Next Article in Journal
Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation
Previous Article in Journal
On the Fractal Langevin Equation
Article Menu

Export Article

Fractal Fract 2019, 3(1), 12; https://doi.org/10.3390/fractalfract3010012

Article
Some New Fractional Trapezium-Type Inequalities for Preinvex Functions
1
Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora 9401, Albania
2
Department of Mathematics, Faculty of Sciences and Arts, Ordu University, 52200 Ordu, Turkey
*
Author to whom correspondence should be addressed.
Received: 28 February 2019 / Accepted: 21 March 2019 / Published: 24 March 2019

Abstract

:
In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research.
Keywords:
trapezium-type integral inequalities; preinvexity; general fractional integrals
MSC:
26A51; 26A33; 26D07; 26D10; 26D15

1. Introduction

The following notations are used throughout this paper. We use I to denote an interval on the real line R = ( , + ) . For any subset K R n , K is the interior of K . The set of integrable functions on the interval [ a 1 , a 2 ] is denoted by L [ a 1 , a 2 ] .
The following inequality obtained by Hermite and Hadamard is one of the most famous inequalities in the literature for convex functions.
Theorem 1.
Let f : I R R be a convex function on I and a 1 , a 2 I with a 1 < a 2 . Then, the following inequality holds:
f a 1 + a 2 2 1 a 2 a 1 a 1 a 2 f ( x ) d x f ( a 1 ) + f ( a 2 ) 2 .
This inequality (1) is known as Hermite–Hadamard or trapezium inequality. As a result of the rich applications in the field of numerical analysis, this result has attracted many mathematicians attention from all over the world. For other recent results which generalize, improve, and extend the inequality (1) through various classes of convex functions interested readers are referred to References [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].
Let us recall some special functions and evoke some basic definitions as follows.
Definition 1
([34]). A set S R n is said to be an invex set with respect to the mapping η : S × S R n , if x + t η ( y , x ) S for every x , y S and t [ 0 , 1 ] .
The invex set S is also termed an η -connected set.
Definition 2.
Let S R n be an invex set with respect to η : S × S R n . A function f : S [ 0 , + ) is said to be preinvex with respect to η , if for every x , y S and t [ 0 , 1 ] ,
f x + t η ( y , x ) ( 1 t ) f ( x ) + t f ( y ) .
Furthermore, let us define a function φ : [ 0 , ) [ 0 , ) satisfying the following conditions:
0 1 φ ( t ) t d t < ,
1 A φ ( s ) φ ( r ) A for 1 2 s r 2
φ ( r ) r 2 B φ ( s ) s 2 for s r
| φ ( r ) r 2 φ ( s ) s 2 | C | r s | φ ( r ) r 2 for 1 2 s r 2
where A , B , C > 0 are independent of r , s > 0 . If φ ( r ) r α is increasing for some α 0 and φ ( r ) r β is decreasing for some β 0 , then φ satisfies (3)–(6), see Reference [35]. Therefore, Sarıkaya and Ertuğral [28] defined the following left-sided and right-sided generalized fractional integral operators, respectively, as follows:
a 1 + I φ f ( x ) = a 1 x φ ( x t ) x t f ( t ) d t , x > a 1 ,
a 2 I φ f ( x ) = x a 2 φ ( t x ) t x f ( t ) d t , x < a 2 .
This fractional integral operators are a new generalization of fractional integrals such as the Riemann–Liouville fractional integral, the k-Riemann–Liouville fractional integral, Katugampola fractional integrals, the conformable fractional integral, Hadamard fractional integrals, etc. To read more about fractional analysis, see References [10,11,22,27].
Motivated by the above literature, the main objective of this paper is firstly to discover in Section 2 an interesting identity in order to establish some new bounds regarding trapezium-type integral inequalities. Then, using this lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals will be obtained. It is pointed out that some new special cases will be deduced from the main results. In Section 3, some applications regarding special means for different real numbers are given. The ideas and techniques described in this paper may stimulate further research in the field of integral inequalities.

2. Main Results

Throughout this study, let P = [ m a 1 , a 2 ] with a 1 < a 2 , m ( 0 , 1 ] be an invex subset with respect to η : P × P R . Additionally, for brevity, we define
Λ m * ( t ) : = 0 t φ η ( a 2 , m a 1 + a 2 x ) u u d u < , η ( a 2 , m a 1 + a 2 x ) > 0
and
Δ m * ( t ) : = 0 t φ η ( m a 1 + a 2 x , m a 1 ) u u d u < , η ( m a 1 + a 2 x , m a 1 ) > 0 .
For establishing some new results regarding general fractional integrals we need to prove the following lemma.
Lemma 1.
Let f : P R be a differentiable mapping on ( m a 1 , a 2 ) . If f L ( P ) , then the following identity for generalized fractional integrals holds:
[ 1 2 Λ m * ( 1 ) × ( m a 1 + a 2 x ) + I φ f m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) + 1 2 Δ m * ( 1 ) × ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) I φ f m a 1 ] f ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) + f ( m a 1 + a 2 x ) 2 = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) 0 1 Λ m * ( 1 t ) f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) d t η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) d t .
We denote
T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) : = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) × 0 1 Λ m * ( 1 t ) f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) d t η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) d t .
Proof. 
Integrating by parts (12), using (9) and (10) and changing the variables of integration, we have
T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) { Λ m * ( 1 t ) f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) η ( a 2 , m a 1 + a 2 x ) | 0 1 + 1 η ( a 2 , m a 1 + a 2 x ) × 0 1 φ η ( a 2 , m a 1 + a 2 x ) ( 1 t ) 1 t f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) d t } η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) { Δ m * ( t ) f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) η ( m a 1 + a 2 x , m a 1 ) | 0 1 1 η ( m a 1 + a 2 x , m a 1 ) × 0 1 φ η ( m a 1 + a 2 x , m a 1 ) t t f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) d t } = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) { Λ m * ( 1 ) f ( m a 1 + a 2 x ) η ( a 2 , m a 1 + a 2 x ) + 1 η ( a 2 , m a 1 + a 2 x ) × ( m a 1 + a 2 x ) + I φ f m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) }
η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) { Δ m * ( 1 ) f m a 1 + η ( m a 1 + a 2 x , m a 1 ) η ( m a 1 + a 2 x , m a 1 ) 1 η ( m a 1 + a 2 x , m a 1 ) × ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) I φ f m a 1 } = [ 1 2 Λ m * ( 1 ) × ( m a 1 + a 2 x ) + I φ f m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) + 1 2 Δ m * ( 1 ) × ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) I φ f m a 1 ] f ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) + f ( m a 1 + a 2 x ) 2 .
This completes the proof of the lemma. □
Remark 1.
Taking m = 1 , η ( m a 1 + a 2 x , m a 1 ) = ( m a 1 + a 2 x ) m a 1 and η ( a 2 , m a 1 + a 2 x ) = a 2 ( m a 1 + a 2 x ) in Lemma 1, we get
T f , Λ 1 * , Δ 1 * ( x ; a 1 , a 2 ) = 1 2 Λ 1 * ( 1 ) × ( a 1 + a 2 x ) + I φ f ( a 2 ) + 1 2 Δ 1 * ( 1 ) × ( a 1 + a 2 x ) I φ f a 1 f ( a 1 + a 2 x ) .
Theorem 2.
Suppose that m ( 0 , 1 ] is a fixed number. Let f : P R be a differentiable mapping on ( m a 1 , a 2 ) . If | f | q is preinvex on P for q > 1 and p 1 + q 1 = 1 , then the following inequality for generalized fractional integrals holds:
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( p ) p | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( p ) p | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q 2 q ,
where
B Λ m * ( p ) : = 0 1 Λ m * ( 1 t ) p d t , C Δ m * ( p ) : = 0 1 Δ m * ( t ) p d t .
Proof. 
From Lemma 1, preinvexity of | f | q , Hölder inequality, and the properties of the modulus, we have
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) 0 1 Λ m * ( 1 t ) | f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) | d t + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) | f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) | d t
η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) 0 1 Λ m * ( 1 t ) p d t 1 p × 0 1 | f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) | q d t 1 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) p d t 1 p × 0 1 | f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) | q d t 1 q η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( p ) p 0 1 ( 1 t ) | f ( m a 1 + a 2 x ) | q + t | f ( a 2 ) | q d t 1 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( p ) p 0 1 ( 1 t ) | f ( m a 1 ) | q + t | f ( m a 1 + a 2 x ) | q d t 1 q = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( p ) p | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( p ) p | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q 2 q .
The proof of this theorem is complete. □
We point out some special cases of Theorem 2.
Corollary 1.
Taking m = 1 , η ( m a 1 + a 2 x , m a 1 ) = ( m a 1 + a 2 x ) m a 1 and η ( a 2 , m a 1 + a 2 x ) = a 2 ( m a 1 + a 2 x ) in Theorem 2, we get
| T f , Λ 1 * , Δ 1 * ( x ; a 1 , a 2 ) | ( x a 1 ) 2 Λ 1 * ( 1 ) B Λ 1 * ( p ) p | f ( a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + ( a 2 x ) 2 Δ 1 * ( 1 ) C Δ 1 * ( p ) p | f ( a 1 ) | q + | f ( a 1 + a 2 x ) | q 2 q .
Corollary 2.
Taking x = a 1 + a 2 2 in Corollary 1, we get
| T f , Λ 1 * , Δ 1 * a 1 + a 2 2 ; a 1 , a 2 | ( a 2 a 1 ) 4 2 q Λ 1 * ( 1 ) { B Λ 1 * ( p ) p | f a 1 + a 2 2 | q + | f ( a 2 ) | q q + C Δ 1 * ( p ) p | f ( a 1 ) | q + | f a 1 + a 2 2 | q q } .
Corollary 3.
Taking p = q = 2 in Theorem 2, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( 2 ) | f ( m a 1 + a 2 x ) | 2 + | f ( a 2 ) | 2 2 + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( 2 ) | f ( m a 1 ) | 2 + | f ( m a 1 + a 2 x ) | 2 2 .
Corollary 4.
Taking φ ( t ) = t in Theorem 2, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 p + 1 p | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + η ( m a 1 + a 2 x , m a 1 ) 2 p + 1 p | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q 2 q .
Corollary 5.
Taking φ ( t ) = t α Γ ( α ) in Theorem 2, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 p α + 1 p | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + η ( m a 1 + a 2 x , m a 1 ) 2 p α + 1 p | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q 2 q .
Corollary 6.
Taking φ ( t ) = t α k k Γ k ( α ) in Theorem 2, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 p α k + 1 p | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q 2 q + η ( m a 1 + a 2 x , m a 1 ) 2 p α k + 1 p | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q 2 q .
Theorem 3.
Suppose that m ( 0 , 1 ] is a fixed number. Let f : P R be a differentiable mapping on ( m a 1 , a 2 ) . If | f | q is preinvex on P for q 1 , then the following inequality for generalized fractional integrals holds:
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( 1 ) 1 1 q D Λ m * | f ( m a 1 + a 2 x ) | q + E Λ m * | f ( a 2 ) | q q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( 1 ) 1 1 q F Δ m * | f ( m a 1 ) | q + G Δ m * | f ( m a 1 + a 2 x ) | q q ,
where
D Λ m * : = 0 1 t Λ m * ( t ) d t , E Λ m * : = 0 1 t Λ m * ( 1 t ) d t ,
F Δ m * : = 0 1 ( 1 t ) Δ m * ( t ) d t , G Δ m * : = 0 1 t Δ m * ( t ) d t ,
and B Λ m * ( 1 ) , C Δ m * ( 1 ) are defined as in Theorem 2.
Proof. 
From Lemma 1, the preinvexity of | f | q , the power mean inequality, and the properties of the modulus, we have
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) 0 1 Λ m * ( 1 t ) | f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) | d t + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) | f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) | d t
η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) 0 1 Λ m * ( 1 t ) d t 1 1 q × 0 1 Λ m * ( 1 t ) | f m a 1 + a 2 x + t η ( a 2 , m a 1 + a 2 x ) | q d t 1 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) 0 1 Δ m * ( t ) d t 1 1 q × 0 1 Δ m * ( t ) | f m a 1 + t η ( m a 1 + a 2 x , m a 1 ) | d t 1 q η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( 1 ) 1 1 q × 0 1 Λ m * ( 1 t ) ( 1 t ) | f ( m a 1 + a 2 x ) | q + t | f ( a 2 ) | q d t 1 q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( 1 ) 1 1 q × 0 1 Δ m * ( t ) ( 1 t ) | f ( m a 1 ) | q + t | f ( m a 1 + a 2 x ) | q d t 1 q = η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) B Λ m * ( 1 ) 1 1 q × D Λ m * | f ( m a 1 + a 2 x ) | q + E Λ m * | f ( a 2 ) | q q + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) C Δ m * ( 1 ) 1 1 q F Δ m * | f ( m a 1 ) | q + G Δ m * | f ( m a 1 + a 2 x ) | q q .
The proof of this theorem is complete. □
We point out some special cases of Theorem 3.
Corollary 7.
Taking m = 1 , η ( m a 1 + a 2 x , m a 1 ) = ( m a 1 + a 2 x ) m a 1 and η ( a 2 , m a 1 + a 2 x ) = a 2 ( m a 1 + a 2 x ) in Theorem 3, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | ( x a 1 ) 2 Λ 1 * ( 1 ) B Λ 1 * ( 1 ) 1 1 q D Λ 1 * | f ( a 1 + a 2 x ) | q + E Λ 1 * | f ( a 2 ) | q q + ( a 2 x ) 2 Δ 1 * ( 1 ) C Δ 1 * ( 1 ) 1 1 q F Δ 1 * | f ( a 1 ) | q + G Δ 1 * | f ( a 1 + a 2 x ) | q q .
Corollary 8.
Taking x = a 1 + a 2 2 in Corollary 7, we get
| T f , Λ 1 * , Δ 1 * a 1 + a 2 2 ; a 1 , a 2 | ( a 2 a 1 ) 4 Λ 1 * ( 1 ) { B Λ 1 * ( 1 ) 1 1 q D Λ 1 * | f a 1 + a 2 2 | q + E Λ 1 * | f ( a 2 ) | q q + C Δ 1 * ( 1 ) 1 1 q F Δ 1 * | f ( a 1 ) | q + G Δ 1 * | f a 1 + a 2 2 | q q } .
Corollary 9.
Taking q = 1 in Theorem 3, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 2 Λ m * ( 1 ) D Λ m * | f ( m a 1 + a 2 x ) | + E Λ m * | f ( a 2 ) | + η ( m a 1 + a 2 x , m a 1 ) 2 Δ m * ( 1 ) F Δ m * | f ( m a 1 ) | + G Δ m * | f ( m a 1 + a 2 x ) | .
Corollary 10.
Taking φ ( t ) = t in Theorem 3, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | η ( a 2 , m a 1 + a 2 x ) 4 3 q 2 | f ( m a 1 + a 2 x ) | q + | f ( a 2 ) | q q + η ( m a 1 + a 2 x , m a 1 ) 4 3 q × | f ( m a 1 ) | q + 2 | f ( m a 1 + a 2 x ) | q q .
Corollary 11.
Taking φ ( t ) = t α Γ ( α ) in Theorem 3, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | 1 α + 1 1 1 q η α + q 1 q ( a 2 , m a 1 + a 2 x ) Γ ( α + 1 ) | f ( m a 1 + a 2 x ) | q α + 2 + β ( 2 , α + 1 ) | f ( a 2 ) | q q + 1 α + 1 1 1 q η α + q 1 q ( m a 1 + a 2 x , m a 1 ) Γ ( α + 1 ) × β ( 2 , α + 1 ) | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q α + 2 q .
Corollary 12.
Taking φ ( t ) = t α k k Γ k ( α ) in Theorem 3, we get
| T f , Λ m * , Δ m * ( x ; a 1 , a 2 ) | k α + k 1 1 q η α k + q 1 q ( a 2 , m a 1 + a 2 x ) Γ k ( α + k ) | f ( m a 1 + a 2 x ) | q α k + 2 + β 2 , α k + 1 | f ( a 2 ) | q q + k α + k 1 1 q η α k + q 1 q ( m a 1 + a 2 x , m a 1 ) Γ k ( α + k ) β 2 , α k + 1 | f ( m a 1 ) | q + | f ( m a 1 + a 2 x ) | q α k + 2 q .

3. Applications to Special Means

Consider the following special means for different real numbers α , β , and α β 0 , as follows:
  • The arithmetic mean:
    A : = A ( α , β ) = α + β 2 ,
  • The harmonic mean:
    H : = H ( α , β ) = 2 1 α + 1 β ,
  • The logarithmic mean:
    L : = L ( α , β ) = β α ln | β | ln | α | ,
  • The generalized log-mean:
    L n : = L n ( α , β ) = β n + 1 α n + 1 ( n + 1 ) ( β α ) 1 n ; n Z { 1 , 0 } .
It is well known that L n is monotonic nondecreasing over n Z with L 1 : = L . In particular, we have the following inequality H L A . Now, using the theory results in Section 2, we give some applications regarding special means for different real numbers.
Proposition 1.
Let a 1 , a 2 R { 0 } . Then, for r 2 ( r Z ) , where q > 1 and p 1 + q 1 = 1 , the following inequality holds:
| L r m a 1 + a 2 x , m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) + L r m a 1 , m a 1 + η ( m a 1 + a 2 x , m a 1 ) 2 A ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) r , ( m a 1 + a 2 x ) r | r η ( a 2 , m a 1 + a 2 x ) 2 p + 1 p × A | m a 1 + a 2 x | q ( r 1 ) , | a 2 | q ( r 1 ) q + r η ( m a 1 + a 2 x , m a 1 ) 2 p + 1 p × A | m a 1 | q ( r 1 ) , | m a 1 + a 2 x | q ( r 1 ) q .
Proof. 
Applying Corollary 4 for f ( t ) = t r , one can obtain the result immediately. □
Proposition 2.
Let a 1 , a 2 R { 0 } . Then, for q > 1 and p 1 + q 1 = 1 , the following inequality holds:
| 1 2 L | m a 1 + a 2 x | , | m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) | + 1 2 L | m a 1 | , | m a 1 + η ( m a 1 + a 2 x , m a 1 ) | 1 H m a 1 + η ( m a 1 + a 2 x , m a 1 ) , m a 1 + a 2 x | η ( a 2 , m a 1 + a 2 x ) 2 p + 1 p 1 H | m a 1 + a 2 x | 2 q , | a 2 | 2 q q + η ( m a 1 + a 2 x , m a 1 ) 2 p + 1 p 1 H | m a 1 | 2 q , | m a 1 + a 2 x | 2 q ) q .
Proof. 
Applying Corollary 4 for f ( t ) = 1 t , one can obtain the result immediately. □
Proposition 3.
Let a 1 , a 2 R { 0 } . Then, for r 2 ( r Z ) and q 1 , the following inequality holds:
| L r m a 1 + a 2 x , m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) + L r m a 1 , m a 1 + η ( m a 1 + a 2 x , m a 1 ) 2 A ( m a 1 + η ( m a 1 + a 2 x , m a 1 ) ) r , ( m a 1 + a 2 x ) r | 2 3 q η ( a 2 , m a 1 + a 2 x ) 4 A 2 | m a 1 + a 2 x | q ( r 1 ) , | a 2 | q ( r 1 ) q + 2 3 q η ( m a 1 + a 2 x , m a 1 ) 4 A | m a 1 | q ( r 1 ) , 2 | m a 1 + a 2 x | q ( r 1 ) q .
Proof. 
Applying Corollary 10 for f ( t ) = t r , one can obtain the result immediately. □
Proposition 4.
Let a 1 , a 2 R { 0 } . Then, for q 1 , the following inequality holds:
| 1 2 L | m a 1 + a 2 x | , | m a 1 + a 2 x + η ( a 2 , m a 1 + a 2 x ) | + 1 2 L | m a 1 | , | m a 1 + η ( m a 1 + a 2 x , m a 1 ) | 1 H m a 1 + η ( m a 1 + a 2 x , m a 1 ) , m a 1 + a 2 x | η ( a 2 , m a 1 + a 2 x ) 4 3 q 4 H | m a 1 + a 2 x | 2 q , 2 | a 2 | 2 q q + η ( m a 1 + a 2 x , m a 1 ) 4 3 q 4 H 2 | m a 1 | 2 q , | m a 1 + a 2 x | 2 q q .
Proof. 
Applying Corollary 10 for f ( t ) = 1 t , one can obtain the result immediately. □
Remark 2.
Applying Theorems 2 and 3 for the appropriate choices of function φ ( t ) = t α Γ ( α ) , t α k k Γ k ( α ) , φ ( t ) = t α exp 1 α α t for α ( 0 , 1 ) , such that | f | q is preinvex, we can deduce some new general fractional integral inequalities using the above special means. The details are left to the interested reader.
Remark 3.
Also, in Remark 2, if we choose η ( y , x ) = y x , x , y P , we can deduce some new general fractional integral inequalities for convex functions using above special means. The details are left to the interested reader.

4. Conclusions

On the basis of a new identity regarding trapezium-type integral inequalities, some new trapezium-type integral inequalities via generalized fractional integral operators are established. Some special cases are consider that are derived from the main results. Furthermore, some applications regarding special means of real numbers are given.

Author Contributions

All authors contributed to each part of this work equally, and they read and approved the final manuscript.

Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Aslani, S.M.; Delavar, M.R.; Vaezpour, S.M. Inequalities of Fejér type related to generalized convex functions with applications. Int. J. Anal. Appl. 2018, 16, 38–49. [Google Scholar]
  2. Chen, F.X.; Wu, S.H. Several complementary inequalities to inequalities of Hermite–Hadamard type for s-convex functions. J. Nonlinear Sci. Appl. 2016, 9, 705–716. [Google Scholar] [CrossRef]
  3. Chu, Y.-M.; Khan, M.A.; Khan, T.U.; Ali, T. Generalizations of Hermite–Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 2016, 9, 4305–4316. [Google Scholar] [CrossRef]
  4. Dahmani, Z. On Minkowski and Hermite–Hadamard integral inequalities via fractional integration. Ann. Funct. Anal. 2010, 1, 51–58. [Google Scholar] [CrossRef]
  5. Delavar, M.R.; Dragomir, S.S. On η-convexity. Math. Inequal. Appl. 2017, 20, 203–216. [Google Scholar] [CrossRef]
  6. Delavar, M.R.; de la Sen, M. Some generalizations of Hermite–Hadamard type inequalities. SpringerPlus 2016, 5, 1661. [Google Scholar] [CrossRef]
  7. Dragomir, S.S.; Agarwal, R.P. Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula. Appl. Math. Lett. 1998, 11, 91–95. [Google Scholar] [CrossRef]
  8. Du, T.S.; Liao, J.G.; Li, Y.J. Properties and integral inequalities of Hadamard–Simpson type for the generalized (s,m)-preinvex functions. J. Nonlinear Sci. Appl. 2016, 9, 3112–3126. [Google Scholar] [CrossRef]
  9. Farid, G.; Rehman, A.U. Generalizations of some integral inequalities for fractional integrals. Ann. Math. Silesianae 2017, 31, 1–14. [Google Scholar] [CrossRef]
  10. Hristov, J. Response functions in linear viscoelastic constitutive equations and related fractional operators. Math. Model. Nat. Phenom. 2019, 14, 1–34. [Google Scholar] [CrossRef]
  11. Katugampola, U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218, 860–865. [Google Scholar] [CrossRef]
  12. Kashuri, A.; Liko, R. Hermite–Hadamard type fractional integral inequalities for generalized (r;s,m,φ)-preinvex functions. Eur. J. Pure Appl. Math. 2017, 10, 495–505. [Google Scholar]
  13. Kashuri, A.; Liko, R. Hermite–Hadamard type inequalities for generalized (s,m,φ)-preinvex functions via k-fractional integrals. Tbil. Math. J. 2017, 10, 73–82. [Google Scholar] [CrossRef]
  14. Kashuri, A.; Liko, R. Hermite–Hadamard type fractional integral inequalities for MT(m,φ)-preinvex functions. Stud. Univ. Babeş-Bolyai Math. 2017, 62, 439–450. [Google Scholar] [CrossRef]
  15. Kashuri, A.; Liko, R.; Dragomir, S.S. Some new Gauss-Jacobi and Hermite–Hadamard type inequalities concerning (n+1)-differentiable generalized ((h1p,h2q);(η1,η2))-convex mappings. Tamkang J. Math. 2018, 49, 317–337. [Google Scholar] [CrossRef]
  16. Khan, M.A.; Chu, Y.-M.; Kashuri, A.; Liko, R. Hermite–Hadamard type fractional integral inequalities for MT(r;g,m,ϕ)-preinvex functions. J. Comput. Anal. Appl. 2019, 26, 1487–1503. [Google Scholar]
  17. Khan, M.A.; Chu, Y.-M.; Kashuri, A.; Liko, R.; Ali, G. New Hermite–Hadamard inequalities for conformable fractional integrals. J. Funct. Sp. 2018, 2018, 6928130. [Google Scholar]
  18. Khan, M.A.; Khurshid, Y.; Ali, T. Hermite–Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comenianae 2017, 79, 153–164. [Google Scholar]
  19. Liu, W.J. Some Simpson type inequalities for h-convex and (α,m)-convex functions. J. Comput. Anal. Appl. 2014, 16, 1005–1012. [Google Scholar]
  20. Liu, W.; Wen, W.; Park, J. Hermite–Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals. J. Nonlinear Sci. Appl. 2016, 9, 766–777. [Google Scholar] [CrossRef]
  21. Luo, C.; Du, T.S.; Khan, M.A.; Kashuri, A.; Shen, Y. Some k-fractional integrals inequalities through generalized λϕm-MT-preinvexity. J. Comput. Anal. Appl. 2019, 27, 690–705. [Google Scholar]
  22. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. [Google Scholar]
  23. Mubeen, S.; Habibullah, G.M. k-Fractional integrals and applications. Int. J. Contemp. Math. Sci. 2012, 7, 89–94. [Google Scholar]
  24. Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S. Hermite–Hadamard inequalities for s-Godunova–Levin preinvex functions. J. Adv. Math. Stud. 2014, 7, 12–19. [Google Scholar]
  25. Omotoyinbo, O.; Mogbodemu, A. Some new Hermite–Hadamard integral inequalities for convex functions. Int. J. Sci. Innov. Technol. 2014, 1, 1–12. [Google Scholar]
  26. Özdemir, M.E.; Dragomir, S.S.; Yildiz, C. The Hadamard’s inequality for convex function via fractional integrals. Acta Mathematica Scientia 2013, 33, 153–164. [Google Scholar] [CrossRef]
  27. Dos Santos, M.A.F. Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting. Physics 2019, 1, 5. [Google Scholar] [CrossRef]
  28. Sarikaya, M.Z.; Ertuğral, F. On the generalized Hermite–Hadamard inequalities. Available online: https://www.researchgate.net/publication/321760443 (accessed on 21 March 2019).
  29. Set, E.; Noor, M.A.; Awan, M.U.; Gözpinar, A. Generalized Hermite–Hadamard type inequalities involving fractional integral operators. J. Inequal. Appl. 2017, 1, 169. [Google Scholar] [CrossRef]
  30. Shi, H.N. Two Schur-convex functions related to Hadamard-type integral inequalities. Publ. Math. Debr. 2011, 78, 393–403. [Google Scholar] [CrossRef]
  31. Wang, H.; Du, T.S.; Zhang, Y. k-fractional integral trapezium-like inequalities through (h,m)-convex and (α,m)-convex mappings. J. Inequal. Appl. 2017, 1, 311. [Google Scholar] [CrossRef]
  32. Zhang, X.M.; Chu, Y.-M.; Zhang, X.H. The Hermite–Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, 1, 507560. [Google Scholar] [CrossRef]
  33. Zhang, Y.; Du, T.S.; Wang, H.; Shen, Y.J.; Kashuri, A. Extensions of different type parameterized inequalities for generalized (m,h)-preinvex mappings via k-fractional integrals. J. Inequal. Appl. 2018, 1, 49. [Google Scholar] [CrossRef] [PubMed]
  34. Weir, T.; Mond, B. Preinvex functions in multiple objective optimization. J. Math. Anal. Appl. 1998, 136, 29–38. [Google Scholar] [CrossRef]
  35. Sarikaya, M.Z.; Yildirim, H. On generalization of the Riesz potential. Indian J. Math. Math. Sci. 2007, 3, 231–235. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Fractal Fract EISSN 2504-3110 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top