1. Introduction

It is common knowledge in mathematical analysis that a function is said to be convex on an interval if

(1.1)

for all x,y∈I and ∈[0,1], if the inequality (1.1) reverses, then f is said to be concave on I.

Let be a convex function on an interval I and a,b∈I with a < b. Then

(1.2)

This inequality is well known in the literature as Hermite-Hadamard integral inequality for convex functions. See ^{[4, 12]} and closely related references therein.

The concept of usually used convexity has been generalized by a number of mathematicians. Some of them can be recited as follows.

Definition 1.1. (^[20]). Let be a function and m∈[0,1]. If

(1.3)

holds for all and ∈[0,1], then we say that f(x) is m-convex on [0,b].

Definition 1.2. (^[11]). Let R be a function and (α,m)∈[0,1]×[0,1]. If

(1.4)

is valid for all x,y∈[0,b] and α∈(0,1], then we say that f(x) is (α,m)-convex on [0,b].

Turkish Journal of Analysis and Number Theory 2.

It is not difficult to see that when (α,m)∈f(α,0), (1,0), (1,m), (1,1), (α,1)g the (α,m)-convex function becomes the α-star-shaped, star-shaped, m-convex, convex, and α-convex functions respectively.

The famous Hermite-Hadamard inequality (1.2) has been refined or generalized by many mathematicians. Some of them can be reformulated as follows.

Theorem 1.1. ([^[14], Theorem 3]). Let be a twice differentiable function such that for a,b∈I with a < b. If is m-convex on [a,b] for some fixed q > 1 and m∈[0,1], then

where and Γ is the classical Euler gamma function which may be defined for R(z) > 0 by

(1.5)

Theorem 1.2. ([^[17], Theorem 4]). Let be an open interval and a,b∈I with a < b, and let be a twice differentiable mapping such that is integrable. If 0 ≤λ≤1 and is convex on [a,b], then

Theorem 1.3. ([^[13], Theorem 3]). Let b* > 0 and be a twice differentiable function such that f’’∈L([a,b]) for a,b∈[0,b*] with a < b. If |f’’(x)|_q is (α,m)-convex on [a,b] for (α,m)∈[0,1]×[0,1] and q≥1, then

(1.6)

In recent years, some other kinds of Hermite-Hadamard type inequalities were generated in ^{[1, 2, 3, 15, 16, 19, 26, 29, 30]}, for example. For more systematic information, please refer to monographs ^{[4, 12]} and related references therein.

In this paper, we will establish some new inequalities of Hermite-Hadamard type for functions whose derivatives of n-th order are (α,m)-convex and deduce some known results in the form of corollaries.

2. A Lemma

For establishing new integral inequalities of Hermite-Hadamard type for functions whose derivatives of n-th order are (α,m)-convex, we need the following lemma.

Lemma 2.1. Let 0 < m≤1 and b > a > 0 satisfying a < mb. If f⁽ⁿ⁾(x) for exists and is integrable on the closed interval [0,b], then

(2.1)

where the sum above takes 0 when n = 1 and n = 2.

Proof. When n = 1, it is easy to deduce the identity (2.1) by performing an integration by parts in the integrals from the right side and changing the variable.

When n = 2, we have

(2.2)

This result is same as [^[13], Lemma 2].

When n = 3, the identity (2.1) is equivalent to

(2.3)

which may be derived from integrating the integral in the second line of (2.3) and utilizing the identity (2.2).

When n ≥ 4, computing the second line in (2.1) by integration by parts yields

which is a recurrent formula

on n where

and

for n≥4. By mathematical induction, the proof of Lemma 2.1 is complete.

Remark 2.1. Similar integral identities to (2.1), produced by replacing f^(k)(a) in (2.1) by f^(k)(b) or by , and corresponding integral inequalities of Hermite-Hadamard type have been established in ^{[10, 22, 23]}.

Remark 2.2. When m = 1, our Lemma 2.1 becomes [^[7], Lemma 2.1].

3. Inequalities of Hermite-Hadamard Type

Now we are in a position to establish some integral inequalities of Hermite-Hadamard type for functions whose derivatives of n-th order are (α,m)-convex.

Theorem 3.1. Let (α,m)∈[0,1]×(0,1] and b > a > 0 with a < mb. If f(x) is n-time differentiable on [0,b] such that and is (α,m)-convex on [0,mb] for n≥2 and p≥1, then

(3.1)

where the sum above takes 0 when n = 2.

Proof. It follows from Lemma 2.1 that

(3.2)

When p = 1, since is (α,m)-convex, we have

Multiplying by the factor t^n-1(n-2t) on both sides of the above inequality and integrating with respect to t∈[0,1] lead to

The proof for the case p = 1 is complete.

When p > 1, by the well-known Hölder integral inequality, we obtain

(3.3)

Using the (α,m)-convexity of produces

(3.4)

Substituting (3.3) and (3.4) into (3.2) yields the inequality (3.1). This completes the proof of Theorem 3.1.

Corollary 3.1. Under conditions of Theorem 3.1,

1. when m = 1, we have

2. when n = 2, we have

3. when m =α= p = 1 and n = 2, we have

4. when m =α= 1 and p = n = 2, we have

Remark 3.1. Under conditions of Theorem 3.1,

1. when n = 2, the inequality (3.1) becomes the one (1.6) in [^[13], Theorem 3];

2. when α = m = 1, Theorem 3.1 becomes [^[7], Theorem 3.1].

Theorem 3.2. Let (α,m)∈[0,1]×(0,1] and b >a > 0 with a < mb. If f(x) is n-time differentiable on [0,b] such that ∈L([0,mb]) and is (α,m)-convex on [0,mb] for n≥2 and p > 1, then

(3.5)

where the sum above takes 0 when n = 2 and .

Proof. It follows from Lemma 2.1 that

(3.6)

By the well-known Hölder integral inequality, we obtain

(3.7)

Making use of the (α,m)-convexity of reveals

(3.8)

Combining (3.7) and (3.8) with (3.6) results in the inequality (3.5). This completes the proof of Theorem 3.2.

Corollary 3.2. Under conditions of Theorem 3.2,

1. when m = 1, we have

2. when n = 2, we have

3. when m =α= p = 1 and n = 2, we have

(3.9)

where .

Theorem 3.3. Let (α,m) ∈[0,1]×(0,1] and b >a > 0 with a < mb. If f(x) is n-time differentiable on [0,b] such that∈L([0,mb]) and is (α,m)-convex on [0,mb] for n≥2 and p > 1, then

(3.10)

where the sum above takes 0 when n = 2.

Proof. Utilizing Lemma 2.1, Hölder integral inequality, and the (α,m)-convexity of yields

This completes the proof of Theorem 3.3.

Corollary 3.3. Under conditions of Theorem 3.3,

1. when m = 1, we have

2. when n = 2, we have

3. when m =α= 1 and n = 2, we have

4. Applications to Special Means

It is well known that, for positive real numbers α and β with α≠β, the quantities

for r≠0, -1 are respectively called the arithmetic, geometric, harmonic, exponential, logarithmic, and generalized logarithmic means.

Basing on inequalities of Hermite-Hadamard type in the above section, we shall derive some inequalities of the above defined means as follows.

Theorem 4.1. Let r∈(-∞, 0)∪[1,∞)\{-1} and b > a > 0. Then, for p,q > 1,

(4.1)

where .

Proof. This follows from applying the inequality (3.9) to the function f(x) = x^r.

Theorem 4.2. Let r∈(-∞, 0)∪[1,∞)\{-1} and b > a > 0. Then, for p ≥1,

Proof. This follows from applying the inequality (3.11) to the function f(x) = x^r.

Theorem 4.3. Let r∈(-∞, 0)∪[1,∞)\{-1} and b > a > 0. Then

Proof. This follows from applying the inequality (3.11) for p = 1 to the function f(x) = x^r.

Theorem 4.4. Let b > a > 0. Then for p,q > 1 we have

(4.2)

where .

Proof. This follows from applying the inequality (3.9) to the function .

Theorem 4.5. Let b > a > 0. Then for p≥1 we have

(4.3)

Proof. This follows from the inequality (3.11) to the function f(x) = x^r.

Theorem 4.6. Let b > a > 0. Then we have

(4.4)

Proof. This follows from applying the inequality (3.11) for p = 1 to the function f(x) = -ln x.

Remark 4.1. This paper is a combined version of the preprints ^{[8, 9]}.

Remark 4.2. Finally, we would like to recommend some newly published articles ^{[5, 6, 18, 21, 24, 25, 27, 28, 31, 32, 33]} which have something to do with this topic.

Acknowledgements

The authors would like to thank Professor Bo-Yan Xi and Dr Shu-Hong Wang at Inner Mongolia University for Nationalities in China for their helpful corrections to and valuable comments on the original version of this paper.

The first author was partially supported by the NNSF under Grant No. 11361038 of China.

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