Inequalities for α-fractional differentiable functions.

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

Introduction

A real-valued function is said to be convex on I if the inequality holds for all and . ψ is said to be concave on I if inequality (1.1) is reversed.

Let be a convex function on the interval I, and with . Then the double inequality is known in the literature as the Hermite-Hadamard inequality for convex functions [1-3]. Both inequalities hold in the reversed direction if ψ is concave on the interval I. In particular, many classical inequalities for means can be derived from (1.2) for appropriate particular selections of the function ψ.

Recently, the improvements, generalizations, refinements and applications for the Hermite-Hadamard inequality have attracted the attention of many researchers [4-22].

Dragomir and Agarwal [23] proved the following results connected with the right hand part of (1.2).

Theorem 1.1

See [23], Lemma 2.1

Let be a differentiable mapping on  . Then the identity holds for all with if , where denotes the interior of I.

Theorem 1.2

See [23], Theorem 2.2

Let be a differentiable mapping on  . Then the inequality holds for with if is convex on .

Making use of Theorem 1.1, Pearce and Pečarić [24] established Theorem 1.3 as follows.

Theorem 1.3

See [24], Theorem 1

Let with , be a differentiable mapping on and . Then the inequality is valid if the mapping is convex on the interval .

Next, we recall several elementary definitions and important results in the theory of conformable fractional calculus, which will be used throughout the article, we refer the interested reader to [25-32].

The conformable fractional derivative of order for a function at is defined by and the fractional derivative at 0 is defined as .

The (left) fractional derivative starting from of a function of order is defined by and we write if . For more details see [26].

Let and be α-differentiable at . Then we have where ψ is differentiable at in equation (1.3). In particular, if ψ is differentiable.

Let and . A function is said to be α-fractional integrable on if the integral exists and is finite. All the α-fractional integrable functions on are denoted by .

It is well known that if are two functions such that ψϕ is differentiable.

Very recently, Anderson [33] established a Hermite-Hadamard type inequality for fractional differentiable functions as follows.

Theorem 1.4

Let and be an α-fractional differentiable function. Then the inequality holds if is increasing on . Moreover, if the function ψ is decreasing on , then one has

Remark 1.5

We clearly see that inequalities (1.4) and (1.5) reduce to inequality (1.2) if .

The main purpose of the article is to present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

Main results

In order to prove our main results we need a lemma, which we present in this section.

Lemma 2.1

Let , with and be an α-fractional differentiable function on . Then the identity holds if .

Proof

Let . Then making use of integration by parts, we get Therefore, Lemma 2.1 follows easily from (2.1). □

Remark 2.2

We clearly see that the identity given in Lemma 2.1 reduces to the identity given in Theorem 1.1 if .

Theorem 2.3

Let , with and be an α-differentiable function. Then the inequality holds if and is convex on .

It follows from Lemma 2.1 and the convexities of the functions and on together with the convexity of on that  □

Remark 2.4

Let . Then inequality (2.2) becomes

Theorem 2.5

Let , , with and be an α-differentiable function on . Then the inequality is valid if and is convex on , where

From Lemma 2.1 and the well-known Hölder mean inequality together with the convexity of on the interval we clearly see that

Therefore, inequality (2.3) follows easily from (2.4)-(2.8). □

Remark 2.6

Let . Then inequality (2.3) becomes with

Theorem 2.7

Let , , with and be an α-differentiable function on . Then the inequality holds if and is concave on , where and are defined as in Theorem 2.5, and and are defined by

It follows from the concavity of and the Hölder mean inequality that which implies that is also concave. Making use of Lemma 2.1 and the Jensen integral inequality, we have

Therefore, inequality (2.9) follows easily from (2.10)-(2.12). □

Remark 2.8

Let . Then inequality (2.9) leads to

Applications to special means of real numbers

Let , , and with . Then the arithmetic mean , logarithmic mean and th generalized logarithmic mean of a and b are defined by respectively. Then from Theorems 2.3 and 2.5 together with the convexities of the functions and on the interval we get several new inequalities for the arithmetic, logarithmic and generalized logarithmic means as follows.

Theorem 3.1

Let with , , and . Then we have where , , , , and are defined as in Theorem 2.5.

Applications to the trapezoidal formula

Let Δ be a division of the interval and consider the quadrature formula where is the trapezoidal version and denotes the associated approximation error. In this section, we are going to derive several new error estimations for the trapezoidal formula.

Theorem 4.1

Let , with , be an α-differentiable function on and Δ be a division of the interval . Then the inequality holds if and is convex on .

Applying Theorem 2.3 on the subinterval of the division Δ, we have

It follows from (4.1) and the convexity of on the interval that  □

Making use of arguments analogous to the proof of Theorem 4.1, we get Theorem 4.2 immediately.

Theorem 4.2

Let , , with , be an α-differentiable function on and Δ be a division of the interval . Then the inequality holds if and is convex on , where , , , , and are defined as in Theorem 2.5.

Conclusion

In this work, we find an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, present some new inequalities for the arithmetic, logarithmic and generalized logarithmic means of two positive real numbers and provide the error estimations for the trapezoidal formula.