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

Abstract

The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings whose first derivative in absolute value aroused to the 𝑞th (𝑞≥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)≤𝑡^𝛼𝑓(𝑥)+𝑚(1−𝑡^𝛼)𝑓(𝑦)(1.1) holds for 𝑥,𝑦∈𝕀, and 𝑡∈[0,1].

Denote by 𝐾^𝛼𝑚(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∈𝐾^𝛼𝑚(𝕀), let us denote (𝑎−𝑚𝑏)𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)=1𝑟𝑓(𝑎)+(𝑟−2)𝑓𝑎+2𝑚𝑏+𝑓(𝑚𝑏)−𝑚𝑏1−𝑎^𝑎𝑚𝑏𝑓(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 𝑞-th (𝑞≥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 𝑓^∈𝐿₁([𝑎,𝑏]), where 𝑎,𝑏∈𝕀 with 𝑎<𝑏 and 𝑏^∗>0. If |𝑓^|^𝑞 is (𝛼,𝑚)-convex on [𝑎,𝑏] for (𝛼,𝑚)∈[0,1]² and 𝑞≥1 with 1/𝑝+1/𝑞=1, then the following inequality holds: (𝑎)||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,2)||≤𝑚𝑏−𝑎216^1/𝑝𝜇||𝑓^(𝑎)||^𝑞+𝑚𝜈||𝑓^(𝑏)||^𝑞^1/𝑞,(1.3) where 𝜇=1(𝛼+2)(𝛼+3),𝜈=𝛼²+5𝛼6(𝛼+2)(𝛼+3),(1.4)(𝑏)||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,6)||≤𝑚𝑏−𝑎221^1/𝑝𝜇₀||𝑓^(𝑎)||^𝑞+𝑚𝜈₀||𝑓^(𝑏)||^𝑞^1/𝑞,(1.5) where 𝜇₀=𝑞𝜈𝑞+𝛼+2ΓΓ((𝛼𝑞++2𝛼)+Γ(1𝑞)₀),=1(𝑞+1)(𝑞+2)−𝑞+𝛼𝑞+2ΓΓ((𝛼𝑞++2𝛼)+Γ(1𝑞)),(1.6) where Γ(𝑥)=^∞0𝑒^−𝑡𝑡^𝑥−1𝑑𝑡,𝑥>0.(1.7)

Theorem 1.2. Under the same notations in Theorem 2.2, if |𝑓^|^𝑞 is (𝛼,𝑚)-convex on [𝑎,𝑏] for (𝛼,𝑚)∈[0,1]² and 𝑞>1 with 1/𝑝+1/𝑞=1, then the following inequality holds: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,2)||≤𝑚𝑏−𝑎8ΓΓ(𝑝(𝑝++31/2))^1/𝑝||𝑓^(𝑎)||^𝑞1𝛼+1+𝑚||𝑓^(𝑏)||^𝑞𝛼𝛼+1^1/𝑞.(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 [𝑎,𝑏] such that 𝑓′∈𝐿_𝑝([𝑎,𝑏]), where 𝑎,𝑏∈𝕀 with 𝑎<𝑏. Then the following inequality holds: ||𝑆^𝑏𝑎(𝑓)(1,1,6)||≤(𝑏−𝑎)^−1/𝑝62^𝑞+1+13(𝑞+1)^1/𝑞‖‖𝑓^‖‖_𝑝.(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 𝐾^𝛼𝑚(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 𝑎,𝑏∈𝕀 with 0≤𝑎<𝑏 and 𝑏^∗>0. If 𝑓^∈𝐿¹([𝑎,𝑏]), then, for 𝑟≥2 and ℎ∈(0,1) with 1/𝑟≤ℎ≤(𝑟−1)/𝑟, the following equality holds: 𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)=¹⁰𝑝(𝑟,𝑡)𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)𝑑𝑡(2.1) for 𝑓∈𝐾^𝛼𝑚([𝑎,𝑏]) and each 𝑡∈[0,1], where 𝑝(𝑟,𝑡)=⎧⎪⎨⎪⎩𝑡𝑡−−1𝑟𝑟−1𝑡∈0,12,𝑟𝑡∈21,1.(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 𝑓^∈𝐿([𝑎,𝑏]), where 𝑎,𝑏∈𝕀 with 0≤𝑎<𝑏<∞ and 𝑏^∗>0. If |𝑓′|∈𝐾^𝛼𝑚([𝑎,𝑏]), for some (𝛼,𝑚)∈(0,1]² and 𝑚𝑏>𝑎, then, for any 𝑟≥2, the following inequality holds: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)||≤𝜇¹¹+𝜇²¹+𝜇³¹+𝜇⁴¹||𝑓^(𝑎)||+𝜈¹¹+𝜈²¹+𝜈³¹+𝜈⁴¹𝑚||𝑓^(𝑏)||,(2.3) where 𝜇¹¹=1(𝛼+1)(𝛼+2)𝑟^𝛼+2𝜇₂₁,=1(𝛼+1)(𝛼+2)𝑟^𝛼+2+(𝛼+1)𝑟−2(𝛼+2)𝜇2^𝛼+2(𝛼+1)(𝛼+2)𝑟₃₁,=(𝑟−1)^𝛼+2(𝛼+1)(𝛼+2)𝑟^𝛼+2+2(𝛼+2)−3𝑟𝜇2^𝛼+2(𝛼+1)(𝛼+2)𝑟₄₁,=(𝑟−1)^𝛼+2(𝛼+1)(𝛼+2)𝑟^𝛼+2+(𝛼+2)−𝑟𝜈(𝛼+1)(𝛼+2)𝑟₁₁,=12𝑟²𝜈₂₁−𝜇¹¹,=(𝑟−2)²8𝑟²𝜈₃₁−𝜇²¹,=(𝑟−2)²8𝑟²𝜈₄₁−𝜇³¹,=12𝑟²−𝜇⁴¹.(2.4)

Proof. From Lemma 2.1 and using the properties of the modulus, we have the following: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)||≤^01/𝑟+1𝑟−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_1/21/𝑟+𝑡−1𝑟||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_{(𝑟−1)/𝑟1/2}+𝑟−𝑟1−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_{1(𝑟−1)/𝑟}𝑡−𝑟−𝑟1||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡.(2.5)
Since |𝑓′| is (𝛼,𝑚)-convex on [𝑎,𝑏], we know that for any 𝑡∈[0,1]||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||≤𝑡^𝛼||𝑓^(𝑎)||+𝑚(1−𝑡^𝛼)||𝑓^(𝑏)||.(2.6)
By (2.5) and (2.6), we get the following: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)||≤^01/𝑟1𝑟−𝑡𝑡^𝛼+||𝑓^(𝑎)||+𝑚(1−𝑡^𝛼)||𝑓^(𝑏)||𝑑𝑡_1/21/𝑟𝑡−1𝑟𝑡^𝛼+||𝑓^(𝑎)||+𝑚(1−𝑡^𝛼)||𝑓^(𝑏)||𝑑𝑡_{(𝑟−1)/𝑟1/2}𝑟−𝑟1−𝑡𝑡^𝛼+||𝑓^(𝑎)||+𝑚(1−𝑡^𝛼)||𝑓^(𝑏)||𝑑𝑡_{1(𝑟−1)/𝑟}𝑡−𝑟−𝑟1𝑡^𝛼≤||𝑓^(𝑎)||+𝑚(1−𝑡^𝛼)||𝑓^(𝑏)||𝑑𝑡_01/2|||1𝑟−𝑡|||𝑡^𝛼𝑑𝑡+^11/2|||𝑟−1𝑟−𝑡|||𝑡^𝛼+𝑑𝑡||𝑓^(𝑎)_01/2|||||1𝑟−𝑡|||(1−𝑡^𝛼)𝑑𝑡+^11/2|||𝑟−1𝑟−𝑡|||(1−𝑡^𝛼=𝜇)𝑑𝑡𝑚||𝑓^(𝑏)₁₁||+𝜇²¹+𝜇³¹+𝜇⁴¹||𝑓^(𝑎)||+𝜈¹¹+𝜈²¹+𝜈³¹+𝜈⁴¹𝑚||𝑓^(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: ||(𝑚𝑏−𝑎)𝑆^𝑏𝑎(𝑓)(1,𝑚,2)||=||||𝑓(𝑎)+𝑓(𝑚𝑏)2−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤𝑚𝑏−𝑎𝑓(𝑥)𝑑𝑥||||8||𝑓^(𝑎)||+𝑚||𝑓^(𝑏)||,(2.8) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have the following ||(𝑚𝑏−𝑎)𝑆^𝑏𝑎(𝑓)(1,𝑚,6)||=||||16𝑓(𝑎)+4𝑓𝑎+2𝑚𝑏+𝑓(𝑚𝑏)−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤5𝑓(𝑥)𝑑𝑥||||72(𝑚𝑏−𝑎)||𝑓^(𝑎)||+𝑚||𝑓^(𝑏)||.(2.9)

Theorem 2.4. Under the same notations in Theorem 2.2, if |𝑓′|^𝑞∈𝐾^𝛼𝑚([𝑎,𝑏]), for some (𝛼,𝑚)∈(0,1]², 𝑚𝑏>𝑎 and 𝑞>1 with 1/𝑝+1/𝑞=1, then, for any 𝑟≥2, the following inequality holds: ||𝑆^𝑏𝑎(≤𝑓)(𝛼,1𝑚,𝑟)||2𝑟²^1/𝑝𝜇¹¹||𝑓^(𝑎)||^𝑞+𝜈¹¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+𝜇⁴¹||𝑓^(𝑎)||^𝑞+𝜈⁴¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+18𝑟−𝑟2²^1/𝑝𝜇²¹||𝑓^(𝑎)||^𝑞+𝜈²¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+𝜇³¹||𝑓^(𝑎)||^𝑞+𝜈³¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞.(2.10)

Proof. From Lemma 2.1 and using the properties of modulus, we have the following: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)||≤^01/𝑟+1𝑟−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_1/21/𝑟+𝑡−1𝑟||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_{(𝑟−1)/𝑟1/2}+𝑟−𝑟1−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡_{1(𝑟−1)/𝑟}𝑡−𝑟−𝑟1||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡.(2.11)
Using the power-mean integral inequality and (𝛼,𝑚)-convexity of |𝑓′|^𝑞 for any 𝑡∈[0,1], we have the following(a)^01/𝑟1𝑟−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||^𝑞𝑑𝑡≤𝜇¹¹||𝑓^(𝑎)||^𝑞+𝜈¹¹𝑚||𝑓^(𝑏)||^𝑞,(2.12)(b)^1/21/𝑟𝑡−1𝑟||𝑓(𝑡𝑎+𝑚(1−𝑡)𝑏)||^𝑞𝑑𝑡≤𝜇²¹||𝑓^(𝑎)||^𝑞+𝜈²¹𝑚||𝑓^(𝑏)||^𝑞,(2.13)(c)^{(𝑟−1)/𝑟1/2}𝑟−𝑟1−𝑡||𝑓(𝑡𝑎+𝑚(1−𝑡)𝑏)||^𝑞𝑑𝑡≤𝜇³¹||𝑓^(𝑎)||^𝑞+𝜈³¹𝑚||𝑓^(𝑏)||^𝑞,(2.14)(d)^{1(𝑟−1)/𝑟}𝑡−𝑟−𝑟1||𝑓(𝑡𝑎+𝑚(1−𝑡)𝑏)||^𝑞𝑑𝑡≤𝜇⁴¹||𝑓^(𝑎)||^𝑞+𝜈⁴¹𝑚||𝑓^(𝑏)||^𝑞.(2.15)
By the similar way as the above inequalities (a)–(d), we have the following:(a′)^01/𝑟1𝑟−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡≤12𝑟²^1/𝑝𝜇¹¹||𝑓^(𝑎)||^𝑞+𝜈¹¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.16)(b′)^1/21/𝑟𝑡−1𝑟||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡≤18𝑟−𝑟2²^1/𝑝𝜇²¹||𝑓^(𝑎)||^𝑞+𝜈²¹𝑚||𝑓^(𝑏)||^1/𝑞,(2.17)(c′)^{(𝑟−1)/𝑟1/2}𝑟−𝑟1−𝑡||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡≤18𝑟−𝑟2²^1/𝑝𝜇³¹||𝑓^(𝑎)||^𝑞+𝜈³¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.18)(d′)^{1(𝑟−1)/𝑟}𝑡−𝑟−𝑟1||𝑓^(𝑡𝑎+𝑚(1−𝑡)𝑏)||𝑑𝑡≤12𝑟²^1/𝑝𝜇⁴¹||𝑓^(𝑎)||^𝑞+𝜈⁴¹𝑚||𝑓^(𝑏)||^𝑞^1/𝑞.(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(𝑎)+𝑓(𝑚𝑏)2−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤6𝑓(𝑥)𝑑𝑥_−1/𝑞||||8(𝑚𝑏−𝑎)||𝑓^(𝑎)||^𝑞+5𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+5||𝑓^(𝑎)||^𝑞+𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.20) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have that ||||16𝑓(𝑎)+4𝑓𝑎+2𝑚𝑏+𝑓(𝑚𝑏)−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤(𝑚𝑏−𝑎)1𝑓(𝑥)𝑑𝑥||||72118^1/𝑞||𝑓^(𝑎)||^𝑞+17𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+17||𝑓^(𝑎)||^𝑞+𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+2528852^2/𝑞×7||𝑓^(𝑎)||^𝑞+11𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+11||𝑓^(𝑎)||^𝑞+7𝑚||𝑓^(𝑏)||^𝑞^1/𝑞.(2.21)

Theorem 2.6. Under the same notations in Theorem 2.2, if |𝑓′|^𝑞∈𝐾^𝛼𝑚([𝑎,𝑏]), for some (𝛼,𝑚)∈(0,1]², 𝑚𝑏>𝑎 and 𝑞>1 with 1/𝑝+1/𝑞=1, then, for any 𝑟≥2, the following inequality holds: ||𝑆^𝑏𝑎(≤𝑓)(𝛼,1𝑚,𝑟)||𝑟^𝑝+1^1/𝑝𝜇¹²||𝑓^(𝑎)||^𝑞+𝜈¹²𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+𝜇⁴²||𝑓^(𝑎)||^𝑞+𝜈⁴²𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+(𝑟−2)_𝑝+12^𝑝+1𝑟^𝑝+1^1/𝑝𝜇²²||𝑓^(𝑎)||^𝑞+𝜈²²𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+𝜇³²||𝑓^(𝑎)||^𝑞+𝜈³²𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.22) where 𝜇¹²=1𝜇𝑟^𝛼+1(𝛼+1)₂₂,=𝑟^𝛼+1−2^𝛼+1𝜇2^𝛼+1𝑟^𝛼+1(𝛼+1)₃₂,=2^𝛼+1(𝑟−1)^𝛼+1−𝑟^𝛼+1𝜇2^𝛼+1𝑟^𝛼+1(𝛼+1)₄₂,=𝑟^𝛼+1−(𝑟−1)^𝛼+1𝜈𝑟^𝛼+1(𝛼+1)₁₂,=1𝑟−𝜇¹²𝜈₂₂,=𝑟−22𝑟+𝜇²²𝜈₃₂,=𝑟−22𝑟+𝜇³²𝜈₄₂,=1𝑟−𝜇⁴².(2.23)

Proof. Suppose that 𝑞>1. From Lemma 2.1, using the Hölder integral inequality, we get the following: ||𝑆^𝑏𝑎(𝑓)(𝛼,𝑚,𝑟)||≤^01/𝑟1𝑟−𝑡^𝑝𝑑𝑡^1/𝑝^01/𝑟||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+^1/21/𝑟𝑡−1𝑟^𝑝𝑑𝑡^1/𝑝^1/21/𝑟||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+^{(𝑟−1)/𝑟1/2}𝑟−1𝑟−𝑡^𝑝𝑑𝑡^1/𝑝^{(𝑟−1)/𝑟1/2}||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+^{1(𝑟−1)/𝑟}𝑡−𝑟−1𝑟^𝑝𝑑𝑡^1/𝑝^{1(𝑟−1)/𝑟}||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞≤1𝑟^𝑝+1^1/𝑝^01/𝑟||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+12^𝑝+1𝑟−𝑟2^𝑝+1^1/𝑝^1/21/𝑟||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+12^𝑝+1𝑟−𝑟2^𝑝+1^1/𝑝^{(𝑟−1)/𝑟1/2}||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞+1𝑟^𝑝+1^1/𝑝^{1(𝑟−1)/𝑟}||𝑓^(𝑡𝑏+𝑚(1−𝑡)𝑎)||^𝑞𝑑𝑡^1/𝑞,(2.24) where we have used the fact that 1/2<(1/(𝑝+1))^1/𝑝<1.
Since |𝑓′|^𝑞∈𝐾^𝑚𝛼([𝑎,𝑏]) for some fixed 𝛼∈(0,1] and 𝑚∈[0,1], we have the followings: ^01/𝑟_1/21/𝑟||𝑓^(𝑡𝑏+(1−𝑡)𝑎)||^𝑞𝑑𝑡≤𝜇¹²||𝑓^(𝑎)||^𝑞+𝜈¹²𝑚||𝑓^(𝑏)||^𝑞,_{(𝑟−1)/𝑟1/2}||𝑓(𝑡𝑏+(1−𝑡)𝑎)||^𝑞𝑑𝑡≤𝜇²²||𝑓^(𝑎)||^𝑞+𝜈²²𝑚||𝑓^(𝑏)||^𝑞,_{1(𝑟−1)/𝑟}||𝑓^(𝑡𝑏+(1−𝑡)𝑎)||^𝑞𝑑𝑡≤𝜇³²||𝑓^(𝑎)||^𝑞+𝜈³²𝑚||𝑓^(𝑏)||^𝑞,||𝑓(𝑡𝑏+(1−𝑡)𝑎)||^𝑞𝑑𝑡≤𝜇⁴²||𝑓^(𝑎)||^𝑞+𝜈⁴²𝑚||𝑓^(𝑏)||^𝑞.(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(𝑎)+𝑓(𝑚𝑏)2−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤1𝑓(𝑥)𝑑𝑥||||4^1+1/𝑞(𝑚𝑏−𝑎)||𝑓^(𝑎)||^𝑞+3𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+3||𝑓^(𝑎)||^𝑞+𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.26) and (ii) if we choose 𝛼=1 and 𝑟=6, then we have ||||16𝑓(𝑎)+4𝑓𝑎+2𝑚𝑏+𝑓(𝑚𝑏)−𝑚𝑏1−𝑎^𝑎𝑚𝑏≤(𝑚𝑏−𝑎)1𝑓(𝑥)𝑑𝑥||||6^1+2/𝑞||𝑓^+1(𝑎)||^𝑞+11𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+11||𝑓^(𝑎)||^𝑞+𝑚||𝑓^(𝑏)||^𝑞^1/𝑞3^2+1/𝑞||𝑓^(𝑎)||^𝑞+2𝑚||𝑓^(𝑏)||^𝑞^1/𝑞+2||𝑓^(𝑎)||^𝑞+𝑚||𝑓^(𝑏)||^𝑞^1/𝑞,(2.27) where we have used the fact that (1/2)^1/𝑞<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 𝑎,𝑏∈ℝ⁺ with 𝑏≥𝑎.(1) The arithmetic mean: 𝐴(𝑎,𝑏)=(𝑎+𝑏)/2.(2)The geometric mean: 𝐺(𝑎,𝑏)=√𝑎𝑏.(3)The logarithmic mean: 𝐿(𝑎,𝑏)=(𝑏−𝑎)/(ln𝑏−ln𝑎) for 𝑎≠𝑏.(4)The harmonic mean: 𝐻(𝑎,𝑏)=2𝑎𝑏/(𝑎+𝑏).(5)The power mean: 𝑀_𝑟(𝑎,𝑏)=((𝑎^𝑟+𝑏^𝑟)/2)^1/𝑟, 𝑟≥1,𝑎,𝑏∈ℝ.(6)The generalized logarithmic mean: 𝐿_𝑛(𝑎,𝑏)=𝑏^𝑛+1−𝑎^𝑛+1(𝑏−𝑎)(𝑛+1)^1/𝑛,𝑎≠𝑏.(3.1)(7)The identric mean:𝐼(𝑎,𝑏)=⎧⎪⎨⎪⎩𝑎1𝑎=𝑏𝑒𝑎𝑏^𝑏𝑎^{1/(𝑏−𝑎)}𝑎≠𝑏.(3.2)

Proposition 3.1. For 𝑛∈(−∞,0)∪[1,∞)⧵{−1} and [𝑎,𝑏]∈[0,𝑏^∗] with 𝑏^∗>0, we have the following inequalities: (𝑎)||𝐴(𝑎^𝑛,𝑚^𝑛𝑏^𝑛)−𝐿^𝑛𝑛(𝑎,𝑚𝑏)||≤𝑚𝑏−𝑎8|𝑛|𝑀^{𝑛−1𝑛−1}𝑎,𝑚^1/(𝑛−1)(𝑏)|||1𝑏.3𝐴(𝑎^𝑛,𝑚^𝑛𝑏^𝑛)+23𝐴^𝑛≤5(𝑎,𝑚𝑏)−𝐿^𝑛𝑛(𝑎,𝑚𝑏)|||72(𝑚𝑏−𝑎)|𝑛|𝑀^{𝑛−1𝑛−1}𝑎,𝑚^1/(𝑛−1)𝑏.(3.3)

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

Proposition 3.2. For [𝑎,𝑏]∈[0,𝑏^∗], we have the following inequalities: (𝑎)||𝐻⁻¹(𝑎,𝑚𝑏)−𝐿⁻¹(𝑎,𝑚𝑏)||≤𝑚𝑏−𝑎8𝑎²𝑏²𝑀²²(𝑏)|||1𝑚^1/2𝑎,𝑏,3𝐻⁻¹(𝑎,𝑚𝑏)+23𝐴⁻¹(𝑎,𝑚𝑏)−𝐿⁻¹(𝑎,𝑚𝑏)|||≤5(𝑚𝑏−𝑎)72𝑎²𝑏²𝑀²²𝑚^1/2𝑎,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 [𝑎,𝑏]∈[0,𝑏^∗], we have the following inequalities: (𝑎)||≤𝐴(𝑎^𝑛3,𝑚^𝑛𝑏^𝑛)−𝐿^𝑛𝑛(𝑎,𝑚𝑏)_−1/𝑞||8(𝑚𝑏−𝑎)|𝑛|×𝐴^1/𝑞𝑎^{(𝑛−1)𝑞}(𝑏)|||1,5𝑚𝑏^{(𝑛−1)𝑞}^1/𝑞+𝐴^1/𝑞5𝑎^{(𝑛−1)𝑞},𝑚𝑏^{(𝑛−1)𝑞}^1/𝑞,3𝐴(𝑎^𝑛,𝑚^𝑛𝑏^𝑛)+23𝐴^𝑛≤1(𝑎,𝑚𝑏)−𝐿^𝑛𝑛(𝑎,𝑚𝑏)|||7291^1/𝑞+25(𝑚𝑏−𝑎)|𝑛|×𝐴^1/𝑞𝑎^{(𝑛−1)𝑞},17𝑚𝑏^{(𝑛−1)𝑞}+𝐴^1/𝑞17𝑎^{(𝑛−1)𝑞},𝑚𝑏^{(𝑛−1)𝑞}288225^1/𝑞×𝐴(𝑚𝑏−𝑎)|𝑛|_1/𝑞7𝑎^{(𝑛−1)𝑞},11𝑚𝑏^{(𝑛−1)𝑞}+𝐴^1/𝑞11𝑎^{(𝑛−1)𝑞},7𝑚𝑏^{(𝑛−1)𝑞}.(3.5)

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

Proposition 3.4. For 𝑛∈(−∞,0)∪[1,∞)⧵{−1} and [𝑎,𝑏]∈[0,𝑏^∗], we have the following inequalities: (𝑎)||𝐴(𝑎^𝑛,𝑚^𝑛𝑏^𝑛)−𝐿^𝑛𝑛(𝑎,𝑏)||≤1×𝐴2^2+1/𝑞|𝑛|(𝑚𝑏−𝑎)_1/𝑞(𝑏)|||1𝑎^{(𝑛−1)𝑞},3𝑚𝑏^{(𝑛−1)𝑞}+𝐴^1/𝑞3𝑎^{(𝑛−1)𝑞},𝑚𝑏^{(𝑛−1)𝑞},3𝐴(𝑎^𝑛,𝑚^𝑛𝑏^𝑛)+23𝐴^𝑛≤1(𝑎,𝑚𝑏)−𝐿^𝑛𝑛(𝑎,𝑚𝑏)|||6^1+1/𝑞31^1/𝑞+1|𝑛|(𝑚𝑏−𝑎)×𝐴^1/𝑞𝑎^{(𝑛−1)𝑞},11𝑚𝑏^{(𝑛−1)𝑞}+𝐴^1/𝑞11𝑎^{(𝑛−1)𝑞},𝑚𝑏^{(𝑛−1)𝑞}3^2+1/𝑞2^1/𝑞|𝑛|(𝑚𝑏−𝑎)×𝐴^1/𝑞𝑎^{(𝑛−1)𝑞},2𝑚𝑏^{(𝑛−1)𝑞}+𝐴^1/𝑞2𝑎^{(𝑛−1)𝑞},𝑚𝑏^{(𝑛−1)𝑞}.(3.6)

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

Proposition 3.5. For [𝑎,𝑏]∈[0,𝑏^∗], we have the following inequalities: (𝑎)||||ln𝐼(𝑎,𝑏)𝐺(𝑎,𝑏)||||≤𝑏𝑎−𝑏𝑎41^1+1/𝑞(𝑎^𝑞(𝑏)||||ln𝐼(𝑎,𝑏)+3𝑏^𝑞)^1/𝑞+(3𝑎^𝑞+𝑏^𝑞)^1/𝑞,𝐺^1/3(𝑎,𝑏)𝐴^2/3(𝑎,𝑏)||||≤𝑏𝑎−𝑏𝑎61^1+2/𝑞+1(𝑎^𝑞+11𝑏^𝑞)^1/𝑞+(11𝑎^𝑞+𝑏^𝑞)^1/𝑞3^2+1/𝑞(𝑎^𝑞+2𝑏^𝑞)^1/𝑞+(2𝑎^𝑞+𝑏^𝑞)^1/𝑞.(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.