Abstract

We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values are 𝑃-convex. Applications to some special means are considered.

1. Introduction

Let 𝑓∶𝐼→ℝ be a convex function defined on the interval 𝐼 of real numbers and ğ‘Ž,𝑏∈𝐼 with ğ‘Ž<𝑏. The following double inequality ğ‘“î‚€ğ‘Ž+𝑏2≤1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“(𝑥)𝑑𝑥≤𝑓(ğ‘Ž)+𝑓(𝑏)2(1.1) is known in the literature as the Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if 𝑓 is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen's inequality. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the function 𝑓. Both inequalities hold in the reversed direction if 𝑓 is concave (see [1]).

It is well known that the Hermite-Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [2–13]) and the references therein. In [13] Dragomir and Agarwal established the following results connected with the right-hand side of (1.1) as well as applied them for some elementary inequalities for real numbers and numerical integration.

Theorem 1.1. Assume that ğ‘Ž,𝑏∈ℝ with ğ‘Ž<𝑏 and 𝑓∶[ğ‘Ž,𝑏]→ℝ is a differentiable function on (ğ‘Ž,𝑏). If |𝑓′| is convex on [ğ‘Ž,𝑏], then the following inequality holds: ||||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.(1.2)

Theorem 1.2. Assume that ğ‘Ž,𝑏∈ℝ with ğ‘Ž<𝑏 and 𝑓∶[ğ‘Ž,𝑏]→ℝ is a differentiable function on (ğ‘Ž,𝑏). Assume 𝑝∈ℝ with 𝑝>1. If |𝑓′|𝑝/(𝑝−1) is convex on [ğ‘Ž,𝑏], then the following inequality holds: ||||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.(1.3)

In [1] Pearce and Pečarić proved the following theorem.

Theorem 1.3. Let 𝑓∶𝐼→ℝ be a differentiable function on 𝐼∘,ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏. If |ğ‘“î…ž|ğ‘ž is convex on [ğ‘Ž,𝑏], for ğ‘žâ‰¥1, then the following inequality holds: ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)4||ğ‘“î…ž(||ğ‘Ž)ğ‘ž+||ğ‘“î…ž(||𝑏)ğ‘ž21/ğ‘ž.(1.4)

Recall that the function 𝑓∶[ğ‘Ž,𝑏]→ℝ is said to be quasiconvex if for every 𝑥,𝑦∈𝐼 we have [].𝑓(𝑡𝑥+(1−𝑡)𝑦)≤max{𝑓(𝑥),𝑓(𝑦)},∀𝑡∈0,1(1.5)

The generalizations of the Theorems 1.1 and 1.2 are introduced by Ion in [14] for quasiconvex functions and are given in [6] to differentiable 𝑃-convex functions. Then, Alomari et al. in [2] improved the results in [14] and Theorem 1.3, for twice differentiable quasiconvex functions.

On the other hand, Dragomir et al. in [11] defined the following class of functions.

Definition 1.4. Let 𝐼⊆ℝ be an interval. The function 𝑓∶𝐼→ℝ is said to be 𝑃-convex (or belong to the class 𝑃(𝐼)) if it is nonnegative and, for all 𝑥,𝑦∈𝐼 and 𝜆∈[0,1], satisfies the inequality 𝑓(𝜆𝑥+(1−𝜆)𝑦)≤𝑓(𝑥)+𝑓(𝑦).(1.6)

Note that 𝑃(𝐼) contain all nonnegative convex and quasiconvex functions. Since then numerous articles have appeared in the literature reflecting further applications in this category, see [3, 6, 12, 15, 16] and references therein. Ozdemir and Yildiz in [15] proved the following results.

Theorem 1.5. Let 𝑓∶𝐼→ℝ be a twice differentiable function on 𝐼∘ and ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏. If |ğ‘“î…žî…ž| is 𝑃-convex, 0≤𝜆≤1, then the following inequality holds: ||||(1−𝜆)ğ‘“ğ‘Ž+𝑏2−𝜆𝑓(ğ‘Ž)+𝑓(𝑏)2+1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤⎧⎪⎨⎪⎩(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2248𝜆3||𝑓−3𝜆+1||+||𝑓(ğ‘Ž)||1(𝑏),0≤𝜆≤2,(ğ‘âˆ’ğ‘Ž)2||𝑓24(3𝜆−1)||+||𝑓(ğ‘Ž)||,1(𝑏)2≤𝜆≤1.(1.7)

Corollary 1.6. If in Theorem 1.5 one chooses 𝜆=1, one obtains ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2||𝑓12||+||𝑓(ğ‘Ž)||.(𝑏)(1.8)

Theorem 1.7. Let 𝑓∶𝐼→ℝ be a twice differentiable function on 𝐼∘ and ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏. If |ğ‘“î…žî…ž|ğ‘ž is 𝑃-convex, 0≤𝜆≤1 and ğ‘žâ‰¥1, then the following inequality holds: ||||(1−𝜆)ğ‘“ğ‘Ž+𝑏2−𝜆𝑓(ğ‘Ž)+𝑓(𝑏)2+1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤⎧⎪⎨⎪⎩(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2488𝜆3||𝑓−3𝜆+1||(ğ‘Ž)ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘žî€¾î€¸1/ğ‘ž1,0≤𝜆≤2,(ğ‘âˆ’ğ‘Ž)2||𝑓48(3𝜆−1)||(ğ‘Ž)ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘žî€¾î€¸1/ğ‘ž,12≤𝜆≤1.(1.9)

Corollary 1.8. If in Theorem 1.7 one chooses 𝜆=1, one obtains ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2||𝑓24||(ğ‘Ž)ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘žî€¾î€¸1/ğ‘ž.(1.10)

The main purpose of this paper is to establish the refinements of results in [15]. Applications for special means are considered.

2. Main Results

In order to prove our main theorems, we need the following Lemma in [5] throughout this paper.

Lemma 2.1. Suppose that 𝑓∶𝐼→ℝ is a twice differentiable function on 𝐼∘, the interior of 𝐼. Assume that ğ‘Ž,𝑏∈𝐼∘, with ğ‘Ž<𝑏 and ğ‘“î…žî…ž, is integrable on [ğ‘Ž,𝑏]. Then, the following equality holds: 𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž=𝑓(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)216101−𝑡2î€¸î‚€ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏+ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏𝑑𝑡.(2.1)

In the following theorem, we will propose some new upper bound for the right-hand side of (1.1) for 𝑃-convex functions, which is better than the inequality had done in [15].

Theorem 2.2. Let 𝑓∶𝐼→ℝ be a twice differentiable function on 𝐼∘ such that |ğ‘“î…žî…ž| is a 𝑃-convex function on 𝐼. Suppose that ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏 and ğ‘“î…žî…žâˆˆğ¿1[ğ‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2||𝑓24|||||𝑓(ğ‘Ž)+2î…žî…žî‚€ğ‘Ž+𝑏2|||+||ğ‘“î…žî…ž||(𝑏).(2.2)

Proof. Since |ğ‘“î…žî…ž| is a 𝑃-convex function, by using Lemma 2.1 we get ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||=||||(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)216101−𝑡2î€¸î‚€ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏+ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏||||≤𝑑𝑡(ğ‘âˆ’ğ‘Ž)21610||1−𝑡2|||||ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏|||+|||ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏|||=𝑑𝑡(ğ‘âˆ’ğ‘Ž)2||𝑓24|||||𝑓(ğ‘Ž)+2î…žî…žî‚€ğ‘Ž+𝑏2|||+||ğ‘“î…žî…ž||.(𝑏)(2.3)

An immediate consequence of Theorem 2.2 is as follows.

Corollary 2.3. Let 𝑓 be as in Theorem 2.2, if in addition

(i) ğ‘“î…žî…ž((ğ‘Ž+𝑏)/2)=0, then one has
||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2||𝑓24||+||𝑓(𝑏)||,(𝑏)(2.4)

(ii) ğ‘“î…žî…ž(ğ‘Ž)=ğ‘“î…žî…ž(𝑏)=0, then one has
||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2|||𝑓12î…žî…žî‚€ğ‘Ž+𝑏2|||.(2.5)

The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.

Theorem 2.4. Let 𝑓∶𝐼→ℝ be a differentiable function on 𝐼∘. Assume that 𝑝∈ℝ, 𝑝>1 such that |ğ‘“î…žî…ž|𝑝/(𝑝−1) is a 𝑃-convex function on 𝐼. Suppose that ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏 and ğ‘“î…žî…žâˆˆğ¿1[ğ‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2√24𝜋21/𝑝Γ(1+𝑝)Γ(3/2+𝑝)1/𝑝||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘žîƒ­,(2.6) where 1/𝑝+1/ğ‘ž=1.

Proof. By assumption, Lemma 2.1 and Hölder's inequality, we have ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤||||(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)216101−𝑡2î€¸î‚€ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏+ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏||||≤𝑑𝑡(ğ‘âˆ’ğ‘Ž)216101−𝑡2𝑝𝑑𝑡1/𝑝×10|||ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏|||ğ‘žî‚¶ğ‘‘ğ‘¡1/ğ‘ž+10|||ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏|||ğ‘žî‚¶ğ‘‘ğ‘¡1/ğ‘žîƒ­â‰¤ğ‘‘ğ‘¡(ğ‘âˆ’ğ‘Ž)2√24𝜋21/𝑝Γ(1+𝑝)Γ((3/2)+𝑝)1/𝑝×||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘žîƒ­,(2.7) where 1/𝑝+1/ğ‘ž=1. We note that the Beta and Gamma functions are defined, respectively, as follows Γ(𝑥)=10𝑒−𝑥𝑡𝑥−1𝛽𝑑𝑡,𝑥>0,(𝑥,𝑦)=10𝑡𝑥−1(1−𝑡)𝑦−1𝑑𝑡,𝑥>0,𝑦>0(2.8) and are used to evaluate the integral ∫10(1−𝑡2)𝑝𝑑𝑡. Indeed, by setting 𝑡2=𝑢, we get 1𝑑𝑡=2𝑢−1/2𝑑𝑢,(2.9) and using property 𝛽(𝑥,𝑦)=Γ(𝑥)Γ(𝑦)Γ(𝑥+𝑦)(2.10) of Beta function, we obtain 101−𝑡2𝑝1𝑑𝑡=210𝑢−1/2(1−𝑢)𝑝1𝑑𝑢=2𝛽12,𝑝+1=2−1Γ(1/2)Γ(1+𝑝)Γ(3/2+𝑝)=2−1√𝜋Γ(1+𝑝)Γ=√(3/2+𝑝)𝜋2Γ(1+𝑝),Γ(3/2+𝑝)(2.11) where √Γ(1/2)=𝜋 and the proof is completed.

The following corollary is an immediate consequence of Theorem 2.4.

Corollary 2.5. Let 𝑓 be as in Theorem 2.4, if in addition

(i) ğ‘“î…žî…ž((ğ‘Ž+𝑏)/2)=0, then one has
||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2√24𝜋21/𝑝Γ(1+𝑝)Γ(3/2+𝑝)1/𝑝||ğ‘“î…žî…ž||+||𝑓(𝑏)||,(𝑏)(2.12)

(ii) ğ‘“î…žî…ž(ğ‘Ž)=ğ‘“î…žî…ž(𝑏)=0, then one has
||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Ž||||≤𝑓(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2√12𝜋21/𝑝Γ(1+𝑝)Γ(3/2+𝑝)1/𝑝|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||.(2.13)

Another similar result may be extended in the following theorem.

Theorem 2.6. Let 𝑓∶𝐼→ℝ be a differentiable function on 𝐼∘. Assume that ğ‘žâ‰¥1 such that |ğ‘“î…žî…ž|ğ‘ž is a 𝑃-convex function on 𝐼. Suppose that ğ‘Ž,𝑏∈𝐼∘ with ğ‘Ž<𝑏 and ğ‘“î…žî…žâˆˆğ¿1[ğ‘Ž,𝑏]. Then, the following inequality holds: ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)2||𝑓24||(ğ‘Ž)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘žîƒ­.(2.14)

Proof. Suppose that ğ‘Ž,𝑏∈𝐼∘. From Lemma 2.1 and using well-known power mean inequality, we get ||||𝑓(ğ‘Ž)+𝑓(𝑏)2−1î€œğ‘âˆ’ğ‘Žğ‘ğ‘Žğ‘“||||≤||||(𝑥)𝑑𝑥(ğ‘âˆ’ğ‘Ž)216101−𝑡2î€¸î‚€ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏+ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏||||≤𝑑𝑡(ğ‘âˆ’ğ‘Ž)216101−𝑡2𝑑𝑡1−1/ğ‘žîƒ¬î‚µî€œ101−𝑡2|||ğ‘“î…žî…žî‚€1+𝑡2ğ‘Ž+1−𝑡2𝑏|||ğ‘žî‚¶ğ‘‘ğ‘¡1/ğ‘ž+101−𝑡2|||ğ‘“î…žî…žî‚€1−𝑡2ğ‘Ž+1+𝑡2𝑏|||ğ‘žî‚¶ğ‘‘ğ‘¡1/ğ‘žîƒ­â‰¤ğ‘‘ğ‘¡(ğ‘âˆ’ğ‘Ž)221631−1/ğ‘žîƒ¬î‚µ23||ğ‘“î…žî…ž||(ğ‘Ž)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¼î‚¶1/ğ‘ž+23||ğ‘“î…žî…ž||(𝑏)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¼î‚¶1/ğ‘žîƒ­=(ğ‘âˆ’ğ‘Ž)2||𝑓24||(ğ‘Ž)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘ž+||ğ‘“î…žî…ž||(𝑏)ğ‘ž+|||ğ‘“î…žî…žî‚€ğ‘Ž+𝑏2|||ğ‘žî‚¶1/ğ‘žîƒ­,(2.15) which completes the proof.

Corollary 2.7. Let 𝑓 be as in Theorem 2.6, if in addition(i)ğ‘“î…žî…ž((ğ‘Ž+𝑏)/2)=0, then (2.4) holds,(ii)ğ‘“î…žî…ž(ğ‘Ž)=ğ‘“î…žî…ž(𝑏)=0, (2.5) holds.

3. Applications to Special Means

Now, we consider the applications of our theorems to the special means. We consider the means for arbitrary real numbers 𝛼, 𝛽 (𝛼≠𝛽). We take the following

(1) Arithmetic mean:
𝐴(𝛼,𝛽)=𝛼+𝛽2,𝛼,𝛽∈ℝ.(3.1)

(2) Logarithmic mean:
𝐿(𝛼,𝛽)=𝛼−𝛽||𝛽||||𝛽||ln|𝛼|−ln,|𝛼|≠,𝛼,𝛽≠0,𝛼,𝛽∈ℝ.(3.2)

(3) Generalized log-mean:
𝐿𝑛𝛽(𝛼,𝛽)=𝑛+1−𝛼𝑛+1(𝑛+1)(𝛽−𝛼)1/𝑛,𝑛∈ℕ,𝛼,𝛽∈ℝ,𝛼≠𝛽.(3.3)

Now, using the results of Section 2, we give some applications for special means of real numbers.

Proposition 3.1. Let ğ‘Ž,𝑏∈ℝ, ğ‘Ž<𝑏, and 𝑛∈ℕ, 𝑛≥2. Then, one has ||𝐿𝑛𝑛(ğ‘Ž,𝑏)−𝐴(ğ‘Žğ‘›,𝑏𝑛)||≤𝑛(𝑛−1)24(ğ‘âˆ’ğ‘Ž)2|ğ‘Ž|𝑛−2|||+2ğ‘Ž+𝑏2|||𝑛−2+||𝑏||𝑛−2.(3.4)

Proof. The assertion follows from Theorem 2.2 applied to the 𝑃-convex function 𝑓(𝑥)=𝑥𝑛, 𝑥∈ℝ.

Proposition 3.2. Let ğ‘Ž,𝑏∈ℝ, ğ‘Ž<𝑏, and 0∉[0,1]. Then, for all 𝑝>1 one has ||𝐿−1î€·ğ‘Ž(ğ‘Ž,𝑏)−𝐴−1,𝑏−1||≤(ğ‘âˆ’ğ‘Ž)28√𝜋21/𝑝Γ(1+𝑝)Γ((3/2)+𝑝)1/𝑝×|ğ‘Ž|−3ğ‘ž+|||ğ‘Ž+𝑏2|||−3ğ‘žî‚¶1/ğ‘ž+||𝑏||−3ğ‘ž+|||ğ‘Ž+𝑏2|||−3ğ‘žî‚¶1/ğ‘žîƒ­,(3.5) where 1/𝑝+1/ğ‘ž=1.

Proof. The assertion follows from Theorem 2.4 applied to the 𝑃-convex function 𝑓(𝑥)=1/𝑥, 𝑥∈[ğ‘Ž,𝑏].

Proposition 3.3. Let ğ‘Ž,𝑏∈ℝ, ğ‘Ž<𝑏, and 𝑛∈ℕ, 𝑛≥2. Then, for all ğ‘žâ‰¥1 one has ||𝐿𝑛𝑛(ğ‘Ž,𝑏)−𝐴(ğ‘Žğ‘›,𝑏𝑛)||≤𝑛(𝑛−1)24(ğ‘âˆ’ğ‘Ž)2|ğ‘Ž|(𝑛−2)ğ‘ž+|||ğ‘Ž+𝑏2|||(𝑛−2)ğ‘žî‚¶1/ğ‘ž+||𝑏||(𝑛−2)ğ‘ž+|||ğ‘Ž+𝑏2|||(𝑛−2)ğ‘žî‚¶1/ğ‘žîƒ­.(3.6)

Proof. The assertion follows from Theorem 2.6 applied to the 𝑃-convex function 𝑓(𝑥)=𝑥𝑛, 𝑥∈ℝ.