International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 809689 | https://doi.org/10.1155/2012/809689

Jaekeun Park, "Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable ( 𝛼 , 𝑚 )-Convex Mappings", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 809689, 12 pages, 2012. https://doi.org/10.1155/2012/809689

Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable ( 𝛼 , 𝑚 )-Convex Mappings

Academic Editor: Jewgeni Dshalalow
Received13 Jul 2011
Revised13 Nov 2011
Accepted28 Nov 2011
Published18 Jan 2012

Abstract

The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings whose first derivative in absolute value aroused to the ğ‘žth (ğ‘žâ‰¥1) power are (𝛼,𝑚)-convex. Some applications to special means of positive real numbers are also given.

1. Introduction

Recall that, for some fixed 𝛼∈(0,1] and 𝑚∈[0,1], a mapping 𝑓∶𝕀⊆[0,∞)→ℝ is said to be (𝛼,𝑚)-convex on an interval 𝕀 if the inequality 𝑓(𝑡𝑥+𝑚(1−𝑡)𝑦)≤𝑡𝛼𝑓(𝑥)+𝑚(1−𝑡𝛼)𝑓(𝑦)(1.1) holds for 𝑥,𝑦∈𝕀, and 𝑡∈[0,1].

Denote by 𝐾𝛼𝑚(𝕀) the set of all (𝛼,𝑚)-convex mappings on 𝕀. For recent results and generalizations concerning 𝑚-convex and (𝛼,𝑚)-convex mappings, see [1–4].

For the simplicities of notations, for 𝑓∈𝐾𝛼𝑚(𝕀), let us denote (ğ‘Žâˆ’ğ‘šğ‘)ğ‘†ğ‘ğ‘Ž1(𝑓)(𝛼,𝑚,𝑟)=𝑟𝑓(ğ‘Ž)+(𝑟−2)ğ‘“ğ‘Ž+𝑚𝑏2−1+𝑓(𝑚𝑏)î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“(𝑥)𝑑𝑥.(1.2)

In [1, 3], Klaričić Bakula and Özdemir et al., proved the following Hadamard’s inequalities for mappings whose second derivative in absolute value aroused to the ğ‘ž-th (ğ‘žâ‰¥1) power are (𝛼,𝑚)-convex.

Theorem 1.1. Let 𝑓∶𝕀⊆[0,𝑏∗]→ℝ be a twice differentiable mapping on the interior 𝕀0 of an interval 𝕀 such that ğ‘“î…žî…žâˆˆğ¿1([ğ‘Ž,𝑏]), where ğ‘Ž,𝑏∈𝕀 with ğ‘Ž<𝑏 and 𝑏∗>0. If |ğ‘“î…žî…ž|ğ‘ž is (𝛼,𝑚)-convex on [ğ‘Ž,𝑏] for (𝛼,𝑚)∈[0,1]2 and ğ‘žâ‰¥1 with 1/𝑝+1/ğ‘ž=1, then the following inequality holds: ||𝑆(ğ‘Ž)ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,2)ğ‘šğ‘âˆ’ğ‘Ž2161/𝑝𝜇||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šğœˆî…žî…ž||(𝑏)ğ‘žî€¾1/ğ‘ž,(1.3) where 1𝜇=𝛼(𝛼+2)(𝛼+3),𝜈=2+5𝛼,6(𝛼+2)(𝛼+3)(1.4)||𝑆(𝑏)ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,6)ğ‘šğ‘âˆ’ğ‘Ž2121/𝑝𝜇0||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž+𝑚𝜈0||ğ‘“î…žî…ž||(𝑏)ğ‘žî€¾1/ğ‘ž,(1.5) where 𝜇0=î‚µğ‘žî‚¶ğ‘ž+𝛼+2Γ(𝛼+2)Γ(ğ‘ž),𝜈Γ(ğ‘ž+𝛼+1)0=1(âˆ’î‚µğ‘žğ‘ž+1)(ğ‘ž+2)î‚¶ğ‘ž+𝛼+2Γ(𝛼+2)Γ(ğ‘ž),Γ(ğ‘ž+𝛼+1)(1.6) where Γ(𝑥)=∞0𝑒−𝑡𝑡𝑥−1𝑑𝑡,𝑥>0.(1.7)

Theorem 1.2. Under the same notations in Theorem 2.2, if |ğ‘“î…žî…ž|ğ‘ž is (𝛼,𝑚)-convex on [ğ‘Ž,𝑏] for (𝛼,𝑚)∈[0,1]2 and ğ‘ž>1 with 1/𝑝+1/ğ‘ž=1, then the following inequality holds: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,2)ğ‘šğ‘âˆ’ğ‘Ž8Γ(𝑝+1)Γ(𝑝+3/2)1/𝑝||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž1||𝑓𝛼+1+ğ‘šî…žî…ž||(𝑏)ğ‘žî‚€ğ›¼ğ›¼+11/ğ‘ž.(1.8)

Note that for (𝛼,𝑚)∈{(0,0),(𝛼,0),(1,0),(1,𝑚),(1,1),(𝛼,1)} one obtains the following classes of functions: increasing, 𝛼-starshaped, starshaped, 𝑚-convex, convex, and 𝛼-convex. For the definitions and elementary properties of these classes, see [4–8].

For recent years, many authors present some new results about Simpson’s inequality for (𝛼,𝑚)-convex mappings and have established error estimations for the Simpson’s inequality: for refinements, counterparts, generalizations, and new Simpson’s type inequalities, see [1–3, 6].

In [9], Dragomir et al. proved the following theorem.

Theorem 1.3. Let 𝑓∶𝕀⊂[0,∞)→ℝ be an absolutely continuous mapping on [ğ‘Ž,𝑏] such that 𝑓′∈𝐿𝑝([ğ‘Ž,𝑏]), where ğ‘Ž,𝑏∈𝕀 with ğ‘Ž<𝑏. Then the following inequality holds: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(1,1,6)(ğ‘âˆ’ğ‘Ž)−1/𝑝62ğ‘ž+1+13(ğ‘ž+1)1/ğ‘žâ€–â€–ğ‘“î…žâ€–â€–ğ‘.(1.9)

The readers can estimate the 𝑒𝑟𝑟𝑜𝑟(𝑓) in the generalized Simpson’s formula without going through its higher derivatives which may not exist, not be bounded, or may be hard to find.

In this paper, the author establishes some generalizations of Hermite-Hadamard and Simpson-like type inequalities based on differentiable (𝛼,𝑚)-convex mappings by using the following new identity in Lemma 2.1 and by using these results, obtain some applications to special means of positive real numbers.

2. Generalizations of Simpson-Like Type Inequalities on 𝐾𝛼𝑚(𝕀)

In order to generalize the classical Simpson-like type inequalities and prove them, we need the following lemma [6].

Lemma 2.1. Let 𝑓∶𝕀⊆[0,𝑏∗]→ℝ be a differentiable mapping on the interior 𝕀0 of an interval 𝕀, where ğ‘Ž,𝑏∈𝕀 with 0â‰¤ğ‘Ž<𝑏 and 𝑏∗>0. If ğ‘“î…žâˆˆğ¿1([ğ‘Ž,𝑏]), then, for 𝑟≥2 and ℎ∈(0,1) with 1/ğ‘Ÿâ‰¤â„Žâ‰¤(𝑟−1)/𝑟, the following equality holds: ğ‘†ğ‘ğ‘Žî€œ(𝑓)(𝛼,𝑚,𝑟)=10𝑝(𝑟,𝑡)ğ‘“î…ž(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡(2.1) for 𝑓∈𝐾𝛼𝑚([ğ‘Ž,𝑏]) and each 𝑡∈[0,1], where ⎧⎪⎨⎪⎩1𝑝(𝑟,𝑡)=𝑡−𝑟1𝑡∈0,2,𝑡−𝑟−1𝑟1𝑡∈2.,1(2.2)

By the similar way as Theorems 1.1–1.3, we obtain the following theorems.

Theorem 2.2. Let 𝑓∶𝕀⊂[0,𝑏∗]→ℝ be a differentiable mapping on 𝕀0 such that ğ‘“î…žâˆˆğ¿([ğ‘Ž,𝑏]), where ğ‘Ž,𝑏∈𝕀 with 0â‰¤ğ‘Ž<𝑏<∞ and 𝑏∗>0. If |𝑓′|∈𝐾𝛼𝑚([ğ‘Ž,𝑏]), for some (𝛼,𝑚)∈(0,1]2 and 𝑚𝑏>ğ‘Ž, then, for any 𝑟≥2, the following inequality holds: ||ğ‘†ğ‘ğ‘Ž||≤𝜇(𝑓)(𝛼,𝑚,𝑟)11+𝜇21+𝜇31+𝜇41||ğ‘“î…ž||+𝜈(ğ‘Ž)11+𝜈21+𝜈31+𝜈41𝑚||ğ‘“î…ž||,(𝑏)(2.3) where 𝜇11=1(𝛼+1)(𝛼+2)𝑟𝛼+2,𝜇21=1(𝛼+1)(𝛼+2)𝑟𝛼+2+(𝛼+1)𝑟−2(𝛼+2)2𝛼+2,𝜇(𝛼+1)(𝛼+2)𝑟31=(𝑟−1)𝛼+2(𝛼+1)(𝛼+2)𝑟𝛼+2+2(𝛼+2)−3𝑟2𝛼+2,𝜇(𝛼+1)(𝛼+2)𝑟41=(𝑟−1)𝛼+2(𝛼+1)(𝛼+2)𝑟𝛼+2+(𝛼+2)−𝑟,𝜈(𝛼+1)(𝛼+2)𝑟11=12𝑟2−𝜇11,𝜈21=(𝑟−2)28𝑟2−𝜇21,𝜈31=(𝑟−2)28𝑟2−𝜇31,𝜈41=12𝑟2−𝜇41.(2.4)

Proof. From Lemma 2.1 and using the properties of the modulus, we have the following: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,𝑟)01/𝑟1𝑟||ğ‘“âˆ’ğ‘¡î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡1/21/𝑟1𝑡−𝑟||ğ‘“î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡(𝑟−1)/𝑟1/2𝑟−1𝑟||ğ‘“âˆ’ğ‘¡î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡1(𝑟−1)/𝑟𝑡−𝑟−1𝑟||ğ‘“î…ž||(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡.(2.5)
Since |𝑓′| is (𝛼,𝑚)-convex on [ğ‘Ž,𝑏], we know that for any 𝑡∈[0,1]||ğ‘“î…ž||(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)≤𝑡𝛼||ğ‘“î…ž||(ğ‘Ž)+𝑚(1−𝑡𝛼)||ğ‘“î…ž||.(𝑏)(2.6)
By (2.5) and (2.6), we get the following: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,𝑟)01/𝑟1𝑟𝑡−𝑡𝛼||ğ‘“î…ž||(ğ‘Ž)+𝑚(1−𝑡𝛼)||ğ‘“î…ž||+(𝑏)𝑑𝑡1/21/𝑟1𝑡−𝑟𝑡𝛼||ğ‘“î…ž||(ğ‘Ž)+𝑚(1−𝑡𝛼)||ğ‘“î…ž||+(𝑏)𝑑𝑡(𝑟−1)/𝑟1/2𝑟−1𝑟𝑡−𝑡𝛼||ğ‘“î…ž||(ğ‘Ž)+𝑚(1−𝑡𝛼)||ğ‘“î…ž||+(𝑏)𝑑𝑡1(𝑟−1)/𝑟𝑡−𝑟−1𝑟𝑡𝛼||ğ‘“î…ž||(ğ‘Ž)+𝑚(1−𝑡𝛼)||ğ‘“î…ž||≤(𝑏)𝑑𝑡01/2|||1𝑟|||𝑡−𝑡𝛼𝑑𝑡+11/2|||𝑟−1𝑟|||𝑡−𝑡𝛼||ğ‘“ğ‘‘ğ‘¡î…ž||+(ğ‘Ž)01/2|||1𝑟|||−𝑡(1−𝑡𝛼)𝑑𝑡+11/2|||𝑟−1𝑟|||−𝑡(1−𝑡𝛼𝑚||𝑓)ğ‘‘ğ‘¡î…ž||=𝜇(𝑏)11+𝜇21+𝜇31+𝜇41||ğ‘“î…ž||+𝜈(ğ‘Ž)11+𝜈21+𝜈31+𝜈41𝑚||ğ‘“î…ž||,(𝑏)(2.7) which completes the proof.

Corollary 2.3. In Theorem 2.2, (i) if we choose 𝛼=1 and 𝑟=2, then we have the following: ||(ğ‘šğ‘âˆ’ğ‘Ž)ğ‘†ğ‘ğ‘Ž||=||||(𝑓)(1,𝑚,2)𝑓(ğ‘Ž)+𝑓(𝑚𝑏)2−1î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“||||≤(𝑥)ğ‘‘ğ‘¥ğ‘šğ‘âˆ’ğ‘Ž8||ğ‘“î…ž||||𝑓(ğ‘Ž)+ğ‘šî…ž||,(𝑏)(2.8) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have the following ||(ğ‘šğ‘âˆ’ğ‘Ž)ğ‘†ğ‘ğ‘Ž||=||||1(𝑓)(1,𝑚,6)6𝑓(ğ‘Ž)+4ğ‘“ğ‘Ž+𝑚𝑏2−1+𝑓(𝑚𝑏)î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“||||≤5(𝑥)𝑑𝑥||𝑓72(ğ‘šğ‘âˆ’ğ‘Ž)||||𝑓(ğ‘Ž)+ğ‘šî…ž||.(𝑏)(2.9)

Theorem 2.4. Under the same notations in Theorem 2.2, if |𝑓′|ğ‘žâˆˆğ¾ğ›¼ğ‘š([ğ‘Ž,𝑏]), for some (𝛼,𝑚)∈(0,1]2, 𝑚𝑏>ğ‘Ž and ğ‘ž>1 with 1/𝑝+1/ğ‘ž=1, then, for any 𝑟≥2, the following inequality holds: ||ğ‘†ğ‘ğ‘Ž||≤1(𝑓)(𝛼,𝑚,𝑟)2𝑟21/𝑝𝜇11||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈11𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+𝜇41||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈41𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡+18𝑟−2𝑟21/𝑝𝜇21||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+𝜈21𝑚||ğ‘“î…ž(||𝑏)ğ‘žî€¸1/ğ‘ž+𝜇31||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+𝜈31𝑚||ğ‘“î…ž(||𝑏)ğ‘žî€¸1/ğ‘žî‚‡.(2.10)

Proof. From Lemma 2.1 and using the properties of modulus, we have the following: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,𝑟)01/𝑟1𝑟||ğ‘“âˆ’ğ‘¡î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡1/21/𝑟1𝑡−𝑟||ğ‘“î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡(𝑟−1)/𝑟1/2𝑟−1𝑟||ğ‘“âˆ’ğ‘¡î…ž||+(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡1(𝑟−1)/𝑟𝑡−𝑟−1𝑟||ğ‘“î…ž||(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡.(2.11)
Using the power-mean integral inequality and (𝛼,𝑚)-convexity of |𝑓′|ğ‘ž for any 𝑡∈[0,1], we have the following(a)01/𝑟1𝑟||ğ‘“âˆ’ğ‘¡î…ž||(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡11||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈11𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,(2.12)(b)1/21/𝑟1𝑡−𝑟||||𝑓(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡21||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈21𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,(2.13)(c)(𝑟−1)/𝑟1/2𝑟−1𝑟||||−𝑡𝑓(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡31||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈31𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,(2.14)(d)1(𝑟−1)/𝑟𝑡−𝑟−1𝑟||||𝑓(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡41||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈41𝑚||ğ‘“î…ž||(𝑏)ğ‘ž.(2.15)
By the similar way as the above inequalities (a)–(d), we have the following:(a′)01/𝑟1𝑟||ğ‘“âˆ’ğ‘¡î…ž||1(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡≤2𝑟21/𝑝𝜇11||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈11𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¾1/ğ‘ž,(2.16)(b′)1/21/𝑟1𝑡−𝑟||ğ‘“î…ž||1(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡≤8𝑟−2𝑟21/𝑝𝜇21||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈21𝑚||ğ‘“î…ž||(𝑏)1/ğ‘ž,(2.17)(c′)(𝑟−1)/𝑟1/2𝑟−1𝑟||ğ‘“âˆ’ğ‘¡î…ž||1(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡≤8𝑟−2𝑟21/𝑝𝜇31||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈31𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¾1/ğ‘ž,(2.18)(d′)1(𝑟−1)/𝑟𝑡−𝑟−1𝑟||ğ‘“î…ž||1(ğ‘¡ğ‘Ž+𝑚(1−𝑡)𝑏)𝑑𝑡≤2𝑟21/𝑝𝜇41||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈41𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¾1/ğ‘ž.(2.19)
By (2.11) and (2.16)–(2.19) the assertion (2.10) holds.

Corollary 2.5. In Theorem 2.4, (i) if we choose 𝛼=1 and 𝑟=2, then we have that ||||𝑓(ğ‘Ž)+𝑓(𝑚𝑏)2−1î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“||||≤6(𝑥)𝑑𝑥−1/ğ‘ž8||𝑓(ğ‘šğ‘âˆ’ğ‘Ž)||(ğ‘Ž)ğ‘ž||𝑓+5ğ‘šî…ž||(𝑏)ğ‘žî€¾1/ğ‘ž+5||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šî…ž||(𝑏)ğ‘žî€¾1/ğ‘žî‚„,(2.20) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have that ||||16𝑓(ğ‘Ž)+4ğ‘“ğ‘Ž+𝑚𝑏2−1+𝑓(𝑚𝑏)î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“||||1(𝑥)𝑑𝑥≤(ğ‘šğ‘âˆ’ğ‘Ž)172181/ğ‘žî‚†î€·||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+17ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+||𝑓17||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡+25228852/ğ‘žÃ—î‚†î€·7||ğ‘“î…ž(||ğ‘Ž)ğ‘ž||𝑓+11ğ‘šî…ž(||𝑏)ğ‘žî€¸1/ğ‘ž+||𝑓11(||ğ‘Ž)ğ‘ž||𝑓+7ğ‘šî…ž(||𝑏)ğ‘žî€¸1/ğ‘ž.(2.21)

Theorem 2.6. Under the same notations in Theorem 2.2, if |𝑓′|ğ‘žâˆˆğ¾ğ›¼ğ‘š([ğ‘Ž,𝑏]), for some (𝛼,𝑚)∈(0,1]2, 𝑚𝑏>ğ‘Ž and ğ‘ž>1 with 1/𝑝+1/ğ‘ž=1, then, for any 𝑟≥2, the following inequality holds: ||ğ‘†ğ‘ğ‘Ž||≤1(𝑓)(𝛼,𝑚,𝑟)𝑟𝑝+11/𝑝𝜇12||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈12𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+𝜇42||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈42𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡+(𝑟−2)𝑝+12𝑝+1𝑟𝑝+11/𝑝𝜇22||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈22𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+𝜇32||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈32𝑚||ğ‘“î…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡,(2.22) where 𝜇12=1𝑟𝛼+1(,𝜇𝛼+1)22=𝑟𝛼+1−2𝛼+12𝛼+1𝑟𝛼+1,𝜇(𝛼+1)32=2𝛼+1(𝑟−1)𝛼+1−𝑟𝛼+12𝛼+1𝑟𝛼+1,𝜇(𝛼+1)42=𝑟𝛼+1−(𝑟−1)𝛼+1𝑟𝛼+1(,𝜈𝛼+1)12=1𝑟−𝜇12,𝜈22=𝑟−22𝑟+𝜇22,𝜈32=𝑟−22𝑟+𝜇32,𝜈42=1𝑟−𝜇42.(2.23)

Proof. Suppose that ğ‘ž>1. From Lemma 2.1, using the Hölder integral inequality, we get the following: ||ğ‘†ğ‘ğ‘Ž||≤(𝑓)(𝛼,𝑚,𝑟)01/𝑟1𝑟−𝑡𝑝𝑑𝑡1/𝑝01/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+1/21/𝑟1𝑡−𝑟𝑝𝑑𝑡1/𝑝1/21/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+(𝑟−1)/𝑟1/2𝑟−1𝑟−𝑡𝑝𝑑𝑡1/𝑝(𝑟−1)/𝑟1/2||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+1(𝑟−1)/𝑟𝑡−𝑟−1𝑟𝑝𝑑𝑡1/𝑝1(𝑟−1)/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘žâ‰¤î‚†1𝑟𝑝+11/𝑝01/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+12𝑝+1𝑟−2𝑟𝑝+11/𝑝1/21/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+12𝑝+1𝑟−2𝑟𝑝+11/𝑝(𝑟−1)/𝑟1/2||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž+1𝑟𝑝+11/𝑝1(𝑟−1)/𝑟||ğ‘“î…ž||(𝑡𝑏+𝑚(1−𝑡)ğ‘Ž)ğ‘žî‚¼ğ‘‘ğ‘¡1/ğ‘ž,(2.24) where we have used the fact that 1/2<(1/(𝑝+1))1/𝑝<1.
Since |𝑓′|ğ‘žâˆˆğ¾ğ‘šğ›¼([ğ‘Ž,𝑏]) for some fixed 𝛼∈(0,1] and 𝑚∈[0,1], we have the followings: 01/𝑟||ğ‘“î…ž||(𝑡𝑏+(1−𝑡)ğ‘Ž)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡12||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈12𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,1/21/𝑟||𝑓||(𝑡𝑏+(1−𝑡)ğ‘Ž)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡22||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈22𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,(𝑟−1)/𝑟1/2||ğ‘“î…ž||(𝑡𝑏+(1−𝑡)ğ‘Ž)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡32||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈32𝑚||ğ‘“î…ž||(𝑏)ğ‘ž,1(𝑟−1)/𝑟||𝑓||(𝑡𝑏+(1−𝑡)ğ‘Ž)ğ‘žğ‘‘ğ‘¡â‰¤ğœ‡42||ğ‘“î…ž||(ğ‘Ž)ğ‘ž+𝜈42𝑚||ğ‘“î…ž||(𝑏)ğ‘ž.(2.25)
Hence, if we combine the inequalities in (2.24)-(2.25), we get the desired result.

Corollary 2.7. In Theorem 2.6, (i) if we choose 𝛼=1 and 𝑟=2, then we have that ||||𝑓(ğ‘Ž)+𝑓(𝑚𝑏)2−1î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘ğ‘“||||≤1(𝑥)𝑑𝑥41+1/ğ‘žî‚†î€·||𝑓(ğ‘šğ‘âˆ’ğ‘Ž)||(ğ‘Ž)ğ‘ž||𝑓+3ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+3||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡,(2.26) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have ||||16𝑓(ğ‘Ž)+4ğ‘“ğ‘Ž+𝑚𝑏2−1+𝑓(𝑚𝑏)î€œğ‘šğ‘âˆ’ğ‘Žğ‘Žğ‘šğ‘||||1𝑓(𝑥)𝑑𝑥≤(ğ‘šğ‘âˆ’ğ‘Ž)61+2/ğ‘žî‚†î€·||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+11ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+||𝑓11||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡+132+1/ğ‘žî‚†î€·||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+2ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘ž+2||ğ‘“î…ž||(ğ‘Ž)ğ‘ž||𝑓+ğ‘šî…ž||(𝑏)ğ‘žî€¸1/ğ‘žî‚‡î‚¹,(2.27) where we have used the fact that (1/2)1/ğ‘ž<1.

3. Applications to Special Means

Now using the results of Section 2, we give some applications to the following special means of positive real numbers ğ‘Ž,𝑏∈ℝ+ with ğ‘â‰¥ğ‘Ž.(1) The arithmetic mean: 𝐴(ğ‘Ž,𝑏)=(ğ‘Ž+𝑏)/2.(2)The geometric mean: √𝐺(ğ‘Ž,𝑏)=ğ‘Žğ‘.(3)The logarithmic mean: 𝐿(ğ‘Ž,𝑏)=(ğ‘âˆ’ğ‘Ž)/(ln𝑏−lnğ‘Ž) for ğ‘Žâ‰ ğ‘.(4)The harmonic mean: 𝐻(ğ‘Ž,𝑏)=2ğ‘Žğ‘/(ğ‘Ž+𝑏).(5)The power mean: 𝑀𝑟(ğ‘Ž,𝑏)=((ğ‘Žğ‘Ÿ+𝑏𝑟)/2)1/𝑟, 𝑟≥1,ğ‘Ž,𝑏∈ℝ.(6)The generalized logarithmic mean: 𝐿𝑛𝑏(ğ‘Ž,𝑏)=𝑛+1âˆ’ğ‘Žğ‘›+1(ğ‘âˆ’ğ‘Ž)(𝑛+1)1/𝑛,ğ‘Žâ‰ ğ‘.(3.1)(7)The identric mean:⎧⎪⎨⎪⎩1𝐼(ğ‘Ž,𝑏)=ğ‘Žğ‘Ž=ğ‘ğ‘’î‚µğ‘ğ‘ğ‘Žğ‘Žî‚¶1/(ğ‘âˆ’ğ‘Ž)ğ‘Žâ‰ ğ‘.(3.2)

Proposition 3.1. For 𝑛∈(−∞,0)∪[1,∞)⧵{−1} and [ğ‘Ž,𝑏]∈[0,𝑏∗] with 𝑏∗>0, we have the following inequalities: ||(ğ‘Ž)𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛)−𝐿𝑛𝑛||≤(ğ‘Ž,𝑚𝑏)ğ‘šğ‘âˆ’ğ‘Ž8|𝑛|𝑀𝑛−1𝑛−1î€·ğ‘Ž,𝑚1/(𝑛−1)𝑏.|||1(𝑏)3𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛2)+3𝐴𝑛(ğ‘Ž,𝑚𝑏)−𝐿𝑛𝑛|||≤5(ğ‘Ž,𝑚𝑏)72(ğ‘šğ‘âˆ’ğ‘Ž)|𝑛|𝑀𝑛−1𝑛−1î€·ğ‘Ž,𝑚1/(𝑛−1)𝑏.(3.3)

Proof. The assertions follow from Corollary 2.3 for 𝑓(𝑥)=𝑥𝑛.

Proposition 3.2. For [ğ‘Ž,𝑏]∈[0,𝑏∗], we have the following inequalities: ||𝐻(ğ‘Ž)−1(ğ‘Ž,𝑚𝑏)−𝐿−1||≤(ğ‘Ž,𝑚𝑏)ğ‘šğ‘âˆ’ğ‘Ž8ğ‘Ž2𝑏2𝑀22𝑚1/2,|||1ğ‘Ž,𝑏(𝑏)3𝐻−12(ğ‘Ž,𝑚𝑏)+3𝐴−1(ğ‘Ž,𝑚𝑏)−𝐿−1|||≤(ğ‘Ž,𝑚𝑏)5(ğ‘šğ‘âˆ’ğ‘Ž)72ğ‘Ž2𝑏2𝑀22𝑚1/2.ğ‘Ž,𝑏(3.4)

Proof. The assertions follow from Corollary 2.3 for 𝑓(𝑥)=1/𝑥.

Proposition 3.3. For 𝑛∈(−∞,0)∪[1,∞)⧵{−1} and [ğ‘Ž,𝑏]∈[0,𝑏∗], we have the following inequalities: ||(ğ‘Ž)𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛)−𝐿𝑛𝑛||≤3(ğ‘Ž,𝑚𝑏)−1/ğ‘ž8𝐴(ğ‘šğ‘âˆ’ğ‘Ž)|𝑛|×1/ğ‘žî€·ğ‘Ž(𝑛−1)ğ‘ž,5𝑚𝑏(𝑛−1)ğ‘žî€¸1/ğ‘ž+𝐴1/ğ‘žî€·5ğ‘Ž(𝑛−1)ğ‘ž,𝑚𝑏(𝑛−1)ğ‘žî€¸1/ğ‘žî‚‡,|||1(𝑏)3𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛2)+3𝐴𝑛(ğ‘Ž,𝑚𝑏)−𝐿𝑛𝑛|||≤1(ğ‘Ž,𝑚𝑏)17291/ğ‘žî€½ğ´(ğ‘šğ‘âˆ’ğ‘Ž)|𝑛|×1/ğ‘žî€·ğ‘Ž(𝑛−1)ğ‘ž,17𝑚𝑏(𝑛−1)ğ‘žî€¸+𝐴1/ğ‘žî€·17ğ‘Ž(𝑛−1)ğ‘ž,𝑚𝑏(𝑛−1)ğ‘ž+252288251/ğ‘ž(Ã—î€½ğ´ğ‘šğ‘âˆ’ğ‘Ž)|𝑛|1/ğ‘žî€·7ğ‘Ž(𝑛−1)ğ‘ž,11𝑚𝑏(𝑛−1)ğ‘žî€¸+𝐴1/ğ‘žî€·11ğ‘Ž(𝑛−1)ğ‘ž,7𝑚𝑏(𝑛−1)ğ‘ž.(3.5)

Proof. The assertions follow from Corollary 2.5 for 𝑓(𝑥)=𝑥𝑛.

Proposition 3.4. For 𝑛∈(−∞,0)∪[1,∞)⧵{−1} and [ğ‘Ž,𝑏]∈[0,𝑏∗], we have the following inequalities: ||(ğ‘Ž)𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛)−𝐿𝑛𝑛||≤1(ğ‘Ž,𝑏)22+1/ğ‘žî‚¶Ã—î€½ğ´|𝑛|(ğ‘šğ‘âˆ’ğ‘Ž)1/ğ‘žî€·ğ‘Ž(𝑛−1)ğ‘ž,3𝑚𝑏(𝑛−1)ğ‘žî€¸+𝐴1/ğ‘žî€·3ğ‘Ž(𝑛−1)ğ‘ž,𝑚𝑏(𝑛−1)ğ‘ž,|||1(𝑏)3𝐴(ğ‘Žğ‘›,𝑚𝑛𝑏𝑛2)+3𝐴𝑛(ğ‘Ž,𝑚𝑏)−𝐿𝑛𝑛|||≤1(ğ‘Ž,𝑚𝑏)61+1/ğ‘žî‚€131/ğ‘ž|𝐴𝑛|(ğ‘šğ‘âˆ’ğ‘Ž)×1/ğ‘žî€·ğ‘Ž(𝑛−1)ğ‘ž,11𝑚𝑏(𝑛−1)ğ‘žî€¸+𝐴1/ğ‘žî€·11ğ‘Ž(𝑛−1)ğ‘ž,𝑚𝑏(𝑛−1)ğ‘ž+132+1/ğ‘žî€·21/ğ‘žî€¸î€½ğ´|𝑛|(ğ‘šğ‘âˆ’ğ‘Ž)×1/ğ‘žî€·ğ‘Ž(𝑛−1)ğ‘ž,2𝑚𝑏(𝑛−1)ğ‘žî€¸+𝐴1/ğ‘žî€·2ğ‘Ž(𝑛−1)ğ‘ž,𝑚𝑏(𝑛−1)ğ‘ž.(3.6)

Proof. The assertions follow from Corollary 2.7 for 𝑓(𝑥)=𝑥𝑛.

Proposition 3.5. For [ğ‘Ž,𝑏]∈[0,𝑏∗], we have the following inequalities: ||||(ğ‘Ž)ln𝐼(ğ‘Ž,𝑏)||||≤𝐺(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Žî‚µ1ğ‘Žğ‘41+1/ğ‘žî‚¶î€½(ğ‘Žğ‘ž+3ğ‘ğ‘ž)1/ğ‘ž+(3ğ‘Žğ‘ž+ğ‘ğ‘ž)1/ğ‘žî€¾,(||||𝑏)ln𝐼(ğ‘Ž,𝑏)𝐺1/3(ğ‘Ž,𝑏)𝐴2/3||||≤(ğ‘Ž,𝑏)ğ‘âˆ’ğ‘Žî‚¸î‚€1ğ‘Žğ‘61+2/ğ‘žî€½(ğ‘Žğ‘ž+11ğ‘ğ‘ž)1/ğ‘ž+(11ğ‘Žğ‘ž+ğ‘ğ‘ž)1/ğ‘žî€¾+132+1/ğ‘žî€½(ğ‘Žğ‘ž+2ğ‘ğ‘ž)1/ğ‘ž+(2ğ‘Žğ‘ž+ğ‘ğ‘ž)1/ğ‘žî€¾î‚¹.(3.7)

Proof. The assertions follow from Corollary 2.7 for 𝑓(𝑥)=−ln𝑥 and 𝑚=1.

Acknowledgment

The author is so indebted to the referee who read carefully through the paper very well and mentioned many scientifically and expressional mistakes.

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Copyright © 2012 Jaekeun Park. 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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