ğœ†âˆ’ğ‘¡ğ‘žî‚€1+𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1−𝑡2||ğ‘“î…ž||(𝑏)ğ‘žî‚=1𝑑𝑡2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€œ10||||1âˆ’ğœ†âˆ’ğ‘¡ğ‘ž1𝑑𝑡+2||ğ‘“î…ž||(ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€œ10𝑡||||1âˆ’ğœ†âˆ’ğ‘¡ğ‘ž=1𝑑𝑡||𝑓2(ğ‘ž+1)||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘žğœ†î€¸î€ºğ‘ž+1+(1−𝜆)ğ‘ž+1+1||𝑓2(ğ‘ž+1)(ğ‘ž+2)||(ğ‘Ž)ğ‘žâˆ’||ğ‘“î…ž||(𝑏)ğ‘žî€¸î€º(1−𝜆)ğ‘ž+2+(ğ‘ž+2−𝜆)ğœ†ğ‘ž+1=12(ğ‘ž+1)(ğ‘ž+2)(ğ‘ž+3−𝜆)(1−𝜆)ğ‘ž+1+(2ğ‘ž+4−𝜆)ğœ†ğ‘ž+1||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+(ğ‘ž+1+𝜆)(1−𝜆)ğ‘ž+1+ğœ†ğ‘ž+2||ğ‘“î…ž||(𝑏)ğ‘žî€¾,10||||ğœ‡âˆ’ğ‘¡ğ‘žî‚€ğ‘¡2||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+2−𝑡2||ğ‘“î…ž(||𝑏)ğ‘žî‚=1𝑑𝑡(2(ğ‘ž+1)(ğ‘ž+2)î€½î€ºğ‘ž+1+𝜇)(1−𝜇)ğ‘ž+1+ğœ‡ğ‘ž+2||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+(ğ‘ž+3−𝜇)(1−𝜇)ğ‘ž+1+(2ğ‘ž+4−𝜇)ğœ‡ğ‘ž+1||ğ‘“î…ž||(𝑏)ğ‘žî€¾.(3.10) Substituting these two equalities into the inequality (3.9) yields (3.8) for ğ‘ž>1.
For ğ‘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 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, 0≤𝜆≤1, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž is convex for ğ‘žâ‰¥1 on [ğ‘Ž,𝑏], then ||||𝜆𝑓(ğ‘Ž)+𝑓(𝑏)2+(1−𝜆)ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž412(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žÃ—î‚†î€·î€º(ğ‘ž+3−𝜆)(1−𝜆)ğ‘ž+1+(2ğ‘ž+4−𝜆)ğœ†ğ‘ž+1||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+(ğ‘ž+1+𝜆)(1−𝜆)ğ‘ž+1+ğœ†ğ‘ž+2||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+(ğ‘ž+1+𝜆)(1−𝜆)ğ‘ž+1+ğœ†ğ‘ž+2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+(ğ‘ž+3−𝜆)(1−𝜆)ğ‘ž+1+(2ğ‘ž+4−𝜆)ğœ†ğ‘ž+1||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡.(3.11)

Corollary 3.7. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž is convex for ğ‘žâ‰¥1 on [ğ‘Ž,𝑏], then ||||12𝑓(ğ‘Ž)+𝑓(𝑏)2+ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Ž814(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žî‚†î€º||𝑓(3ğ‘ž+6)||(ğ‘Ž)ğ‘ž||𝑓+(ğ‘ž+2)||(𝑏)ğ‘žî€»1/ğ‘ž+||𝑓(ğ‘ž+2)||(ğ‘Ž)ğ‘ž||𝑓+(3ğ‘ž+6)||(𝑏)ğ‘žî€»1/ğ‘žî‚‡,||||16𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî‚¸11218(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žÃ—î‚†î€ºî€·11+6ğ‘ž+(3ğ‘ž+8)2ğ‘ž+1||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+1+(3ğ‘ž+4)2ğ‘ž+1||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘ž+1+(3ğ‘ž+4)2ğ‘ž+1||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+11+6ğ‘ž+(3ğ‘ž+8)2ğ‘ž+1||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘žî‚‡.(3.12)

Corollary 3.8. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)| is convex on [ğ‘Ž,𝑏], then ||||12𝑓(ğ‘Ž)+𝑓(𝑏)2+ğ‘“ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî€·||𝑓16||+||𝑓(ğ‘Ž)||,||||1(𝑏)6𝑓(ğ‘Ž)+𝑓(𝑏)+4ğ‘“ğ‘Ž+𝑏2−1î‚î‚„î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥2(ğ‘âˆ’ğ‘Ž)||𝑓27(||||ğ‘“ğ‘Ž)+32(||.𝑏)(3.13)

Corollary 3.9. Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼∘, ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏, and 𝑓′∈𝐿[ğ‘Ž,𝑏]. If |ğ‘“î…ž(𝑥)|ğ‘ž is convex for ğ‘žâ‰¥1 on [ğ‘Ž,𝑏] and 𝑓(ğ‘Ž)+𝑓(𝑏)2=ğ‘“ğ‘Ž+𝑏2(3.14) then ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||=||||𝑓(𝑥)ğ‘‘ğ‘¥ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî‚¸1164(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žÃ—î‚†î€º||𝑓(2ğ‘ž+5)||(ğ‘Ž)ğ‘ž||𝑓+(2ğ‘ž+3)||(𝑏)ğ‘žî€»1/ğ‘ž+||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+(4ğ‘ž+7)||(𝑏)ğ‘žî€»1/ğ‘ž+||𝑓(4ğ‘ž+7)||(ğ‘Ž)ğ‘ž+||ğ‘“î…ž||(𝑏)ğ‘žî€»1/ğ‘ž+||𝑓(2ğ‘ž+3)||(ğ‘Ž)ğ‘ž||𝑓+(2ğ‘ž+5)||(𝑏)ğ‘žî€»1/ğ‘žî‚‡.(3.15) In particular, when ğ‘ž=1, one has ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||=||||𝑓(𝑥)ğ‘‘ğ‘¥ğ‘Ž+𝑏2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘âˆ’ğ‘Žî€·||𝑓16||+||𝑓(ğ‘Ž)||.(𝑏)(3.16)

4. Applications to Means

For two positive numbers ğ‘Ž>0 and 𝑏>0, define 𝐴(ğ‘Ž,𝑏)=ğ‘Ž+𝑏2√,𝐺(ğ‘Ž,𝑏)=ğ‘Žğ‘,𝐻(ğ‘Ž,𝑏)=2ğ‘Žğ‘,âŽ§âŽªâŽ¨âŽªâŽ©î‚¸ğ‘ğ‘Ž+𝑏𝐿(ğ‘Ž,𝑏)=𝑠+1âˆ’ğ‘Žğ‘ +1(𝑠+1)(ğ‘âˆ’ğ‘Ž)1/ğ‘ âŽ§âŽªâŽ¨âŽªâŽ©1,aâ‰ ğ‘ğ‘Ž,ğ‘Ž=𝑏,𝐼(ğ‘Ž,𝑏)=ğ‘’î‚µğ‘ğ‘ğ‘Žğ‘Žî‚¶1/(ğ‘âˆ’ğ‘Ž)𝐻,ğ‘Žâ‰ ğ‘ğ‘Ž,ğ‘Ž=𝑏,𝜔,ğ‘ âŽ§âŽªâŽ¨âŽªâŽ©î‚¸ğ‘Ž(ğ‘Ž,𝑏)=𝑠+(ğ‘Žğ‘)𝑠/2𝜔+𝑏𝑠𝜔+21/𝑠√,𝑠≠0ğ‘Žğ‘,𝑠=0(4.1) for 0≤𝜔<∞. These means are, respectively, called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive number ğ‘Ž and 𝑏.

Applying Theorem 3.1 to 𝑓(𝑥)=𝑥𝑠 for 𝑠≠0 and 𝑥>0 leads to the following inequalities for means.

Theorem 4.1. Let ğ‘Ž,𝑏>0, ğ‘žâ‰¥1, either 𝑠>1 and (𝑠−1)ğ‘žâ‰¥1 or 𝑠<0, then ||||𝐴(ğ‘Žğ‘ ,𝑏𝑠)+𝐴𝑠(ğ‘Ž,𝑏)2−𝐿𝑠||||≤(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Ž16|𝑠|ğ‘žî‚¸14(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žÃ—î‚†î€º(2ğ‘ž+5)ğ‘Ž(𝑠−1)ğ‘ž+(2ğ‘ž+3)𝑏(𝑠−1)ğ‘žî€»1/ğ‘ž+î€ºğ‘Ž(𝑠−1)ğ‘ž+(4ğ‘ž+7)𝑏(𝑠−1)ğ‘žî€»1/ğ‘ž+(4ğ‘ž+7)ğ‘Ž(𝑠−1)ğ‘ž+𝑏(𝑠−1)ğ‘žî€»1/ğ‘ž+(2ğ‘ž+3)ğ‘Ž(𝑠−1)ğ‘ž+(2ğ‘ž+5)𝑏(𝑠−1)ğ‘žî€»1/ğ‘žî‚‡.(4.2) In particular, if 𝑠≥2 or 𝑠<0, then ||||𝐴(ğ‘Žğ‘ ,𝑏𝑠)+𝐴𝑠(ğ‘Ž,𝑏)2−𝐿𝑠||||≤(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Ž8î€·ğ‘Ž|𝑠|𝐴𝑠−1,𝑏𝑠−1.(4.3)

Taking 𝑓(𝑥)=ln𝑥 for 𝑥>0 in Theorem 3.1 results in the following inequalities for means.

Theorem 4.2. For ğ‘Ž,𝑏>0 and ğ‘žâ‰¥1, one has |||ln𝐺(ğ‘Ž,𝑏)+ln𝐴(ğ‘Ž,𝑏)2|||≤−ln𝐼(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Žî‚¸1164(ğ‘ž+1)(ğ‘ž+2)1/ğ‘žîƒ¬î‚µ2ğ‘ž+5ğ‘Žğ‘ž+2ğ‘ž+3ğ‘ğ‘žî‚¶1/ğ‘ž+1ğ‘Žğ‘ž+4ğ‘ž+7ğ‘ğ‘žî‚¶1/ğ‘ž+4ğ‘ž+7ğ‘Žğ‘ž+1ğ‘ğ‘žî‚¶1/ğ‘ž+2ğ‘ž+3ğ‘Žğ‘ž+2ğ‘ž+5ğ‘ğ‘žî‚¶1/ğ‘žîƒ­.(4.4) In particular, |||ln𝐴(ğ‘Ž,𝑏)+ln𝐺(ğ‘Ž,𝑏)2|||≤−ln𝐼(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Ž81.𝐻(ğ‘Ž,𝑏)(4.5)

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

Theorem 4.3. For 𝑏>ğ‘Ž>0, 𝜔≥0, and 𝑠≥4 or 0≠𝑠<1, one has |||||𝐻𝑠𝜔,𝑠(ğ‘Ž,𝑏)𝐻(ğ‘Žğ‘ ,𝑏𝑠)+𝐻(𝑠/2)+1𝜔,(𝑠/2)+1î‚€ğ‘ğ‘Ž+ğ‘Žğ‘î‚,1−𝐻𝑠𝜔,ğ‘ îƒ©ğ¿î€·ğ‘Ž2,𝑏2𝐺2|||||≤(ğ‘Ž,𝑏),1ğ‘âˆ’ğ‘Ž4îƒ¬î€·ğ‘Ž|𝑠|𝐴(ğ‘Ž,𝑏)𝐴2(𝑠−1),𝑏2(𝑠−1)(𝜔+2)𝐺2𝑠+î€·ğ‘Ž(ğ‘Ž,𝑏)𝜔𝐴(ğ‘Ž,𝑏)𝐴𝑠−2,𝑏𝑠−22(𝜔+2)𝐺𝑠.(ğ‘Ž,𝑏)(4.6)

Proof. Let 𝑓(𝑥)=(𝑥𝑠+𝜔𝑥𝑠/2+1)/(𝜔+2) for 𝑥>0 and 𝑠∉(1,4). Then ğ‘“î…žğ‘ (𝑥)=𝑥𝜔+2𝑠−1+𝜔2𝑥𝑠/2−1,||ğ‘“î…ž||(𝑥)=𝑥(𝑠/2)−3|𝑠|𝜔𝜔+28(𝑠−2)(𝑠−4)+(𝑠−1)(𝑠−2)𝑥𝑠/2≥0.(4.7) In virtue of Corollary 3.3, it follows that ||||12𝑓(𝑏/ğ‘Ž)+𝑓(ğ‘Ž/𝑏)2+𝑓𝑏/ğ‘Ž+ğ‘Ž/𝑏2−1𝑏/ğ‘Žâˆ’ğ‘Ž/𝑏𝑏/ğ‘Žğ‘Ž/𝑏𝑓||||=||||1(𝑥)𝑑𝑥212𝑏𝑠+𝜔(ğ‘Žğ‘)𝑠/2+ğ‘Žğ‘ ğ‘Žğ‘ +ğ‘Ž(𝜔+2)𝑠+𝜔(ğ‘Žğ‘)𝑠/2+𝑏𝑠𝑏s+(𝜔+2)(𝑏/ğ‘Ž+ğ‘Ž/𝑏)𝑠+𝜔(𝑏/ğ‘Ž+ğ‘Ž/𝑏)𝑠/2+1−1𝜔+2𝜔+2(𝑏/ğ‘Ž)𝑠+1−(ğ‘Ž/𝑏)𝑠+1(𝑠+1)(𝑏/ğ‘Žâˆ’ğ‘Ž/𝑏)+𝜔(𝑏/ğ‘Ž)𝑠/2+1−(ğ‘Ž/𝑏)𝑠/2+1(||||=|||||𝐻𝑠/2+1)(𝑏/ğ‘Žâˆ’ğ‘Ž/𝑏)+1𝑠𝜔,𝑠(ğ‘Ž,𝑏)𝐻(ğ‘Žğ‘ ,𝑏𝑠)+𝐻(𝑠/2)+1𝜔,(𝑠/2)+1î‚€ğ‘ğ‘Ž+ğ‘Žğ‘î‚,1−𝐻𝑠𝜔,ğ‘ îƒ©ğ¿î€·ğ‘Ž2,𝑏2𝐺2|||||.(ğ‘Ž,𝑏),1(4.8) On the other hand, we have 𝑏/ğ‘Žâˆ’ğ‘Ž/𝑏|||𝑓16î…žî‚€ğ‘Žğ‘î‚|||+|||ğ‘“î…žî‚€ğ‘ğ‘Žî‚|||=𝑏2âˆ’ğ‘Ž2𝑏16(𝜔+2)ğ‘Žğ‘|𝑠|ğ‘Žî‚ğ‘ âˆ’1+î‚€ğ‘Žğ‘î‚ğ‘ âˆ’1+𝜔2𝑏|𝑠|ğ‘Žî‚ğ‘ /2−1+î‚€ğ‘Žğ‘î‚ğ‘ /2−1=(ğ‘âˆ’ğ‘Ž)|𝑠|4îƒ¬î€·ğ‘Žğ´(ğ‘Ž,𝑏)𝐴2(𝑠−1),𝑏2(𝑠−1)(𝜔+2)𝐺2𝑠+î€·ğ‘Ž(ğ‘Ž,𝑏)𝜔𝐴(ğ‘Ž,𝑏)𝐴𝑠−2,𝑏𝑠−22(𝜔+2)𝐺𝑠.(ğ‘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.