# Hermite-Hadamard-Fejer Inequality Related to Generalized Convex Functions via Fractional Integrals.

1. Introduction and PreliminariesIn 1906, L. Fejer [1] proved the following integral inequalities which are known in the literature as Fejer inequality:

[mathematical expression not reproducible], (1)

where f : [a; b] [right arrow] R is convex and g : [a, b] [right arrow] [0, [infinity]) = [R.sup.+] is integrable and symmetric to x = ((a + b)/2)(g(x) = g(a + b - x), [for all]x [member of] [a, b]).

If we consider in (1) that g [equivalent to] 1, then we recapture the known Hermite-Hadamard inequality.

[mathematical expression not reproducible]. (2)

On the other hand, the concept of ([[eta].sub.1], [[eta].sub.2])-convex has been introduced in [2, 3] as a generalization of preinvex functions [4-6] and [eta]-convex functions [7-9]. We remind these introductory concepts in the following with some geometric interpretation.

(Preinvex Functions)

Definition 1. A set I [subset] R is invex with respect to a real bifunction [eta] : I x I [right arrow] R, if

x, y [member of] l, [lambda] [member of] [0, 1] [right arrow] y + [lambda][eta] (x, y) [member of] I. (3)

If I is an invex set with respect to g, then a function f: I [right arrow] R is said to be preinvex if x, y [member of] I and [lambda] [member of] [0, 1] implies

f(y + [lambda][eta] (x, y)) [less than or equal to] [lambda]f (x) + (1 - [lambda])f (y). (4)

In fact, in an invex set for any x, y [member of] I, there is a path starting from y to y + [eta](x, y) which is contained in I. The point x is not necessarily the end point of the path. If for every x, y [member of] I we need that x should be an end point of the path, then I reduces to a convex set.

([eta]-Convex Functions)

Definition 2. Consider a convex set I [subset or equal to] R and a bifunction [eta] : f(I) x f(I) [right arrow] R. A function f: I [right arrow] R is called convex with respect to [eta] (briefly [eta]-convex), if

f([lambda]x + (1 - [lambda])y) [less than or equal to] f(y) + [lambda][eta] (f (x), f (y)), (5)

for all x, y [member of] I and [lambda] [member of] [0, 1].

Geometrically, it says that if a function is [eta]-convex on I, then, for any x, y [member of] l, its graph is on or under the path starting from (y, f(y)) and ending at (x, f(y) + [eta] (f(x), f(y))). If f(x) should be the end point of the path for every x, y [member of] I, then we have [eta](x, y) = x - y and the function reduces to a convex one.

(([[eta].sub.1], [[eta].sub.2])-Convex Functions)

Definition 3. Let I [subset] R be an invex set with respect to [[eta].sub.1] : 1 x 1 [right arrow] R. Consider f: I [right arrow] R and [[eta].sub.2] : f(I) x f(I) [right arrow] R. The function f is said to be ([[eta].sub.1], [[eta].sub.2])-convex if

f(x + [lambda][[eta].sub.1] (y, x)) [less than or equal to] f(x) + [lambda][[eta].sub.2] (f (y), f (x)), (6)

for all x, y [member of] I and X [member of] [0, 1].

Remark 4. An ([[eta].sub.1], [[eta].sub.2])-convex function reduces to

(i) a [eta]-convex function if we consider [[eta].sub.1] (x, y) = x - y for all x, y [member of] I,

(ii) a preinvex function if we consider [[eta].sub.2](x, y) = x - y for all x, y [member of] f(I),

(iii) a convex function if satisfying (i) and (ii).

Example 5 (see [2]). Consider the function f: [R.sup.+] [right arrow] [R.sup.+] by

[mathematical expression not reproducible]. (7)

Define two bifunctions [[eta].sub.1] : [R.sup.+] x [R.sup.+] [right arrow] R and [[eta].sub.2] : [R.sup.+] x [R.sup.+] [right arrow] [R.sup.+] by

[mathematical expression not reproducible], (8)

and

[mathematical expression not reproducible]. (9)

Then f is an ([[eta].sub.1], [[eta].sub.2])-convex function. But f is not preinvex with respect to [[eta].sub.1] and it is not convex (consider x = 0, y = 2, and [lambda] > 0).

Also we need the following short preliminaries about the fractional calculus theory which are used throughout the paper.

Definition 6 (see [10]). Consider f [member of] [L.sup.1] [a, b]. The Riemann-Liouville integrals [mathematical expression not reproducible] and [mathematical expression not reproducible] of order a [greater than or equal to] 0 with a [greater than or equal to] 0 are defined by

[mathematical expression not reproducible], (10)

and

[mathematical expression not reproducible], (11)

respectively, where

[mathematical expression not reproducible] (12)

is Gamma function and [mathematical expression not reproducible].

Fejer inequality for convex functions related to fractional integrals has been obtained in [11] as the following theorem.

Theorem 7. Let f: [a, b] [right arrow] R be convex function with a < b and f [member of] [L.sup.1] [a, b]. If g : [a, b] [right arrow] R is nonnegative, integrable, and symmetric to (a + b)/2, then the following inequalities hold for fractional integrals:

[mathematical expression not reproducible], (13)

with [alpha] > 0.

Motivated by above works and results, in this paper, we obtain Fejer inequality for ([[eta].sub.1], [[eta].sub.2])-convex functions via fractional integrals. Also we give some mid-point and trapezoidtype inequalities related to Hermite-Hadamard inequality when the absolute value of the derivative of the considered function is ([[eta].sub.1], [[eta].sub.2])-convex functions. Furthermore, we prove that when a positive ([[eta].sub.1], [[eta].sub.2])-convex function is increasing, there exists a refinement for classic Hermite-Hadamard inequality via fractional integrals.

2. Fejer Inequality

In this section, we obtain ([[eta].sub.1], [[eta].sub.2])-convex version of the Fejer inequality related to fractional integrals. For convenience, we separate this inequality to the left and right.

Theorem 8 (Fejer's left inequality). Let I [subset or equal to] R be an invex set with respect to [[eta].sub.1] such that

[mathematical expression not reproducible], (14)

for all [x.sub.1], [x.sub.2] el and [t.sub.1], [t.sub.2] [member of] [0, 1]. Also let f : I [right arrow] R be an ([[eta].sub.1], [[eta].sub.2])-convex function, where [[eta].sub.2] is an integrable bifunction on f(I) x f(I). For any a, b [member of] I with [[eta].sub.1] (b, a) > 0, suppose that f [member of] [L.sup.1] [a, a + [[eta].sub.1](b, a)] and the function g: [a, a + [[eta].sub.1](b, a)] [right arrow] [R.sup.+] is integrable and symmetric to a + (1/2) [[eta].sub.1](b, a). Then, for [alpha] > 0, the following inequality holds:

[mathematical expression not reproducible]. (15)

Proof. Using condition (14) and the ([[eta].sub.1], [[eta].sub.2])-convexity of f, we have

[mathematical expression not reproducible], (16)

and with the same argument as above we have

[mathematical expression not reproducible]. (17)

By the use of Definition 6 and two changes of variable

x = [2a + (1 - t) [[eta].sub.1] (b, a)/2] (18)

and

x = [2a + (1 - t) [[eta].sub.1] (b, a)/2], (19)

in (16) and (17), respectively, we obtain the following inequalities:

[mathematical expression not reproducible]. (20)

The simple form of (20) along with the fact that g is symmetric to a+(1/2)[[eta].sub.1] (b, a) leads to the following relations:

[mathematical expression not reproducible]. (21)

Also with the same argument as above we have

[mathematical expression not reproducible]. (22)

Now adding [K.sub.1] to [K.sub.2] implies the result.

To obtain the right part of the Fejer inequality related to fractional integrals, we need a primary lemma.

Lemma 9. Let I [subset or equal to] R be an invex set with respect to [[eta].sub.1] : I x I [right arrow] R and a, b [member of] I with [[eta].sub.1] (b, a) > 0. If g : [a, a + [[eta].sub.1] (b, a)] [right arrow] R is integrable and symmetric to a+(1/2) [[eta].sub.1](b, a), then

[mathematical expression not reproducible]. (23)

Proof. Since g is symmetric to a+(1/2) [[eta].sub.1](b, a), we have g(2a+ [[eta].sub.1] (b, a) - x) = g(x) for all x [member of] [a, a + [[eta].sub.1](b, a)]. Then

[mathematical expression not reproducible]. (24)

Theorem 10 (Fejer's right inequality). Let I [subset or equal to] R be an invex set with respect to [[eta].sub.1] and let f: I [right arrow] R be an ([[eta].sub.1], [[eta].sub.2])-convex function, where [[eta].sub.1] is an integrable bifunction on f(I) x f(I). For any a, b [member of] I with [[eta].sub.1](b, a) > 0, suppose that the function g : [a, a + [[eta].sub.1](b,a)] [right arrow] [R.sup.+] is integrable and symmetric to a + (1/2) [[eta].sub.1](b, a) and f [member of] [L.sup.1][a, a + [[eta].sub.1] (b, a)]. Then, for [alpha] > 0, the following inequality holds:

[mathematical expression not reproducible]. (25)

Proof. From ([[eta].sub.1], [[eta].sub.2])-convexity of f, using the changes of variables x = a + t[[eta].sub.1](b, a) and x = a + (1 - t) [[eta].sub.1](b, a), respectively, we obtain the two following inequalities:

[mathematical expression not reproducible], (26)

and

[mathematical expression not reproducible]. (27)

Now adding (26) to (27) with the fact that g is symmetric to a + (1/2)[[eta].sub.1](b, a) implies that

[mathematical expression not reproducible]. (28)

Now by the use of Lemma 9 we have

[mathematical expression not reproducible], (29)

which implies the respected inequality.

Corollary 11. If in Theorems 8 and 10 we consider

(i) [alpha] = 1, then the following inequality holds, which is the classical form of Fejer inequality related to ([[eta].sub.1], [[eta].sub.2])-convex functions:

[mathematical expression not reproducible]. (30)

(ii) g [equivalent to] 1, then we get Hermite-Hadamard inequality for ([[eta].sub.1], [[eta].sub.2])-convex functions as follows:

[mathematical expression not reproducible], (31)

which is a generalization of inequality (2.1) in [12].

Corollary 12. If in Theorems 8 and 10 we set [[eta].sub.1] (x, y) = x - y for all x, y [member of] I, then we obtain Fejer inequality for fractional integrals related to [eta]-convex functions.

[mathematical expression not reproducible]. (32)

Corollary 13. If in Theorems 8 and 10 we set [[eta].sub.2](x, y) = x - y for all x, y [member of] f(I), then we obtain Fejer inequality for fractional integrals related to preinvex functions.

[mathematical expression not reproducible]. (33)

Corollary 14 (see [11]). With all conditions of Corollaries 12 and 13, we have the classic Fejer inequality for fractional integrals.

[mathematical expression not reproducible]. (34)

Corollary 15 (see [12]). If in (34) we consider 0 [equivalent to] 1, then we recapture Hermite-Hadamard inequality for fractional integrals in convex case.

[mathematical expression not reproducible]. (35)

3. Mid-Point and Trapezoid-Type Inequalities

In this section, we obtain, respectively, the mid-point and trapezoid-type inequalities related to (31) when the absolute value of the derivative of the considered function is ([[eta].sub.1], [[eta].sub.2])-convex. In fact, by mid-point-type inequality we mean estimating the difference between left and middle parts of (31) and by trapezoid-type inequality we mean estimating the difference between right and middle parts of (31).

The following lemma is generalization of Lemma 1 obtained in [13] to the preinvex case.

Lemma 16. Let I c R be an open invex set with respect to [[eta].sub.1] : I x I [right arrow] R and let f: I [right arrow] R be a differentiable function. For any a, b [member of] I with [[eta].sub.1] (b, a) > 0, if f [member of] [L.sup.1] [a, a + [[eta].sub.1] (b, a)], then the following equality for fractional integrals holds:

[mathematical expression not reproducible], (36)

where

[mathematical expression not reproducible]. (37)

Proof. Integrating by parts in [I.sub.1] implies that

[mathematical expression not reproducible]. (38)

Similarly, we have

[mathematical expression not reproducible]. (39)

Now, by adding all of above equalities, we get to the desired result.

[mathematical expression not reproducible].

The mid-point-type inequality related to (31) is obtained in the following.

Theorem 17. Let 1 [subset or equal to] R be an open invex set with respect to [[eta].sub.1] : I x I [right arrow] R and let f : I [right arrow] R be a differentiable function. Suppose that [absolute value of f'] is an ([[eta].sub.1], [[eta].sub.2])-convex function on I and, for any a, b [member of] I with [[eta].sub.1](b, a) > 0, f' [member of] [L.sup.1] [a, a + [[eta].sub.1] (b, a)]. Then

[mathematical expression not reproducible]. (41)

Proof. Using Lemma 16, we get

[mathematical expression not reproducible]. (42)

Now, using ([[eta].sub.1], [[eta].sub.2])-convexity of [absolute value of f'], we obtain

[mathematical expression not reproducible]. (43)

Analogously,

[mathematical expression not reproducible]. (44)

Also, using the fact that

[mathematical expression not reproducible]. (45)

for all [alpha] [member of] (0, 1] and [t.sub.1], [t.sub.2] [member of] [0, 1], we have

[mathematical expression not reproducible], (46)

and

[mathematical expression not reproducible]. (47)

Now adding all of above inequalities implies the required result.

Corollary 18. If in Theorem 17 we consider [[eta].sub.1] (x, y) = x - y for all x, y [member of] I, then

[mathematical expression not reproducible]. (48)

Furthermore, if we set [[eta].sub.2](x, y) = x - y for all x, y [member of] f(I), then we recapture Theorem 2 in [13].

The following result has been obtained in [14].

Lemma 19. Let I [subset or equal to] R be an open invex set with respect to [[eta].sub.1] : I x I [right arrow] R. Also, suppose that f : I [right arrow] R is a differentiable function. For any a, b [member of] I with [[eta].sub.1] (b, a) > 0, if f [member of] [L.sup.1][a, a + [[eta].sub.1] (b, a)], then the following equality for fractional integral holds:

[mathematical expression not reproducible]. (49)

Now we give the trapezoid-type inequality related to (31).

Theorem 20. Let I [subset or equal to] R be an open invex set with respect to [[eta].sub.1] : I x I [right arrow] R and let f : I [right arrow] R be a differentiable function. Suppose that [absolute value of f'] is an ([[eta].sub.1], [[eta].sub.2])-convex function on I and, for any a, b [member of] I with [[eta].sub.1](b, a) > 0, f' [member of] [L.sup.1][a, a + [[eta].sub.1](b, a)]. Then the following inequality for fractional integrals holds:

[mathematical expression not reproducible]. (50)

Proof. Using Lemma 19 and ([[eta].sub.1], [[eta].sub.2])-convexity of [absolute value of f'],we get

[mathematical expression not reproducible], (51)

where the last equality can be obtained after some calculations in corresponding integrals and utilizing them.

Corollary 21. If in Theorem 20 we consider

(i) [[eta].sub.1](x, y) = x - y for all x, y [member of] I, then we obtain

[mathematical expression not reproducible]. (52)

(ii) [[eta].sub.2](x, y) = x - y for all x, y [member of] f(I), then we have Theorem 2.5 in [14].

(iii) conditions of (i) and (ii) together, then we recapture Theorem 3 in [12].

As the last result by using Theorem 2 in [15], we obtain a refinement of Hermite-Hadamard inequality in connection with fractional integrals related to the increasing ([[eta].sub.1], [[eta].sub.2])convex functions.

Theorem 22 (see [15]). If [f.sub.1] and [f.sub.2] are positive increasing functions on [0, 1], then

[mathematical expression not reproducible]. (53)

Also if [f.sub.1] and [f.sub.2] are positive decreasing functions on [0, 1] and K is an upper bound for [f.sub.1] and [f.sub.2], then K - [f.sub.1] and K - [f.sub.2] are positive increasing functions and we have

[mathematical expression not reproducible], (54)

which gives again

[mathematical expression not reproducible]. (55)

Theorem 23. Let I c R be an invex set with respect to [[eta].sub.1] : I x I [right arrow] R. Also let f : I [right arrow] R be an increasing positive ([[eta].sub.1], [[eta].sub.2])-convex function. If, for each a, b [member of] I with [[eta].sub.1](b, a) > 0, f [member of] [L.sup.1][a,a + [[eta].sub.1](b,a)] and [[eta].sub.2] is integrable on f(I) x f(I), then, for [alpha] > 1, the following inequalities hold:

[mathematical expression not reproducible]. (56)

Proof. From ([[eta].sub.1], [[eta].sub.2])-convexity of f, we have,

[mathematical expression not reproducible], (57)

and

[mathematical expression not reproducible]. (58)

By adding these inequalities, we get

[mathematical expression not reproducible]. (59)

Multiplying both sides of (59) by [t.sup.[alpha]-1] and integrating the resulting inequality with respect to t over [0,1] and using Theorem 22, we obtain

[mathematical expression not reproducible]. (60)

Using the changes of variables x = a + t[[eta].sub.1] (b, a) and x = a + (1 - t)[[eta].sub.1](b, a), respectively, in above integrals, we have

[mathematical expression not reproducible]. (61)

Since

[mathematical expression not reproducible], (62)

and

[mathematical expression not reproducible], (63)

we get

[mathematical expression not reproducible]. (64)

Furthermore, since f is ([[eta].sub.1], [[eta].sub.2])-convex, then

[mathematical expression not reproducible], (65)

which completes the proof.

Corollary 24. Let f : [a, b] [right arrow] R be an increasing positive convex function. Then, for [alpha] > 1, the following inequalities hold:

[mathematical expression not reproducible]. (66)

4. 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 these generalizations is the notion of ([[eta].sub.1], [[eta].sub.2])-convex functions which can generalize many inequalities related to convex functions such as Hermite-Hadamard inequality, Fejer inequality, and trapezoid-type and mid-point-type inequalities.

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/5864091

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

M. De La Sen is very grateful to the Spanish Government for its support by the European Regional Development Fund (ERDF) through Grant DPI2015-64766-R and to UPV/EHU for its support by Grant PGC 17/33.

References

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[2] S. M. Aslani, M. R. Delavar, and S. M. Vaezpour, "Inequalities of Fejer type related to generalized convex functions with applications," International Journal of Analysis and Applications, vol. 16, no. 1, pp. 38-49, 2018.

[3] M. S. Aslani, R. M. Delavar, and S. M. Vaezpour, "Hermite-Hadamard type integral inequalities for generalized convex functions," Journal of Inequalities and Special Functions, vol. 8, no. 1, pp. 17-33, 2018.

[4] A. Ben-Israel and B. Mond, "What is invexity?" The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, vol. 28, no. 1, pp. 1-9, 1986.

[5] M. A. Hanson and B. Mond, "Convex transformable programming problems and invexity," Journal of Information & Optimization Sciences. A Journal Devoted to Advances in Information Sciences, Optimization Sciences and Related Aspects, vol. 8, no. 2, pp. 201-207, 1987.

[6] V. Jeyakumar, "Strong and weak invexity in mathematical programming," European Journal of Operational Research, vol. 55, pp. 109-125, 1985.

[7] M. Eshaghi Gordji, M. Rostamian Delavar, and M. De La Sen, "On [phi]-convex functions," Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 173-183, 2016.

[8] M. R. Delavar and S. S. Dragomir, "On [eta]-Convexity," Journal of Inequalities and Applications, vol. 20, no. 1, pp. 203-216, 2017.

[9] M. Rostamian Delavar and M. De La Sen, "Some generalizations of Hermite-Hadamard type inequalities," SpringerPlus, vol. 5, no. 1, article no. 1661, 2016.

[10] R. Gorenflo and F. Mainardi, "Fractional calculus: integral and differential equations of fractional order," in Fractals and Fractional Calculus in Continuum Mechanics, Springer, Vienna, Austria, 1997.

[11] I. Iscan, "Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals," Studia Universitatis Babes-Bolyai Mathematica, vol. 60, no. 3, pp. 355-366, 2015.

[12] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, "Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities," Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2403-2407, 2013.

[13] M. Iqbal, M. I. Bhatti, and K. Nazeer, "Generalization of inequalities analogous to Hermite-HADamard inequality via fractional integrals," Bulletin of the Korean Mathematical Society, vol. 52, no. 3, pp. 707-716, 2015.

[14] I. Iscan, "Hermite-Hadamards inequalities for preinvex functions via fractional integrals and related fractional inequalities," American Journal of Mathematical Analysis, vol. 1, no. 3, pp. 33-38, 2013.

[15] M. D. Rostamian and F. Sajadian, "Hermite-Hadamard type integral inequalities for log-[eta]-convex functions," Mathematics and Computer Science, vol. 1, no. 4, pp. 86-92, 2016.

M. Rostamian Delavar (iD), (1) S. Mohammadi Aslani, (2) and M. De La Sen (iD) (3)

(1) Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

(2) Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

(3) Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa, Aptdo. 644, 48080 Bilbao, Spain

Correspondence should be addressed to M. Rostamian Delavar; m.rostamian@ub.ac.ir

Received 16 May 2018; Accepted 26 June 2018; Published 1 August 2018

Academic Editor: S. K. Q. Al-Omari

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Title Annotation: | Research Article |
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Author: | Delavar, M. Rostamian; Aslani, S. Mohammadi; De La Sen, M. |

Publication: | Journal of Mathematics |

Geographic Code: | 4EXHU |

Date: | Jan 1, 2018 |

Words: | 3640 |

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