Some New Classes of Preinvex Functions and Inequalities - MDPI

mathematics Article

Some New Classes of Preinvex Functions and Inequalities Muhammad Aslam Noor ∗ , Khalida Inayat Noor and Saima Rashid Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan; [email protected] (K.I.N.); [email protected] (S.R.) * Correspondence: [email protected] Received: 3 November 2018; Accepted: 24 December 2018; Published: 29 December 2018

Abstract: In this article, we introduce some new class of preinvex functions involving two arbitrary auxiliary functions. We derive some new integral inequalities for these classes of preinvex functions. We also discuss some special cases which can be deduced from our main results. Keywords: convex function; preinvex function; Hermite–Hadamard inequality; Holder’s inequality and power-mean inequality MSC: 26D15; 26D10; 90C23

1. Introduction Several branches of mathematical and engineering science have been developed by using the crucial and significant concepts of convex analysis. Inequalities present a very active and fascinating field of research. In recent years, a wide class of integral inequalities is being derived via different concepts of convexity. These integral inequalities are useful in Physics, where upper bounds for natural phenomena described by integrals such as mechanical work (virtual work) are required. Integral inequalities are closely related to the convex functions and their variant forms. Convexity theory is an effective and powerful technique for studying a wide class of problems which arise in various branches of pure and applied sciences. Several new classes of convex functions and convex sets have been introduced and investigated. Various new inequalities related to these new classes of convex functions have been derived by researchers—see, for example, Refs. [1–13] and the reference therein. However, it is amazing that convexity allows many diversified applications in every branch of pure and applied sciences. It is said that f : I = [ a, b] ⊆ R → R is a convex function, if and only if it satisfies the inequality f

a+b 2

≤

1 b−a

Z b a

f ( x )dx ≤

f ( a) + f (b) , 2

(1)

which is called the Hermite–Hadamard inequality for convex function (see [12–16]). For the novel applications of the Hermite-Hadamard Inequality (1) (see [7]). Hanson [8] introduced and investigated another class of generalized convex functions, which is called invex functions. Ben-Israel and Mond [17] introduced the concepts of invex sets and preinvex functions. They have shown that the differentiable preininvex functions are invex functions, but the converse may not be true. These preinvex functions are not convex functions, but they enjoy some nice properties, which convex functions have. It is known that the invex functions and preinvex functions are equivalent under some suitable conditions,

Mathematics 2019, 7, 29; doi:10.3390/math7010029

www.mdpi.com/journal/mathematics

Mathematics 2019, 7, 29

2 of 16

see [10]. Noor [11,13] has shown that a function f is a preinvex function on Ω = [ a, a + η (b, a)], if and only if, it satisfies the inequality f

2a + η (b, a) 2

1 ≤ η (b, a)

a+Zη (b,a)

f ( x )dx ≤

a

f ( a) + f (b) , 2

(2)

which is called Hermite-Hadamard-Noor type inequality. Several integral inequalities for various type of preinvex functions have been obtained in recent years. For more details, see [11–15,18–31] and the references therein. In this paper, we introduce a new class of preinvex functions with respect to two nonnegative arbitrary functions h1 and h2 , which is called (h1 , h2 )-preinvex function. We establish some new Hermite-Hadamarad inequality for (h1 , h2 )-preinvex function. Some special cases are also discussed which can be obtained from our results. 2. Preliminaries We now define some new classes of preinvex functions involving two arbitrary functions. Let Ω ⊂ R be a set and η (·, ·) : Ω × Ω −→ R be a continuous bifunction. First of all, we recall the following well known concepts and results. Definition 1 ([9,17,19]). A set Ω ⊂ R is said to be invex set with respect to the bifunction η (·, ·), if and only if x + tη (y, x ) ∈ Ω,

∀ x, y ∈ Ω, t ∈ [0, 1].

The invex set Ω is also called η-connected set. Note that, if η (b, a) = b − a, this means that every convex set is an invex set, but the converse is not true (see [9]). From now onward, the set Ω is an invex set, unless otherwise it is specified. We now consider a new class of preinvex function with respect to two arbitrary functions h1 and h2 . Definition 2. Let h1 , h2 : (0, 1) ⊆ J −→ R be two nonnegative functions and Ω be an invex set. A function f : Ω −→ R is said to be a (h1 , h2 )- preinvex function, if f ( x + tη (y, x )) ≤ h1 (1 − t)h2 (t) f ( x ) + h1 (t)h2 (1 − t) f (y), ∀ x, y ∈ Ω, t ∈ [0, 1].

(3)

Note that for t = 12 , we have Jensen type (h1 , h2 )-preinvex function, that is, f

2x + η (y, x ) 2

≤ h1

1 1 h2 [ f ( x ) + f (y)] . 2 2

(4)

We now discuss several special cases. (I). If h1 (t) = ts and h2 (t) = ts in Definition 2, then we have a new class of s-preinvex functions. Definition 3. Let s ∈ [0, 1] be a real number and Ω be an invex set. We say that f : Ω −→ R is a s-preinvex function, if f ( x + tη (y, x )) ≤ ts (1 − t)s [ f ( x ) + f (y)],

∀ x, y ∈ Ω, t ∈ [0, 1].

Definition 4. Let s1 , s2 ∈ [0, 1] be two real numbers and Ω be an invex set. We say that f : Ω −→ R is an (s1 , s2 )-preinvex functions, if f ( x + tη (y, x )) ≤ ts1 (1 − t)s2 f ( x ) + (1 − t)s1 ts2 f (y),

∀ x, y ∈ Ω, t ∈ [0, 1].


3 of 16

(II). If h1 (t) = t−s and h2 (t) = t−s in Definition 2, then we have a new class of preinvex functions, which is called Godunova–Levin (s1 , s2 )-preinvex functions. Definition 5. Let s1 , s2 ∈ [0, 1] be two real numbers and Ω be an invex set. We say that f : Ω −→ R is a (s1 , s2 )-preinvex functions, if f ( x + tη (y, x )) ≤

1 1 f (x) + f ( y ), t s1 (1 − t ) s2 (1 − t ) s1 t s2

∀ x, y ∈ Ω, t ∈ [0, 1].

For appropriate and suitable choice of functions h1 , h2 and the bifunction, one can obtain several new and known classes of preinvex functions and convex functions as special cases. See, for example, [1,3,4,11,21,22,24,27–31]. This shows that the concept of (h1 , h2 )-preinvex function is quite a general and unifying one. We need the following result. Lemma 1. Let f : Ω = [ a, a + η (b, a)] ⊂ R −→ R be a continuous function. Then, for some fixed α, β > 0, a+Zη (b,a)

(u − a) ( a + η (b, a) − u) f (u)du = (η (b, a)) α

β

α + β +1

a

Z1

tα (1 − t) β f ( a + tη (b, a))dt.

0

Proof. Using the change of variables, u = a + tη (b, a), we have a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a

=

Z1

η (b, a)(tη (b, a))α ((1 − t)η (b, a)) β f ( a + tη (b, a))dt

0

= (η (b, a))

α + β +1

Z1

tα (1 − t) β f ( a + tη (b, a))dt,

0

which is the required result. We recall the special functions which are known as Gamma function and Beta function, respectively: Γ( x )

=

Z∞

e−t t x−1 dt,

0

B( x, y) =

Z1 0

t x−1 (1 − t)y−1 dt =

Γ( x )Γ(y) , Γ( x + y)

x, y > 0.

In addition, we recall the well known fact about the bifunction η (., .) : η ( a + t2 η (b, a), a + t1 η (b, a)) = (t2 − t1 )η (b, a),

∀ a, b ∈ Ω, t1 , t2 ∈ [0, 1].

(5)

3. Main Results In this section, we establish several new Hermite–Hadamard type inequalities for (h1 , h2 )-preinvex functions.


4 of 16

Theorem 1. Let f : Ω = [ a, a + η (b, a)] ⊂ R −→ R be (h1 , h2 )-preinvex function with η (b, a) > 0 and h1 21 6= 0, h2 12 6= 0. If f ∈ L[ a, a + η (b, a)] and Equation (5) holds, then we have 1 f 1 2h1 ( 2 )h2 ( 12 )

2a + η (b, a) 2

a+Zη (b,a)

1 ≤ η (b, a)

≤ f ( a)

Z1

f ( x )dx

a

h1 (t)h2 (1 − t)dt + f (b)

0

Z1

h2 (t)h1 (1 − t)dt.

0

Proof. Let f be an (h1 , h2 )-preinvex function. Then, from inequality (2), we have f

2x + η (y, x ) 2

≤ h1

1 2

h2

1 2

[ f ( x ) + f (y)], ∀ x, y ∈ Ω.

(6)

Substituting x = a + (1 − t)η (b, a) and y = a + tη (b, a) in (6) and using (5), we have

2x + η (y, x ) 2 2 a + (1 − t)η (b, a) + η a + tη (b, a), a + (1 − t)η (b, a) = f 2 2 a + (1 − t)η (b, a) + (2t − 1)η (b, a) ≤ f . 2

f

From inequality (6), we have f

2a + η (b, a) 2

1 1 ≤ h1 h2 [ f ( x ) + f (y)] 2 2 1 1 = h1 h2 [ f ( a + (1 − t)η (b, a)) + f ( a + tη (b, a))] . 2 2

Integrating the above inequality with respect to t over [0,1], we have f

2a + η (b, a) 2

1 1 h ≤ h1 ( ) h2 ( ) 2 2

Z1

f ( a + (1 − t)η (b, a))dt +

f

2a + η (b, a) 2

1 1 1 = h1 h2 2 2 η (b, a) 1 1 1 = 2h1 h2 2 2 η (b, a)

i f ( a + tη (b, a))dt .

0

0

Using the change of variable technique, x = a + tη (b, a),

Z1

a+Zη (b,a)

w = a + (1 − t)η (b, a), we have

f ( x )dx −

a a+Zη (b,a) a

Za a+η (b,a)

f ( x )dx.

f (w)dw


5 of 16

Thus, 2a + η (b, a) 1 1 f ≤ 2 η (b, a) 2h1 12 h2 12

≤

Z1

a+Zη (b,a)

f ( x )dx.

a

(h1 (t)h2 (1 − t) f ( a) + h1 (1 − t)h2 (t) f (b)) dt

0

= f ( a)

Z1

h1 (t)h2 (1 − t)dt + f (b)

Z1

h1 (1 − t)h2 (t)dt,

0

0

which is the required result. We now discuss the new special cases of Theorem 1. (III). If h1 (t) = ts1 and h2 (t) = ts2 , then under the assumptions of Theorem 1, then we have a new result for Breckner type of (s1 , s2 )-preinvex functions: 1 21 − s 1 − s 2

f

2a + η (b, a) 2

≤

1 η (b, a)

Z a+η (b,a) a

f ( x )dx

≤[ f ( a) + f (b)]B(s1 + 1, s2 + 1), where B( x, y) denotes the beta as a special function. (IV). If h(t1 ) = t−s1 and h(t2 ) = t−s2 , under the assumptions of Theorem 1, then we have a new result for Godunova–Levin type of (s1 , s2 )-preinvex function: 1 21+s1 +s2

f

2a + η (b, a) 2

1 ≤ η (b, a)

a+Zη (b,a)

f ( x )dx

a

≤[ f ( a) + f (b)]B(1 − s1 , 1 − s2 ). Theorem 2. Let f : Ω = [ a, a + η (b, a)] ⊂ R −→ R be a continuous function. If f is a (h1 , h2 )-preinvex function, then, for some fixed α, β > 0, a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du ≤ (η (b, a))α+ β+1 [Ψ1 (t) f ( a) + Ψ2 (t) f (b)],

a

where Ψ1 ( t ) : =

Z1

tα (1 − t) β h1 (1 − t)h2 (t)dt,

0

Ψ2 ( t ) : =

Z1 0

tα (1 − t) β h1 (t)h2 (1 − t)dt.


6 of 16

Proof. Using Lemma 1 and the fact that f is a (h1 , h2 )-preinvex function, we have a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a

= (η (b, a))α+ β+1

Z1

tα (1 − t) β f ( a + tη (b, a))dt

0

≤ (η (b, a))α+ β+1

Z1

tα (1 − t) β [h1 (1 − t)h2 (t) f ( a) + h2 (1 − t)h1 (t) f (b)] dt

0

= (η (b, a))

α + β +1

[Ψ1 (t) f ( a) + Ψ2 (t) f (b)],

which completes the proof. The next results are special cases of Theorem 2. (V). If h1 (t) = ts1 and h2 (t) = ts2 under the assumption of Theorem 2, then we have a new result for Breckner type of (s1 , s2 )-preinvex functions, we have a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du ≤ (η (b, a))α+ β+1 [γ1 (t) f ( a) + γ2 (t) f (b)],

a

where γ1 (t) := B(α + s2 + 1, β + s1 + 1), γ2 (t) := B(α + s1 + 1, β + s2 + 1). (VI). If h(t1 ) = t−s1 and h(t2 ) = t−s2 , under the assumption of Theorem 2, then we have a new result for Godunova–Levin type of (s1 , s2 )-preinvex functions, we have a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du ≤ (η (b, a))α+ β+1 [δ1 (t) f ( a) + δ2 (t) f (b)],

a

where δ1 (t) := B(α − s2 + 1, β − s1 + 1), δ2 (t) := B(α − s1 + 1, β − s2 + 1). r

Theorem 3. Let f : Ω = [ a, a + η (b, a)] ⊂ R −→ R be a continuous function. If | f | r−1 is a (h1 , h2 )-preinvex function, then for some fixed α, β > 0, a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a

h i r −1 r r r ≤ (η (b, a))α+ β+1 (B(rα + 1, rβ + 1) | f ( a)| r−1 θ1 (t) + | f (b)| r−1 θ2 (t) , where θ1 ( t ) =

Z1 0

h1 (1 − t)h2 (t)dt,

θ2 ( t ) =

Z1 0

h1 (t)h2 (1 − t)dt.


7 of 16

r

Proof. Using Lemma 1, Holder’s inequality and the fact that | f | r−1 is a (h1 , h2 )-preinvex function, we have a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a

= (η (b, a))

α + β +1

Z1

tα (1 − t) β f ( a + tη (b, a))dt

0



≤ (η (b, a))α+ β+1 

Z1

 1r  trα (1 − t)rβ dt 

Z1

0

 r−r 1

| f ( a + tη (b, a))|

r r −1

dr 

0 1

≤ (η (b, a))α+ β+1 (B(rα + 1, rβ + 1)) r

Z1

r

[h1 (1 − t)h2 (t)| f ( a)| r−1

0

r−r 1

r

+ h1 (t)h2 (1 − t)| f (b)| r−1 ]dt Z1 r 1 h1 (1 − t)h2 (t)dt ≤ (η (b, a))α+ β+1 (B(rα + 1, rβ + 1)) r | f ( a)| r−1 0 r

+ | f (b)| r−1

Z1

r−r 1 h1 (t)h2 (1 − t)dt

0

h i r −1 r r 1 r = (η (b, a))α+ β+1 (B(rα + 1, rβ + 1)) r | f ( a)| r−1 θ1 (t) + | f (b)| r−1 θ2 (t) , which completes the proof. The following results are some special cases of Theorem 3. (VII). If h1 (t) = ts1 and h2 (t) = ts2 in Theorem 3, then we have a Breckner type of (s1 , s2 )-preinvex function: a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a 1

≤ η (b, a)α+ β+1 B(rα + 1, rβ + 1) r [B(s1 + 1, s2 + 1)]

r −1 r

h

r

r

| f ( a)| r−1 + | f (b)| r−1

i r −1 r

.

(VIII). If h(t1 ) = t−s1 and h(t2 ) = t−s2 in Theorem 3, then we have Godunova–Levin type (s1 , s2 )-preinvex function: a+Zη (b,a)

(u − a)α ( a + η (b, a) − u) β f (u)du

a 1

≤ η (b, a)α+ β+1 B(rα + 1, rβ + 1) r [B(1 − s1 , 1 − s2 )]

r −1 r

h

r

r

| f ( a)| r−1 + | f (b)| r−1

i r −1 r

.

We now derive some some Hermite-Hadamard type inequalities for differentiable (h1 , h2 )-preinvex function. For this, we need the following result which can be proved using integration by parts. For the sake of completeness, we include its proof.


8 of 16

Lemma 2. Let f : Ω = [ a, a + η (b, a)] ⊂ R → R be a differentiable preinvex function on the interior Ω◦ of 0 Ω with η (b, a) > 0. If f ∈ L[ a, a + η (b, a)] is (h1 , h2 )-preinvex function and λ ∈ [0, 1], then

(1 − λ ) f

2a + η (b, a) 2

Z1

η (b, a) = 2

f ( a) + f ( a + η (b, a)) 1 +λ − 2 η (b, a)

a+Zη (b,a)

f ( x )dx

a

0

µ(t) f ( a + tη (b, a))dt,

0

where µ(t) =

   

t ∈ [0, 21 ),

2t − λ,

(7)

t ∈ [ 12 , 1].

   2t − 2 + λ,

Proof. Consider η (b, a) I= 2

Z1

0

µ(t) f ( a + tη (b, a))dt

0 1

η (b, a) = 2

Z2

0

(2t − λ) f ( a + tη (b, a))dt +

0

Z1

0

(2t − 2 + λ) f ( a + tη (b, a))dt

1 2

= I1 + I2 . Now, 1

η (b, a) I1 = 2

Z2

0

(2t − λ) f ( a + tη (b, a))dt

0

1 Z12 2 1 = (2t − λ) f ( a + tη (b, a)) − f ( a + tη (b, a))dt 2 0 0

1

1−λ = f 2

2a + η (b, a) 2

λ + f ( a) − 2

Z2

f ( a + tη (b, a)).dt.

0

Similarly, Z 1 0 η (b, a) I2 = (2t − 2 + λ)( f ( a + tη (b, a))dt 1 2 2 1 Z1 1 = (2t − 2 + λ)( f ( a + tη (b, a)) − f ( a + tη (b, a))dt 2 1 2

1−λ = f 2

2a + η (b, a) 2

1 2

λ + f ( a + η (b, a) − 2

Z1 1 2

f ( a + tη (b, a)).


9 of 16

Thus I1 + I2

= (1 − λ ) f

2a + η (b, a) 2

+λ

f ( a) + f ( a + η (b, a) 2

1 − η (b, a)

a+Zη (b,a)

f ( x )dx

a

the required result. Theorem 4. Let f : Ω = [ a, a + η (b, a)] ⊂ R → R be a differentiable preinvex function on the interior Ω◦ 0 0 of Ω with η (b, a) > 0. If f ∈ L[ a, a + η (b, a)] and | f |q is a (h1 , h2 )-preinvex function on Ω for q ≥ 1 and λ ∈ [0, 1], then

1 2a + η (b, a) f ( a) + f ( a + η (b, a) − (1 − λ ) f +λ 2 2 η (b, a) " h 0 η (b, a) 1− 1 = (ζ 1 ( a, b; λ)) q ζ 2 ( a, b; λ, h1 , h2 )| f ( a)|q 2 0

i 1q

0

iq

+ ζ 3 ( a, b; λ, h1 , h2 )| f (b)|q + ζ 6 ( a, b; λ, h1 , h2 )| f (b)|q

+ (ζ 4 ( a, b; λ)) # 1

1− 1q

h

a+Zη (b,a)

f ( x )dx

a

0

ζ 5 ( a, b; λ, h1 , h2 )| f ( a)|q

,

where µ(t) is defined by (7) and 1

ζ 1 ( a, b; λ)

Z2

=

|µ(t)|dt,

(8)

0 1

ζ 2 ( a, b; λ, h1 , h2 )

=

Z2

h1 (1 − t)h2 (t)|µ(t)|dt,

(9)

h1 (t)h2 (1 − t)|µ(t)|dt,

(10)

0 1

ζ 3 ( a, b; λ, h1 , h2 )

=

Z2 0

ζ 4 ( a, b; λ, h)

=

Z1

|µ(t)|dt,

(11)

1 2

ζ 5 ( a, b; λ, h1 , h2 )

=

Z1

h1 (1 − t)h2 (t)|µ(t)|dt,

(12)

h1 (t)h2 (1 − t)|µ(t)|dt.

(13)

1 2

ζ 6 ( a, b; λ, h1 , h2 )

=

Z1 1 2


10 of 16

Proof. Using Lemma 2 and the power mean inequality, we have

(1 − λ ) f

η (b, a) ≤ 2

2a + η (b, a) 2

" Z21

f ( a) + f ( a + η (b, a) 1 +λ − 2 η (b, a)

0

|2t − λ|| f ( a + tη (b, a))|dt +

0

+

Z1

"

Z2

a

Z2

|2t − λ|dt

0

|2t − 2 + λ|| f ( a + tη (b, a))|dt ! 1q

0

q

|2t − λ||( f ( a + tη (b, a))|) dt

0

Z1

|2t − 2 + λ|dt

1 2

! 1q #

0

|2t − 2 + λ||( f ( a + tη (b, a))|)dt

1 2 1

"

Z2

!1− 1q

|2t − λ|dt

0

1

Z2

h 0 |2t − λ| h1 (1 − t)h2 (t)| f ( a)|q

0

0

q

i

+ h2 (1 − t)h1 (t)| f (b)| dt

! 1q

  +

Z1

 1− 1 q

 |2t − 2 + λ|dt

1 2

h

#

0

1

!1− 1q

!1− 1q

η (b, a) ≤ 2

f ( x )dx

1 2 1

η (b, a) ≤ 2

Z1

a+Zη (b,a)

0

q

Z1

|2t − 2 + λ|

1 2

0

q

i

1 q

! #

h1 (1 − t)h2 (t)| f ( a)| + h2 (1 − t)h1 (t)| f (b)| dt

" i 1q h 0 0 η (b, a) 1− 1 = (ζ 1 ( a, b; λ)) q ζ 2 ( a, b; λ, h1 , h2 )| f ( a)|q + ζ 3 ( a, b; λ, h1 , h2 )| f (b)|q 2 # i 1q h 0 0 1− 1q + (ζ 4 ( a, b; λ)) ζ 5 ( a, b; λ, h1 , h2 )| f ( a)|q + ζ 6 ( a, b; λ, h1 , h2 )| f (b)|q the required result. For appropriate and suitable choice of q and λ, we obtain several new results for the example midpoint, Trapezoidal, three point Trapezoidal rule and Simpson’s rule.

Corollary 1. If q = 1, then, under the assumption of Theorem 4, we have a+Zη (b,a) 2a + η (b, a) f ( a) + f ( a + η (b, a) 1 +λ − f ( x )dx (1 − λ ) f 2 2 η (b, a) a " i 0 0 η (b, a) h ζ 2 ( a, b; λ, h1 , h2 )| f ( a)| + ζ 3 ( a, b; λ, h1 , h2 )| f (b)| ≤ 2 # h i 0 0 + ζ 5 ( a, b; λ, h1 , h2 )| f ( a)| + ζ 6 ( a, b; λ, h1 , h2 )| f (b)| .


11 of 16

Corollary 2. If λ = 0, then under the assumption of Theorem 4, we have two points midpoint, which is a+Zη (b,a) 2a + η (b, a) 1 − f ( x )dx f 2 η (b, a) a " i 0 0 η (b, a) h ≤ ζ 2 ( a, b; 0, h1 , h2 )| f ( a)|q + ζ 3 ( a, b; 0, h1 , h2 )| f (b)|q 2 0

h

0

+ ζ 5 ( a, b; 0, h1 , h2 )| f ( a)|q + ζ 6 ( a, b; 0, h1 , h2 )| f (b)|q

i

# 1q ,

where ζ 2 ( a, b; 0, h1 , h2 ), ζ 3 ( a, b; 0, h1 , h2 ), ζ 5 ( a, b; 0, h1 , h2 ) and ζ 6 ( a, b; 0, h1 , h2 ) are given by (9), (10), (12) and (13), respectively. Corollary 3. If λ = 1, then, under the assumption of Theorem 4, we have Trapezoidal rule, which is f ( a) + f ( a + η (b, a)) 1 − 2 η (b, a)

a+Zη (b,a) a

f ( x )dx

" h i 1q 0 0 η (b, a) 1− 1 ≤ (ζ 1 ( a, b; 1)) q ζ 2 ( a, b; 1, h1 , h2 )| f ( a)|q + ζ 3 ( a, b; 1, h1 , h2 )| f (b)|q 2

+ (ζ 4 ( a, b; 1))

1− 1q

h

0

0

ζ 5 ( a, b; 1, h1 , h2 )| f ( a)|q + ζ 6 ( a, b; 1, h1 , h2 )| f (b)|q

i

# 1q ,

where ζ 1 ( a, b; 1), ζ 2 ( a, b; 1, h1 , h2 ), ζ 3 ( a, b; 1, h1 , h2 ), ζ 4 ( a, b; 1, h), ζ 5 ( a, b; 1, h1 , h2 ) and ζ 6 ( a, b; 1, h1 , h2 ) are given by (8)–(13), respectively. Corollary 4. If λ = 12 , then, under the assumption of Theorem 4, we have the three points Trapezoidal rule a+Zη (b,a) 1 2a + η (b, a) 1 f ( x )dx f ( a) + 2 f + f ( a + η (b, a)) − 4 2 η (b, a) a " 1 i 1q 0 0 1 1− q h 1 1 ≤ η (b, a) ζ 1 ( a, b; ) ζ 2 ( a, b; , h1 , h2 )| f ( a)|q + ζ 3 ( a, b; , h1 , h2 )| f (b)|q 2 2 2 # 1 i1 0 0 1 1− q h 1 1 q q q + ζ 4 ( a, b; ) ζ 5 ( a, b; , h1 , h2 )| f ( a)| + ζ 6 ( a, b; , h1 , h2 )| f (b)| , 2 2 2 where ζ 1 ( a, b; 12 ), ζ 2 ( a, b; 12 , h1 , h2 ), ζ 3 ( a, b; 12 , h1 , h2 ), ζ 4 ( a, b; 12 , h), ζ 5 ( a, b; 12 , h1 , h2 ) and ζ 6 ( a, b; 12 , h1 , h2 ) are given by (8)–(13), respectively.


12 of 16

Corollary 5. If λ = 31 , then, under the assumption of Theorem 4, we have Simpson’s rule a+Zη (b,a) 1 2a + η (b, a) 1 f ( a) + 4 f + f ( a + η (b, a)) − f ( x )dx 6 2 η (b, a) a " 1 i 1q 0 0 η (b, a) 1 1 1− q h 1 ≤ ζ 2 ( a, b; , h1 , h2 )| f ( a)|q + ζ 3 ( a, b; , h1 , h2 )| f (b)|q ζ 1 ( a, b; ) 2 3 3 3 # 1− 1 h i1 q 0 0 1 1 1 q q q ζ 5 ( a, b; , h1 , h2 )| f ( a)| + ζ 6 ( a, b; , h1 , h2 )| f (b)| + ζ 4 ( a, b; ) , 3 3 3 where ζ 1 ( a, b; 13 ), ζ 2 ( a, b; 13 , h1 , h2 ), ζ 3 ( a, b; 13 , h1 , h2 ), ζ 4 ( a, b; 13 , h), ζ 5 ( a, b; 13 , h1 , h2 ) and ζ 6 ( a, b; 13 , h1 , h2 ) are given by (8)–(13), respectively. Theorem 5. Let f : Ω = [ a, a + η (b, a)] ⊂ R → R be a differentiable preinvex function on the interior 0 0 Ω◦ of Ω with η (b, a) > 0. If f ∈ L[ a, a + η (b, a)] and | f |q is a (h1 , h2 )-preinvex function on Ω for p, q ≥ 1, 1p + 1q = 1 and λ ∈ [0, 1], then

2a + η (b, a) 2

a+Zη (b,a)

f ( a) + f ( a + η (b, a) 1 (1 − λ ) f +λ − f ( x )dx 2 η (b, a) a q 0 0 2a + η ( b,a ) ! 1q " | f ( a)|q + f Z1 2 1 η (b, a) p h1 (1 − t)h2 (t)dt (ζ 7 ( a, b, p; λ)) ≤ 2 2 0 0 ! 1q # 0 2a+η (b,a) q q f + | f ( a + η (b, a))| Z1 1 2 , h2 (1 − t)h1 (t)dt + (ζ 8 ( a, b, p; λ)) p 2 0

where 1

ζ 7 ( a, b, p; λ) =

Z2 0

p

|µ(t)| dt,

ζ 8 ( a, b, p; λ) =

Z1 1 2

|µ(t)| p dt.


13 of 16

Proof. Using Lemma 2 and the Holder’s integral inequality, we have

(1 − λ ) f

≤

η (b, a) 2

+

Z1

2a + η (b, a) 2

"

1 Z2

f ( a) + f ( a + η (b, a) 1 +λ − 2 η (b, a)

a+Zη (b,a)

f ( x )dx

a

!

0

|2t − λ||( f ( a + tη (b, a))|dt)

0

!#

0

|2t − 2 + λ||( f ( a + tη (b, a))|dt)

1 2 1

η (b, a) ≤ 2

+

Z1

"

! 1p

Z2

! 1q

0

p

|2t − λ| dt

q

| f ( a + tη (b, a))| dt

0

!p

|2t − 2 + λ|

dt

i 1p

! 1q #

0

|( f ( a + tη (b, a))|q dt

1 2 1

η (b, a) ≤ 2

+

Z1

"

! 1p "

Z2

p

|2t − λ| dt

0

! 1p "

|2t − 2 + λ| p dt

1 2

2a+η (b,a) Z2

1 η (b, a)

1 η (b, a)

!# 1q

0

q

| f ( x )| dx

a a+Zη (b,a)

!# 1q #

0

| f ( x )|q dx

.

2a+η (b,a) 2

0

Using the definition of (h1 , h2 )-preinvex function of | f |q , we obtain the inequality (3): 2a+η (b,a) Z2

2 η (b, a)

"

0

0

q

| f ( x )| dx ≤ | f ( a)|

a

q

Z1

h1 (1 − t)h2 (t)dt

0

# 0 2a + η (b, a) q Z1 h2 (1 − t)h1 (t)dt + f 2

(14)

0

and 2 η (b, a)

a+Zη (b,a) 2a+η (b,a) 2

0 2a + η (b, a) q Z1 | f ( x )| dx ≤ f h1 (1 − t)h2 (t)dt 2 0

"

q

0

0

+ | f ( a + η (b, a))|q

Z1

(15) #

h2 (1 − t)h1 (t)dt

0

0 0 q | f ( a)|q + f 2a+η2(b,a) Z1

! 1q 1 η (b, a) (ζ 7 ( a, b, p; λ)) p h1 (1 − t)h2 (t)dt = 2 2 0 0 ! 1q # 0 2a+η (b,a) q f + | f ( a + η (b, a))|q Z1 1 2 + (ζ 8 ( a, b, p; λ)) p h2 (1 − t)h1 (t)dt . 2 "

0

(16)


14 of 16

A combination of (14)–(15) gives the inequality (16). For appropriate and suitable choice of λ, we obtain several new results for two points midpoint, Trapezoidal rule, three point Trapezoidal rule and Simpson’s rule. Theorem 6. Let f : Ω = [ a, a + η (b, a)] ⊂ R → R be a differentiable preinvex function on the interior 0 0 Ω◦ of Ω with η (b, a) > 0. If f ∈ L[ a, a + η (b, a)] and | f |q is a (h1 , h2 )-preinvex function on Ω for p, q ≥ 1, 1p + 1q = 1 and λ ∈ [0, 1], then 2a + η (b, a) f ( a) + f ( a + η (b, a) 1 +λ − (1 − λ ) f 2 2 η (b, a)

a

f ( x )dx

0 0 η (b, a) λ p+1 + (1 − λ) p+1 1p × ζ 9 ( a, b; q, h1 , h2 ) | f ( a)|q + | f (b)|q 2 2( p + 1) 1q 0 0 q q , + ζ 10 ( a, b; q, h1 , h2 ) | f ( a)| + | f (b)|

≤

a+Zη (b,a)

1q

where 1

ζ 9 ( a, b; λ, h1 , h2 ) =

Z2

h1 (t)h2 (1 − t)dt,

ζ 10 ( a, b; λ, h1 , h2 ) =

0

Z1

h1 (t)h2 (1 − t)dt.

1 2

Proof. Using Lemma 2 and the Holder’s integral inequality, we have

(1 − λ ) f " η (b,a) 2

≤

2a+η (b,a) 2

+ λ f (a)+ f (2a+η (b,a) −

1

R2

0

|2t − λ|| f ( a + tη (b, a)|)dt +

η (b,a) 2

≤

+

R1

! 1p

1

R2

|2t − λ| p dt

≤

+

R1

R1

|2t − 2 + λ| p dt

! 1q

0

| f ( a + tη (b, a))|q dt ! 1q #

0

| f ( a + tη (b, a))|q dt

1 2

! 1p

! 1q

1

R2

|2t − λ| p dt

0

[h1 (1 − t)h2 (t)| f

0

( a)|q

+ h2 (1 − t)h1 (t)| f

0

(b)|q ]dt

0

! 1p

|2t − 2 + λ| p dt

1 2

≤

|2t − 2 + λ|| f ( a + tη (b, a))|dt

0

! 1p

1

R2

0

1

R2

0

1 2

"

f ( x )dx

a

1 2

"

η (b,a) 2

a+ηR(b,a)

#

R1

0

1 η (b,a)

R1

0

0

[h1 (1 − t)h2 (t)| f ( a)|q + h2 (1 − t)h1 (t)| f (b)|q ]dt

1 2

η (b,a) 2

×

λ p+1 +(1−λ) p+1 2( p +1)

! 1p

h

1q 0 0 ζ 9 ( a, b; q, h1 , h2 ) | f ( a)|q + | f (b)|q

+ ζ 10 ( a, b; q, h1 , h2 ) | f the required result.

0

( a)|q

+ |f

0

(b)|q

1q i

,

! 1q #


15 of 16

For appropriate and suitable choice of λ, we obtain several new results for two points midpoint, Trapezoidal rule, three points Trapezoidal rule and Simpson’s rule. 4. Conclusions In this paper, we have established several Hermite–Hadamard type inequalities for (h1 , h2 )-preinvex functions. These results can be viewed as refinement and significant improvements of the previously known and new classes of preinvex functions. The ideas and techniques of this paper may be extended for other classes of convex functions and their variant forms. Author Contributions: Authors contributions are as: conceptualization, M.A.N. and K.I.N.; methodology, S.R.; validation, M.A.N., K.I.N. and S.R.; resources, K.I.N. writing—original draft preparation, M.A.N. and S.R.; writing—review and editing, M.A.N.; supervision, M.A.N.; project administration, K.I.N.; funding acquisition, S.R. Funding: This research is supported by the Higher Education Commission(HEC) of Pakistan Project Grant No.: 20-1966/R&D/11-2553. Acknowledgments: The authors would like to thank the Rector, COMSATS University Islamabad, Pakistan for providing excellent research and academic environments. The authors are grateful to the referees for their very constructive and valuable comments. Conflicts of Interest: All the authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Breckner, W.W. Stetigkeitsaussagen fiir eine Klasse verallgemeinerter convexer funktionen in topologischen linearen Raumen. Publ. Inst. Math. 1978, 23, 13-20. Niculescu, C.P.; Persson, L.E. Convex Functions and Their Applications; Springer: New York, NY, USA, 2018. Sarikaya, M.Z.; Saglam, A.; Yildrim, H. On some Hadamard- type inequalities for h-convex functions. J. Math. Inequal. 2008, 2, 335–341. [CrossRef] Barani, A.; Dragomir, S.S.; Ghazanfari, A. On Hermite–Hadamard inequality for function whose derivatives absolute values are preinvex. J. Inequal. Appl. 2012, 2012, 247. [CrossRef] Hadamard, J. Etude sur les proprietes des fonctions entieres et en particulier dune fonction consideree par Riemann. J. Math. Pure Appl. 1983, 58, 171–215. Hermite, C. Sur deux limites d’une integrale definie. Mathesis 1883, 3, 82. Kattri, S. Three proofs of the inequality e < (1 + n1 )n+0.5 . Am. Math. Mon. 2010, 117, 273–277. Hanson, M.A. On sufficiency of the Kun-Tucker conditions. J. Math. Anal. Appl. 1981, 80, 545–550. [CrossRef] Ben-Israel, A.; Mond, B. What is invexity? J. Aust. Math. Soc. 1984, 28, 1–9. [CrossRef] Noor, M.A. Numerical and Convex Analysis; Lecture Notes; COMSATS Insitute of Information Technology: Islamabad, Pakistan, 2018. Noor, M.A. Hermite–Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2007, 2, 126–131. Noor, M.A.; Noor, K.I.; Iftikhar, S. On integral inequalities for differentiable relative harmonic preinvex functions (survey). TWMS J. Pure Appl. Math. 2016, 7, 3–19. Noor, M.A. Hadamard integral inequalities for product of two preinvex functions. Nonlinear Anal. Forum 2009, 14, 167–173. Pitea, A.; Postolache, M. Duality theorems for a new class of multitime multiobjective variational problems. J. Glob. Optim. 2012, 54, 47–58. [CrossRef] Pitea, A.; Postolache, M. Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions. Optim. Lett. 2012, 6, 459–470. [CrossRef] Dragomir, S.S.; Peccaric, J.; Persson, L.E. Some inequalities of Hadamard type. Soochow J. Inequal. Math. 1995, 21, 335–341. Weir, T.; Mond, B. Preinvex functions in multiobjective optimization. J. Math. Anal. Appl. 1988, 136, 29–38. [CrossRef] Pitea, A.; Postolache, M. Minimization of vectors of curvilinear functionals on the second order jet bundle. Sufficient efficiency conditions. Optim. Lett. 2012, 6, 1657–1669. [CrossRef]


19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

16 of 16

Mohan, S.R.; Neogy, S.K. On invex set and preinvex functions. J. Math. Anal. Appl. 1995, 189, 901–908. [CrossRef] Agarwal, R.P.; Cernone, P.; Dragomir, S.S. On simpson inequality and applications. J. Inequal. Appl. 2000, 5, 533–579. Godunova, E.K.; Levin, V.I. Neravenstva dlja funkcii sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkii, Vycislitel. Mat. i. Fiz. M ezvuzov. Sb. Nauc. Trudov MGPI Moskva 1985,9, 138–148. Hwang, D.Y. Some inequalities for n-times differentiable mappings and applications. Kyungpook Math. J. 2003, 43, 335–343. Iscan, I. On Hermite–Hadamard and Simpson like inequalities for differentiable harmonically convex functions. J. Math. 2014, 10, 346305. Latif, M.A.; Dragomir, S.S. Some Hermite–Hadamard type inequalities for fuctions whose partial derivatives in absolut value are preinvex on the coordinates. Facta Univ. Math. Inform. 2013, 28, 257–270. Noor, M.A. On variational-like inequalities. Optimization 1994, 30, 323–330. [CrossRef] Noor, M.A. Some developments in genereal variational inequalities . Appl. Math. Comput. 2004, 152, 199-277. Noor, M.A. On Hermite–Hadamard integral inequalities for involving two log preinvex functions. J. Inequal. Pure Appl. Math. Anal. 2007, 3, 75–81. Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J. On Hermite–Hadamard inequalities for h preinvex functions. Filomat 2014, 28, 1463–1474. [CrossRef] Set, E. New inequalities of ostrowski type for mappings whose derivatives are s- convex in the second sense via fractional integrals. Comput. Appl. 2012, 63, 1147–1154. [CrossRef] Tunc, M.; Gov, E.; Sanal, U. On tgs-convex function and their inequalities. Facta Univ. Math. Inform. 2015, 36, 679–691. Verosanec, S. On h-convexity. J. Math. Anal. Appl. 2007, 326, 303–311. [CrossRef] c 2018 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/).