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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 809689, 12 pages
http://dx.doi.org/10.1155/2012/809689
Research Article

Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable (𝛼,π‘š)-Convex Mappings

Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea

Received 13 July 2011; Revised 13 November 2011; Accepted 28 November 2011

Academic Editor: JewgeniΒ Dshalalow

Copyright Β© 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.

Abstract

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

1. Introduction

Recall that, for some fixed π›Όβˆˆ(0,1] and π‘šβˆˆ[0,1], a mapping π‘“βˆΆπ•€βŠ†[0,∞)→ℝ is said to be (𝛼,π‘š)-convex on an interval 𝕀 if the inequality 𝑓(𝑑π‘₯+π‘š(1βˆ’π‘‘)𝑦)≀𝑑𝛼𝑓(π‘₯)+π‘š(1βˆ’π‘‘π›Ό)𝑓(𝑦)(1.1) holds for π‘₯,π‘¦βˆˆπ•€, and π‘‘βˆˆ[0,1].

Denote by πΎπ›Όπ‘š(𝕀) the set of all (𝛼,π‘š)-convex mappings on 𝕀. For recent results and generalizations concerning π‘š-convex and (𝛼,π‘š)-convex mappings, see [1–4].

For the simplicities of notations, for π‘“βˆˆπΎπ›Όπ‘š(𝕀), let us denote (π‘Žβˆ’π‘šπ‘)π‘†π‘π‘Ž1(𝑓)(𝛼,π‘š,π‘Ÿ)=π‘Ÿξ‚†ξ‚€π‘“(π‘Ž)+(π‘Ÿβˆ’2)π‘“π‘Ž+π‘šπ‘2ξ‚ξ‚‡βˆ’1+𝑓(π‘šπ‘)ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“(π‘₯)𝑑π‘₯.(1.2)

In [1, 3], KlaričiΔ‡ Bakula and Γ–zdemir et al., proved the following Hadamard’s inequalities for mappings whose second derivative in absolute value aroused to the π‘ž-th (π‘žβ‰₯1) power are (𝛼,π‘š)-convex.

Theorem 1.1. Let π‘“βˆΆπ•€βŠ†[0,π‘βˆ—]→ℝ be a twice differentiable mapping on the interior 𝕀0 of an interval 𝕀 such that π‘“ξ…žξ…žβˆˆπΏ1([π‘Ž,𝑏]), where π‘Ž,π‘βˆˆπ•€ with π‘Ž<𝑏 and π‘βˆ—>0. If |π‘“ξ…žξ…ž|π‘ž is (𝛼,π‘š)-convex on [π‘Ž,𝑏] for (𝛼,π‘š)∈[0,1]2 and π‘žβ‰₯1 with 1/𝑝+1/π‘ž=1, then the following inequality holds: ||𝑆(π‘Ž)π‘π‘Ž||≀(𝑓)(𝛼,π‘š,2)π‘šπ‘βˆ’π‘Ž2ξ‚€161/π‘ξ€½πœ‡||π‘“ξ…žξ…ž||(π‘Ž)π‘ž||𝑓+π‘šπœˆξ…žξ…ž||(𝑏)π‘žξ€Ύ1/π‘ž,(1.3) where 1πœ‡=𝛼(𝛼+2)(𝛼+3),𝜈=2+5𝛼,6(𝛼+2)(𝛼+3)(1.4)||𝑆(𝑏)π‘π‘Ž||≀(𝑓)(𝛼,π‘š,6)π‘šπ‘βˆ’π‘Ž2ξ‚€121/π‘ξ€½πœ‡0||π‘“ξ…žξ…ž||(π‘Ž)π‘ž+π‘šπœˆ0||π‘“ξ…žξ…ž||(𝑏)π‘žξ€Ύ1/π‘ž,(1.5) where πœ‡0=ξ‚΅π‘žξ‚Άπ‘ž+𝛼+2Ξ“(𝛼+2)Ξ“(π‘ž),πœˆΞ“(π‘ž+𝛼+1)0=1(βˆ’ξ‚΅π‘žπ‘ž+1)(π‘ž+2)ξ‚Άπ‘ž+𝛼+2Ξ“(𝛼+2)Ξ“(π‘ž),Ξ“(π‘ž+𝛼+1)(1.6) where ξ€œΞ“(π‘₯)=∞0π‘’βˆ’π‘‘π‘‘π‘₯βˆ’1𝑑𝑑,π‘₯>0.(1.7)

Theorem 1.2. Under the same notations in Theorem 2.2, if |π‘“ξ…žξ…ž|π‘ž is (𝛼,π‘š)-convex on [π‘Ž,𝑏] for (𝛼,π‘š)∈[0,1]2 and π‘ž>1 with 1/𝑝+1/π‘ž=1, then the following inequality holds: ||π‘†π‘π‘Ž||≀(𝑓)(𝛼,π‘š,2)π‘šπ‘βˆ’π‘Ž8ξ‚»Ξ“(𝑝+1)ξ‚ΌΞ“(𝑝+3/2)1/𝑝||π‘“ξ…žξ…ž||(π‘Ž)π‘ž1||𝑓𝛼+1+π‘šξ…žξ…ž||(𝑏)π‘žξ‚€π›Όπ›Ό+11/π‘ž.(1.8)

Note that for (𝛼,π‘š)∈{(0,0),(𝛼,0),(1,0),(1,π‘š),(1,1),(𝛼,1)} one obtains the following classes of functions: increasing, 𝛼-starshaped, starshaped, π‘š-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,π‘š)-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 π‘“βˆΆπ•€βŠ‚[0,∞)→ℝ be an absolutely continuous mapping on [π‘Ž,𝑏] such that π‘“β€²βˆˆπΏπ‘([π‘Ž,𝑏]), where π‘Ž,π‘βˆˆπ•€ with π‘Ž<𝑏. Then the following inequality holds: ||π‘†π‘π‘Ž||≀(𝑓)(1,1,6)(π‘βˆ’π‘Ž)βˆ’1/𝑝6ξ‚»2π‘ž+1+1ξ‚Ό3(π‘ž+1)1/π‘žβ€–β€–π‘“ξ…žβ€–β€–π‘.(1.9)

The readers can estimate the π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ(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,π‘š)-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 πΎπ›Όπ‘š(𝕀)

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

Lemma 2.1. Let π‘“βˆΆπ•€βŠ†[0,π‘βˆ—]→ℝ be a differentiable mapping on the interior 𝕀0 of an interval 𝕀, where π‘Ž,π‘βˆˆπ•€ with 0β‰€π‘Ž<𝑏 and π‘βˆ—>0. If π‘“ξ…žβˆˆπΏ1([π‘Ž,𝑏]), then, for π‘Ÿβ‰₯2 and β„Žβˆˆ(0,1) with 1/π‘Ÿβ‰€β„Žβ‰€(π‘Ÿβˆ’1)/π‘Ÿ, the following equality holds: π‘†π‘π‘Žξ€œ(𝑓)(𝛼,π‘š,π‘Ÿ)=10𝑝(π‘Ÿ,𝑑)π‘“ξ…ž(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑(2.1) for π‘“βˆˆπΎπ›Όπ‘š([π‘Ž,𝑏]) and each π‘‘βˆˆ[0,1], where ⎧βŽͺ⎨βŽͺ⎩1𝑝(π‘Ÿ,𝑑)=π‘‘βˆ’π‘Ÿξ‚ƒ1π‘‘βˆˆ0,2ξ‚„,π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚€1π‘‘βˆˆ2ξ‚„.,1(2.2)

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

Theorem 2.2. Let π‘“βˆΆπ•€βŠ‚[0,π‘βˆ—]→ℝ be a differentiable mapping on 𝕀0 such that π‘“ξ…žβˆˆπΏ([π‘Ž,𝑏]), where π‘Ž,π‘βˆˆπ•€ with 0β‰€π‘Ž<𝑏<∞ and π‘βˆ—>0. If |𝑓′|βˆˆπΎπ›Όπ‘š([π‘Ž,𝑏]), for some (𝛼,π‘š)∈(0,1]2 and π‘šπ‘>π‘Ž, then, for any π‘Ÿβ‰₯2, the following inequality holds: ||π‘†π‘π‘Ž||β‰€ξ€½πœ‡(𝑓)(𝛼,π‘š,π‘Ÿ)11+πœ‡21+πœ‡31+πœ‡41ξ€Ύ||π‘“ξ…ž||+ξ€½πœˆ(π‘Ž)11+𝜈21+𝜈31+𝜈41ξ€Ύπ‘š||π‘“ξ…ž||,(𝑏)(2.3) where πœ‡11=1(𝛼+1)(𝛼+2)π‘Ÿπ›Ό+2,πœ‡21=1(𝛼+1)(𝛼+2)π‘Ÿπ›Ό+2+(𝛼+1)π‘Ÿβˆ’2(𝛼+2)2𝛼+2,πœ‡(𝛼+1)(𝛼+2)π‘Ÿ31=(π‘Ÿβˆ’1)𝛼+2(𝛼+1)(𝛼+2)π‘Ÿπ›Ό+2+2(𝛼+2)βˆ’3π‘Ÿ2𝛼+2,πœ‡(𝛼+1)(𝛼+2)π‘Ÿ41=(π‘Ÿβˆ’1)𝛼+2(𝛼+1)(𝛼+2)π‘Ÿπ›Ό+2+(𝛼+2)βˆ’π‘Ÿ,𝜈(𝛼+1)(𝛼+2)π‘Ÿ11=12π‘Ÿ2βˆ’πœ‡11,𝜈21=(π‘Ÿβˆ’2)28π‘Ÿ2βˆ’πœ‡21,𝜈31=(π‘Ÿβˆ’2)28π‘Ÿ2βˆ’πœ‡31,𝜈41=12π‘Ÿ2βˆ’πœ‡41.(2.4)

Proof. From Lemma 2.1 and using the properties of the modulus, we have the following: ||π‘†π‘π‘Ž||β‰€ξ€œ(𝑓)(𝛼,π‘š,π‘Ÿ)01/π‘Ÿξ‚€1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚||π‘“ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚||π‘“ξ…ž||(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑.(2.5)
Since |𝑓′| is (𝛼,π‘š)-convex on [π‘Ž,𝑏], we know that for any π‘‘βˆˆ[0,1]||π‘“ξ…ž||(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)≀𝑑𝛼||π‘“ξ…ž||(π‘Ž)+π‘š(1βˆ’π‘‘π›Ό)||π‘“ξ…ž||.(𝑏)(2.6)
By (2.5) and (2.6), we get the following: ||π‘†π‘π‘Ž||β‰€ξ€œ(𝑓)(𝛼,π‘š,π‘Ÿ)01/π‘Ÿξ‚€1π‘Ÿξ‚ξ€½π‘‘βˆ’π‘‘π›Ό||π‘“ξ…ž||(π‘Ž)+π‘š(1βˆ’π‘‘π›Ό)||π‘“ξ…ž||ξ€Ύ+ξ€œ(𝑏)𝑑𝑑1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚ξ€½π‘‘π›Ό||π‘“ξ…ž||(π‘Ž)+π‘š(1βˆ’π‘‘π›Ό)||π‘“ξ…ž||ξ€Ύ+ξ€œ(𝑏)𝑑𝑑(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚ξ€½π‘‘βˆ’π‘‘π›Ό||π‘“ξ…ž||(π‘Ž)+π‘š(1βˆ’π‘‘π›Ό)||π‘“ξ…ž||ξ€Ύ+ξ€œ(𝑏)𝑑𝑑1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚ξ€½π‘‘π›Ό||π‘“ξ…ž||(π‘Ž)+π‘š(1βˆ’π‘‘π›Ό)||π‘“ξ…ž||ξ€Ύβ‰€ξ‚»ξ€œ(𝑏)𝑑𝑑01/2|||1π‘Ÿ|||π‘‘βˆ’π‘‘π›Όξ€œπ‘‘π‘‘+11/2|||π‘Ÿβˆ’1π‘Ÿ|||π‘‘βˆ’π‘‘π›Όξ‚Ό||π‘“π‘‘π‘‘ξ…ž||+ξ‚»ξ€œ(π‘Ž)01/2|||1π‘Ÿ|||βˆ’π‘‘(1βˆ’π‘‘π›Όξ€œ)𝑑𝑑+11/2|||π‘Ÿβˆ’1π‘Ÿ|||βˆ’π‘‘(1βˆ’π‘‘π›Όξ‚Όπ‘š||𝑓)π‘‘π‘‘ξ…ž||=ξ€½πœ‡(𝑏)11+πœ‡21+πœ‡31+πœ‡41ξ€Ύ||π‘“ξ…ž||+ξ€½πœˆ(π‘Ž)11+𝜈21+𝜈31+𝜈41ξ€Ύπ‘š||π‘“ξ…ž||,(𝑏)(2.7) which completes the proof.

Corollary 2.3. In Theorem 2.2, (i) if we choose 𝛼=1 and π‘Ÿ=2, then we have the following: ||(π‘šπ‘βˆ’π‘Ž)π‘†π‘π‘Ž||=||||(𝑓)(1,π‘š,2)𝑓(π‘Ž)+𝑓(π‘šπ‘)2βˆ’1ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“||||≀(π‘₯)𝑑π‘₯π‘šπ‘βˆ’π‘Ž8ξ€½||π‘“ξ…ž||||𝑓(π‘Ž)+π‘šξ…ž||ξ€Ύ,(𝑏)(2.8) and (ii) if we choose 𝛼=1 and π‘Ÿ=6, then we have the following ||(π‘šπ‘βˆ’π‘Ž)π‘†π‘π‘Ž||=||||1(𝑓)(1,π‘š,6)6𝑓(π‘Ž)+4π‘“π‘Ž+π‘šπ‘2ξ‚ξ‚‡βˆ’1+𝑓(π‘šπ‘)ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“||||≀5(π‘₯)𝑑π‘₯ξ€½||𝑓72(π‘šπ‘βˆ’π‘Ž)ξ…ž||||𝑓(π‘Ž)+π‘šξ…ž||ξ€Ύ.(𝑏)(2.9)

Theorem 2.4. Under the same notations in Theorem 2.2, if |𝑓′|π‘žβˆˆπΎπ›Όπ‘š([π‘Ž,𝑏]), for some (𝛼,π‘š)∈(0,1]2, π‘šπ‘>π‘Ž and π‘ž>1 with 1/𝑝+1/π‘ž=1, then, for any π‘Ÿβ‰₯2, the following inequality holds: ||π‘†π‘π‘Ž||≀1(𝑓)(𝛼,π‘š,π‘Ÿ)2π‘Ÿ21/π‘ξ‚†ξ€·πœ‡11||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈11π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·πœ‡41||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈41π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡+ξ‚»18ξ‚€π‘Ÿβˆ’2π‘Ÿξ‚2ξ‚Ό1/π‘ξ‚†ξ€·πœ‡21||π‘“ξ…ž(||π‘Ž)π‘ž+𝜈21π‘š||π‘“ξ…ž(||𝑏)π‘žξ€Έ1/π‘ž+ξ€·πœ‡31||π‘“ξ…ž(||π‘Ž)π‘ž+𝜈31π‘š||π‘“ξ…ž(||𝑏)π‘žξ€Έ1/π‘žξ‚‡.(2.10)

Proof. From Lemma 2.1 and using the properties of modulus, we have the following: ||π‘†π‘π‘Ž||β‰€ξ€œ(𝑓)(𝛼,π‘š,π‘Ÿ)01/π‘Ÿξ‚€1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚||π‘“ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||+ξ€œ(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚||π‘“ξ…ž||(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑.(2.11)
Using the power-mean integral inequality and (𝛼,π‘š)-convexity of |𝑓′|π‘ž for any π‘‘βˆˆ[0,1], we have the following(a)ξ€œ01/π‘Ÿξ‚€1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)π‘žπ‘‘π‘‘β‰€πœ‡11||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈11π‘š||π‘“ξ…ž||(𝑏)π‘ž,(2.12)(b)ξ€œ1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚||||𝑓(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)π‘žπ‘‘π‘‘β‰€πœ‡21||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈21π‘š||π‘“ξ…ž||(𝑏)π‘ž,(2.13)(c)ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚||||βˆ’π‘‘π‘“(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)π‘žπ‘‘π‘‘β‰€πœ‡31||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈31π‘š||π‘“ξ…ž||(𝑏)π‘ž,(2.14)(d)ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚||||𝑓(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)π‘žπ‘‘π‘‘β‰€πœ‡41||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈41π‘š||π‘“ξ…ž||(𝑏)π‘ž.(2.15)
By the similar way as the above inequalities (a)–(d), we have the following:(aβ€²)ξ€œ01/π‘Ÿξ‚€1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||1(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑≀2π‘Ÿ21/π‘ξ€½πœ‡11||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈11π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Ύ1/π‘ž,(2.16)(bβ€²)ξ€œ1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚||π‘“ξ…ž||ξ‚»1(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑≀8ξ‚€π‘Ÿβˆ’2π‘Ÿξ‚2ξ‚Ό1/π‘ξ€½πœ‡21||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈21π‘š||π‘“ξ…ž||ξ€Ύ(𝑏)1/π‘ž,(2.17)(cβ€²)ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚||π‘“βˆ’π‘‘ξ…ž||ξ‚»1(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑≀8ξ‚€π‘Ÿβˆ’2π‘Ÿξ‚2ξ‚Ό1/π‘ξ€½πœ‡31||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈31π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Ύ1/π‘ž,(2.18)(dβ€²)ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚||π‘“ξ…ž||1(π‘‘π‘Ž+π‘š(1βˆ’π‘‘)𝑏)𝑑𝑑≀2π‘Ÿ21/π‘ξ€½πœ‡41||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈41π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Ύ1/π‘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 π‘Ÿ=2, then we have that ||||𝑓(π‘Ž)+𝑓(π‘šπ‘)2βˆ’1ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“||||≀6(π‘₯)𝑑π‘₯βˆ’1/π‘ž8||𝑓(π‘šπ‘βˆ’π‘Ž)ξ…ž||(π‘Ž)π‘ž||𝑓+5π‘šξ…ž||(𝑏)π‘žξ€Ύ1/π‘ž+ξ€½5||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+π‘šξ…ž||(𝑏)π‘žξ€Ύ1/π‘žξ‚„,(2.20) and (ii) if we choose 𝛼=1 and π‘Ÿ=6, then we have that ||||16𝑓(π‘Ž)+4π‘“π‘Ž+π‘šπ‘2ξ‚ξ‚‡βˆ’1+𝑓(π‘šπ‘)ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“||||ξ‚Έξ‚€1(π‘₯)𝑑π‘₯≀(π‘šπ‘βˆ’π‘Ž)172181/π‘žξ‚†ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+17π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·||𝑓17ξ…ž||(π‘Ž)π‘ž||𝑓+π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡+ξ‚€25228852/π‘žΓ—ξ‚†ξ€·7||π‘“ξ…ž(||π‘Ž)π‘ž||𝑓+11π‘šξ…ž(||𝑏)π‘žξ€Έ1/π‘ž+ξ€·||𝑓11ξ…ž(||π‘Ž)π‘ž||𝑓+7π‘šξ…ž(||𝑏)π‘žξ€Έ1/π‘ž.(2.21)

Theorem 2.6. Under the same notations in Theorem 2.2, if |𝑓′|π‘žβˆˆπΎπ›Όπ‘š([π‘Ž,𝑏]), for some (𝛼,π‘š)∈(0,1]2, π‘šπ‘>π‘Ž and π‘ž>1 with 1/𝑝+1/π‘ž=1, then, for any π‘Ÿβ‰₯2, the following inequality holds: ||π‘†π‘π‘Ž||≀1(𝑓)(𝛼,π‘š,π‘Ÿ)π‘Ÿπ‘+11/π‘ξ‚†ξ€·πœ‡12||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈12π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·πœ‡42||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈42π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡+ξ‚»(π‘Ÿβˆ’2)𝑝+12𝑝+1π‘Ÿπ‘+1ξ‚Ό1/π‘ξ‚†ξ€·πœ‡22||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈22π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·πœ‡32||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈32π‘š||π‘“ξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡,(2.22) where πœ‡12=1π‘Ÿπ›Ό+1(,πœ‡π›Ό+1)22=π‘Ÿπ›Ό+1βˆ’2𝛼+12𝛼+1π‘Ÿπ›Ό+1,πœ‡(𝛼+1)32=2𝛼+1(π‘Ÿβˆ’1)𝛼+1βˆ’π‘Ÿπ›Ό+12𝛼+1π‘Ÿπ›Ό+1,πœ‡(𝛼+1)42=π‘Ÿπ›Ό+1βˆ’(π‘Ÿβˆ’1)𝛼+1π‘Ÿπ›Ό+1(,πœˆπ›Ό+1)12=1π‘Ÿβˆ’πœ‡12,𝜈22=π‘Ÿβˆ’22π‘Ÿ+πœ‡22,𝜈32=π‘Ÿβˆ’22π‘Ÿ+πœ‡32,𝜈42=1π‘Ÿβˆ’πœ‡42.(2.23)

Proof. Suppose that π‘ž>1. From Lemma 2.1, using the HΓΆlder integral inequality, we get the following: ||π‘†π‘π‘Ž||β‰€ξ‚»ξ€œ(𝑓)(𝛼,π‘š,π‘Ÿ)01/π‘Ÿξ‚€1π‘Ÿξ‚βˆ’π‘‘π‘ξ‚Όπ‘‘π‘‘1/π‘ξ‚»ξ€œ01/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+ξ‚»ξ€œ1/21/π‘Ÿξ‚€1π‘‘βˆ’π‘Ÿξ‚π‘ξ‚Όπ‘‘π‘‘1/π‘ξ‚»ξ€œ1/21/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+ξ‚»ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2ξ‚€π‘Ÿβˆ’1π‘Ÿξ‚βˆ’π‘‘π‘ξ‚Όπ‘‘π‘‘1/π‘ξ‚»ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+ξ‚»ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿξ‚€π‘‘βˆ’π‘Ÿβˆ’1π‘Ÿξ‚π‘ξ‚Όπ‘‘π‘‘1/π‘ξ‚»ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰€ξ‚†1π‘Ÿπ‘+11/π‘ξ‚»ξ€œ01/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+ξ‚»12𝑝+1ξ‚€π‘Ÿβˆ’2π‘Ÿξ‚π‘+1ξ‚Ό1/π‘ξ‚»ξ€œ1/21/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+ξ‚»12𝑝+1ξ‚€π‘Ÿβˆ’2π‘Ÿξ‚π‘+1ξ‚Ό1/π‘ξ‚»ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž+1π‘Ÿπ‘+11/π‘ξ‚»ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+π‘š(1βˆ’π‘‘)π‘Ž)π‘žξ‚Όπ‘‘π‘‘1/π‘ž,(2.24) where we have used the fact that 1/2<(1/(𝑝+1))1/𝑝<1.
Since |𝑓′|π‘žβˆˆπΎπ‘šπ›Ό([π‘Ž,𝑏]) for some fixed π›Όβˆˆ(0,1] and π‘šβˆˆ[0,1], we have the followings: ξ€œ01/π‘Ÿ||π‘“ξ…ž||(𝑑𝑏+(1βˆ’π‘‘)π‘Ž)π‘žπ‘‘π‘‘β‰€πœ‡12||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈12π‘š||π‘“ξ…ž||(𝑏)π‘ž,ξ€œ1/21/π‘Ÿ||𝑓||(𝑑𝑏+(1βˆ’π‘‘)π‘Ž)π‘žπ‘‘π‘‘β‰€πœ‡22||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈22π‘š||π‘“ξ…ž||(𝑏)π‘ž,ξ€œ(π‘Ÿβˆ’1)/π‘Ÿ1/2||π‘“ξ…ž||(𝑑𝑏+(1βˆ’π‘‘)π‘Ž)π‘žπ‘‘π‘‘β‰€πœ‡32||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈32π‘š||π‘“ξ…ž||(𝑏)π‘ž,ξ€œ1(π‘Ÿβˆ’1)/π‘Ÿ||𝑓||(𝑑𝑏+(1βˆ’π‘‘)π‘Ž)π‘žπ‘‘π‘‘β‰€πœ‡42||π‘“ξ…ž||(π‘Ž)π‘ž+𝜈42π‘š||π‘“ξ…ž||(𝑏)π‘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 π‘Ÿ=2, then we have that ||||𝑓(π‘Ž)+𝑓(π‘šπ‘)2βˆ’1ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘π‘“||||≀1(π‘₯)𝑑π‘₯41+1/π‘žξ‚†ξ€·||𝑓(π‘šπ‘βˆ’π‘Ž)ξ…ž||(π‘Ž)π‘ž||𝑓+3π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·3||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡,(2.26) and (ii) if we choose 𝛼=1 and π‘Ÿ=6, then we have ||||16𝑓(π‘Ž)+4π‘“π‘Ž+π‘šπ‘2ξ‚ξ‚‡βˆ’1+𝑓(π‘šπ‘)ξ€œπ‘šπ‘βˆ’π‘Žπ‘Žπ‘šπ‘||||ξ‚Έξ‚€1𝑓(π‘₯)𝑑π‘₯≀(π‘šπ‘βˆ’π‘Ž)61+2/π‘žξ‚†ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+11π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·||𝑓11ξ…ž||(π‘Ž)π‘ž||𝑓+π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡+ξ‚€132+1/π‘žξ‚†ξ€·||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+2π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘ž+ξ€·2||π‘“ξ…ž||(π‘Ž)π‘ž||𝑓+π‘šξ…ž||(𝑏)π‘žξ€Έ1/π‘žξ‚‡ξ‚Ή,(2.27) where we have used the fact that (1/2)1/π‘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 π‘Ž,π‘βˆˆβ„+ with 𝑏β‰₯π‘Ž.(1) The arithmetic mean: 𝐴(π‘Ž,𝑏)=(π‘Ž+𝑏)/2.(2)The geometric mean: √𝐺(π‘Ž,𝑏)=π‘Žπ‘.(3)The logarithmic mean: 𝐿(π‘Ž,𝑏)=(π‘βˆ’π‘Ž)/(lnπ‘βˆ’lnπ‘Ž) for π‘Žβ‰ π‘.(4)The harmonic mean: 𝐻(π‘Ž,𝑏)=2π‘Žπ‘/(π‘Ž+𝑏).(5)The power mean: π‘€π‘Ÿ(π‘Ž,𝑏)=((π‘Žπ‘Ÿ+π‘π‘Ÿ)/2)1/π‘Ÿ, π‘Ÿβ‰₯1,π‘Ž,π‘βˆˆβ„.(6)The generalized logarithmic mean: 𝐿𝑛𝑏(π‘Ž,𝑏)=𝑛+1βˆ’π‘Žπ‘›+1ξ‚Ή(π‘βˆ’π‘Ž)(𝑛+1)1/𝑛,π‘Žβ‰ π‘.(3.1)(7)The identric mean:⎧βŽͺ⎨βŽͺ⎩1𝐼(π‘Ž,𝑏)=π‘Žπ‘Ž=π‘π‘’ξ‚΅π‘π‘π‘Žπ‘Žξ‚Ά1/(π‘βˆ’π‘Ž)π‘Žβ‰ π‘.(3.2)

Proposition 3.1. For π‘›βˆˆ(βˆ’βˆž,0)βˆͺ[1,∞)⧡{βˆ’1} and [π‘Ž,𝑏]∈[0,π‘βˆ—] with π‘βˆ—>0, we have the following inequalities: ||(π‘Ž)𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›)βˆ’πΏπ‘›π‘›||≀(π‘Ž,π‘šπ‘)π‘šπ‘βˆ’π‘Ž8|𝑛|π‘€π‘›βˆ’1π‘›βˆ’1ξ€·π‘Ž,π‘š1/(π‘›βˆ’1)𝑏.|||1(𝑏)3𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›2)+3𝐴𝑛(π‘Ž,π‘šπ‘)βˆ’πΏπ‘›π‘›|||≀5(π‘Ž,π‘šπ‘)72(π‘šπ‘βˆ’π‘Ž)|𝑛|π‘€π‘›βˆ’1π‘›βˆ’1ξ€·π‘Ž,π‘š1/(π‘›βˆ’1)𝑏.(3.3)

Proof. The assertions follow from Corollary 2.3 for 𝑓(π‘₯)=π‘₯𝑛.

Proposition 3.2. For [π‘Ž,𝑏]∈[0,π‘βˆ—], we have the following inequalities: ||𝐻(π‘Ž)βˆ’1(π‘Ž,π‘šπ‘)βˆ’πΏβˆ’1||≀(π‘Ž,π‘šπ‘)π‘šπ‘βˆ’π‘Ž8π‘Ž2𝑏2𝑀22ξ€·π‘š1/2ξ€Έ,|||1π‘Ž,𝑏(𝑏)3π»βˆ’12(π‘Ž,π‘šπ‘)+3π΄βˆ’1(π‘Ž,π‘šπ‘)βˆ’πΏβˆ’1|||≀(π‘Ž,π‘šπ‘)5(π‘šπ‘βˆ’π‘Ž)72π‘Ž2𝑏2𝑀22ξ€·π‘š1/2ξ€Έ.π‘Ž,𝑏(3.4)

Proof. The assertions follow from Corollary 2.3 for 𝑓(π‘₯)=1/π‘₯.

Proposition 3.3. For π‘›βˆˆ(βˆ’βˆž,0)βˆͺ[1,∞)⧡{βˆ’1} and [π‘Ž,𝑏]∈[0,π‘βˆ—], we have the following inequalities: ||(π‘Ž)𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›)βˆ’πΏπ‘›π‘›||≀3(π‘Ž,π‘šπ‘)βˆ’1/π‘ž8𝐴(π‘šπ‘βˆ’π‘Ž)|𝑛|Γ—1/π‘žξ€·π‘Ž(π‘›βˆ’1)π‘ž,5π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ1/π‘ž+𝐴1/π‘žξ€·5π‘Ž(π‘›βˆ’1)π‘ž,π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ1/π‘žξ‚‡,|||1(𝑏)3𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›2)+3𝐴𝑛(π‘Ž,π‘šπ‘)βˆ’πΏπ‘›π‘›|||≀1(π‘Ž,π‘šπ‘)17291/π‘žξ€½π΄(π‘šπ‘βˆ’π‘Ž)|𝑛|Γ—1/π‘žξ€·π‘Ž(π‘›βˆ’1)π‘ž,17π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ+𝐴1/π‘žξ€·17π‘Ž(π‘›βˆ’1)π‘ž,π‘šπ‘(π‘›βˆ’1)π‘ž+ξ‚€ξ€Έξ€Ύ252288251/π‘ž(Γ—ξ€½π΄π‘šπ‘βˆ’π‘Ž)|𝑛|1/π‘žξ€·7π‘Ž(π‘›βˆ’1)π‘ž,11π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ+𝐴1/π‘žξ€·11π‘Ž(π‘›βˆ’1)π‘ž,7π‘šπ‘(π‘›βˆ’1)π‘ž.ξ€Έξ€Ύ(3.5)

Proof. The assertions follow from Corollary 2.5 for 𝑓(π‘₯)=π‘₯𝑛.

Proposition 3.4. For π‘›βˆˆ(βˆ’βˆž,0)βˆͺ[1,∞)⧡{βˆ’1} and [π‘Ž,𝑏]∈[0,π‘βˆ—], we have the following inequalities: ||(π‘Ž)𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›)βˆ’πΏπ‘›π‘›||≀1(π‘Ž,𝑏)22+1/π‘žξ‚ΆΓ—ξ€½π΄|𝑛|(π‘šπ‘βˆ’π‘Ž)1/π‘žξ€·π‘Ž(π‘›βˆ’1)π‘ž,3π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ+𝐴1/π‘žξ€·3π‘Ž(π‘›βˆ’1)π‘ž,π‘šπ‘(π‘›βˆ’1)π‘ž,|||1ξ€Έξ€Ύ(𝑏)3𝐴(π‘Žπ‘›,π‘šπ‘›π‘π‘›2)+3𝐴𝑛(π‘Ž,π‘šπ‘)βˆ’πΏπ‘›π‘›|||≀1(π‘Ž,π‘šπ‘)61+1/π‘žξ‚€131/π‘ž|𝐴𝑛|(π‘šπ‘βˆ’π‘Ž)Γ—1/π‘žξ€·π‘Ž(π‘›βˆ’1)π‘ž,11π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ+𝐴1/π‘žξ€·11π‘Ž(π‘›βˆ’1)π‘ž,π‘šπ‘(π‘›βˆ’1)π‘ž+ξ‚€1ξ€Έξ€Ύ32+1/π‘žξ€·21/π‘žξ€Έξ€½π΄|𝑛|(π‘šπ‘βˆ’π‘Ž)Γ—1/π‘žξ€·π‘Ž(π‘›βˆ’1)π‘ž,2π‘šπ‘(π‘›βˆ’1)π‘žξ€Έ+𝐴1/π‘žξ€·2π‘Ž(π‘›βˆ’1)π‘ž,π‘šπ‘(π‘›βˆ’1)π‘ž.ξ€Έξ€Ύ(3.6)

Proof. The assertions follow from Corollary 2.7 for 𝑓(π‘₯)=π‘₯𝑛.

Proposition 3.5. For [π‘Ž,𝑏]∈[0,π‘βˆ—], we have the following inequalities: ||||(π‘Ž)ln𝐼(π‘Ž,𝑏)||||≀𝐺(π‘Ž,𝑏)π‘βˆ’π‘Žξ‚΅1π‘Žπ‘41+1/π‘žξ‚Άξ€½(π‘Žπ‘ž+3π‘π‘ž)1/π‘ž+(3π‘Žπ‘ž+π‘π‘ž)1/π‘žξ€Ύ,(||||𝑏)ln𝐼(π‘Ž,𝑏)𝐺1/3(π‘Ž,𝑏)𝐴2/3||||≀(π‘Ž,𝑏)π‘βˆ’π‘Žξ‚Έξ‚€1π‘Žπ‘61+2/π‘žξ€½(π‘Žπ‘ž+11π‘π‘ž)1/π‘ž+(11π‘Žπ‘ž+π‘π‘ž)1/π‘žξ€Ύ+ξ‚€132+1/π‘žξ€½(π‘Žπ‘ž+2π‘π‘ž)1/π‘ž+(2π‘Žπ‘ž+π‘π‘ž)1/π‘žξ€Ύξ‚Ή.(3.7)

Proof. The assertions follow from Corollary 2.7 for 𝑓(π‘₯)=βˆ’lnπ‘₯ and π‘š=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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