Abstract

Using some results about generalized Hermite-Hadamard-Fejér type inequalities related to -convex functions, we give some examples and applications for trapezoid and midpoint type inequalities for differentiable -convex functions.

1. Introduction and Preliminaries

The celebrated Hermite-Hadamard-Fejér inequality (simply Fejér inequality) for convex functions has been proved in [1] as the following.

Theorem 1. Let be a convex function. Thenwhere is integrable and symmetric about ().

If in (1) we consider , then we obtain Hermite-Hadamard inequality.For various types of (1) and more results related to generalized Hermite-Hadamard-Fejér inequality, see [28] and references therein.

On the other hand, the concept of -convex functions, firstly named by -convex functions, as generalization of convex functions has been introduced in [9].

Let be an interval in real line . Consider for appropriate .

Definition 2 (see [9, 10]). A function is called convex with respect to (briefly -convex), iffor all and .

In fact, the above definition geometrically says that if a function is -convex on , then its graph between any is on or under the path starting from and ending at . If should be the end point of the path for every , then we have and the function reduces to a convex one. Note that by taking in (1) we get for any and which implies that for any . Also if we take = 1 in (1) we get for any .

There are simple examples about -convexity of a function.

Example 3 (see [9, 10]). Consider a function defined as and define a bifunction as , for all It is not hard to check that is a -convex function but not a convex one.
Define the function as and a bifunction as Then is -convex but is not convex.

The following characterization of -convexity holds [11].

Theorem 4. A function is -convex if and only if, for any with ,

The following result is of importance.

Theorem 5 (see [9, 11]). Suppose that is a -convex function and is bounded from above on . Then satisfies a Lipschitz condition on any closed interval contained in the interior of . Hence, is absolutely continuous on and continuous on .

Note. As a consequence of Theorem 5, a -convex function where is bounded from above on is integrable. For more results about -convex functions, see [912].

Motivated by the above works, in this paper we give some applications and examples for trapezoidal and midpoint type inequalities when the intended function is differentiable. Furthermore we consider integral quadrature formula and give an error estimate related to trapezoidal and midpoint formula. Furthermore some examples support our results.

The following two theorems have been proved in [13] which improve the right part and the lefty side of (1), respectively.

Theorem 6 (see [13]). Let be a -convex function which is bounded from above on . If is integrable on , then we have inequalitiesIf is symmetric on , then from inequality (10) we get

Theorem 7 (see [13]). Let be a -convex function with bounded from above on . If is integrable on , thenMoreover, if is symmetric on , then

Remark 8. (a) Inequality (11) gives a refinement for the right side and inequality (13) gives a refinement for the left side of (1), respectively.
(b) If in Theorems 6 and 7 we consider for all (see Theorem  3.6 in [9]), then respectively, we have which is a refinement for the right side of Hermite-Hadamard inequality related to -convex functions, and which is a refinement for the left side of Hermite-Hadamard inequality related to -convex functions.

2. Main Results

As an application of Theorem 6, we can obtain some estimation results for the difference between the right and middle part of (2) and also for the difference between the left and middle part of (2), respectively.

The following identity for an absolutely continuous function holds (see [14]):

Theorem 9. Let be a differentiable function, . Suppose that the function is a -convex function and is bounded from above on , and then we have

Proof. From (16) we getSince is a -convex function and with is symmetric, then by inequality (11) we haveA simple calculation shows that then by (19) we get the desired result (17).

Also there exists another identity for absolutely continuous functions as the following (see [15]).where is considered as So we can obtain a result for the difference between the left and middle part of (2).

Theorem 10. Let be a differentiable function, . Suppose that the function is a -convex function and is bounded from above on , and then

Proof. From (21) we have It follows thatwhich is a symmetric function on the interval Since is a -convex function and with is symmetric, then by inequality (11) we haveSincethen by (26) we get the desired result (23).

Example 11. Consider the function defined as and a bifunction as It is clear that the function is a -convex function. From inequality (23) in Theorem 10, for with , we getFurther calculations show that where Also with the same argument as above, from (17) in Theorem 9, for with , we can obtain the following inequality: which implies that where is defined as above.

Using the following identities for twice differentiable functions , we can obtain some results similar to Theorems 9 and 10.see [14];see [16], where then we have the following.

Theorem 12. Let the function be twice differentiable, . Suppose that the function is a -convex function and is bounded from above on , and then we have the inequalities

Proof. Taking the modulus on (36) along with implies the expected result (39).

Theorem 13. With the assumptions of Theorem 12, we have the inequalities

Proof. Taking the modulus on (37) along with implies the expected result (41).

Example 14. Consider the function defined as and the bifunction defined in Example 11. It is easy to see that the function is a -convex function. From inequality (41) in Theorem 13, for with , we getIt follows that where is defined in Example 11.

In [13], we estimated the difference of the left and middle section of (1) as the following.

Theorem 15 (see [13]). Suppose that is a differentiable mapping, . If is a continuous mapping symmetric about and is a -convex mapping on with bounded from above on , thenwhere

Theorem 16 (see [10]). Suppose that is a differentiable mapping, . If is a continuous function and symmetric about and is a -convex function with bounded from above on , then

If in Theorem 16 we set and , then we have the following result.

Using of Theorems 15 and 16 can result in some numerical inequalities. This fact is shown in the following example.

Example 17. Consider all assumptions of Example 11 and which is symmetric to . From inequality (46) in Theorem 15, for with , we getwhere Further calculations in (49) show that where

As an application of Theorems 15 and 16, we give an error estimate for trapezoidal and midpoint formula that are generalization of Proposition in [17] and Proposition in [18], respectively.

Suppose that is a partition of interval . Consider quadrature formula where and is the approximation error for trapezoidal and midpoint formula.

Theorem 18. Suppose that is a differentiable function, is a continuous function and symmetric with respect to , and is a -convex function where is bounded from above on . Then

Proof. It is enough to apply Theorem 15 on the subinterval () of the partition for interval and to sum all achieved inequalities over and then use triangle inequality.

Theorem 19. Suppose that is a differentiable mapping, is a continuous mapping symmetric about , and is a -convex mapping on with a bounded from above. Thenwhere for

Proof. It is enough to apply Theorem 16 on the subinterval () of the partition for interval and to sum all achieved inequalities over and then use triangle inequality.

Example 20. Suppose that is the partition of interval . Consider the functions and the bifunction as defined in Example 17 for . So the approximation error related to the trapezoidal and midpoint formula can be calculated as the following. From (55) we get Also from (56) we have where Then with some calculations we get Note that we can decrease the approximation error by choosing .

3. Conclusions

The convexity of a function is the basis for many inequalities in mathematics. Note that, in new problems related to the convexity, generalized notions about convex functions are required to obtain applicable results. One of this generalizations is the notion of -convex functions which can generalize many inequalities related to convex functions such as Hermite-Hadamard inequality, Fejér inequality, and trapezoid and midpoint type inequality.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are very grateful to Professor S. S. Dragomir for his valuable comments and suggestions about properties of -convex functions. M. De La Sen is grateful to the Spanish Government and to the European Fund of Regional Development (FEDER) through Grant DPI2015-64766-R and to UPV/EHU through Grant PGC 17/33.