Extensions of different type parameterized inequalities for generalized \((m,h)\) preinvex mappings via k fractional integrals

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R E S E A R C H

Open Access

Extensions of different type

parameterized inequalities for generalized

(

m

,

h

)

-preinvex mappings via

k

-fractional

integrals

Yao Zhang

1

_{, Ting-Song Du}

1,2*

_{, Hao Wang}

1

_{, Yan-Jun Shen}

3

_{and Artion Kashuri}

4

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

College of Science, China Three Gorges University, Yichang, China

2_{Three Gorges Mathematical}

Research Center, China Three Gorges University, Yichang, China Full list of author information is available at the end of the article

Abstract

The authors discover a generalk-fractional integral identity with multi-parameters for twice diﬀerentiable functions. By using this integral equation, the authors derive some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for generalized (m,h)-preinvex functions throughk-fractional integrals. By taking the special parameter values for various suitable choices of functionh, some interesting results are also obtained.

MSC: 26A33; 26A51; 26D07; 26D20; 41A55

Keywords: Hermite–Hadamard’s inequality; Simpson’s inequality; Generalized (m,h)-preinvex functions;k-fractional integrals

1 Introduction

The subsequent inequalities are notable in the literature as Hermite–Hadamard’s inequal-ity and Simpson’s inequalinequal-ity, respectively.

Theorem 1.1 Suppose that f :I⊆R→Ris a convex function deﬁned on the interval I of real numbers and a,b∈I along with a<b.The following double inequality holds:

f

a+b

2

≤ 1

b–a

b

a

f(x) dx≤f(a) +f(b)

2 .

Theorem 1.2 Assume that f : [a,b]→Ris a four times continuously diﬀerentiable map-ping on(a,b)andf(4)

∞= supx∈(a,b)|f(4)(x)|<∞.Then the following inequality holds:

1₃f(a) +₂f(b)+ 2f

a+b

2

– 1

b–a

b

a

f(x) dx≤ 1

2880f (4)

∞(b–a)4.

Hermite–Hadamard’s inequalities and Simpson’s inequalities have remained an area of great interest owing to their extensive applications in mathematics and other sciences. Many researchers generalized these inequalities. For recent results, for example, see [1– 8] and the references mentioned in these papers.

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In 2013, Sarikaya et al. established the subsequent interesting Hermite–Hadamard’s in-equalities by utilizing Riemann–Liouville fractional integrals.

Theorem 1.3([9]) Let f : [a,b]→Rbe a positive function along with0≤a<b,and let f ∈L1_[_a_,_b_]._{Suppose that f is a convex function on}_[_a_,_b_],_{then the following inequalities for}

fractional integrals hold:

f

a+b

2

≤ (μ+ 1) 2(b–a)μ J

μ

a+f(b) +J_bμ–f(a)

≤f(a) +f(b)

2 , (1.1)

where the symbols J_aμ+f and J_bμ–f denote respectively the left-sided and right-sided Riemann–

Liouville fractional integrals of orderμ> 0deﬁned by

J_aμ+f(x) = 1

(μ)

x

a

(x–t)μ–1_f₍_t_{) d}_t_, _a_<_x_,

and

J_bμ–f(x) = 1

(μ)

b

x

(t–x)μ–1f(t) dt, x<b.

Here,(μ)is the gamma function and its deﬁnition is(μ) =₀∞e–t_tμ–1_d_t_._{It is to be noted}

that J0

a+f(x) =J_b0–f(x) =f(x).

In the case ofμ= 1, the fractional integral recaptures the classical integral.

Because of the extensive application of Riemann–Liouville fractional integrals, some au-thors extended their studies to fractional Hermite–Hadamard’s inequalities via mappings of diﬀerent classes. For example, refer to [10–12] for convex mappings, to [13] fors-convex mappings, to [14] for (s,m)-convex mappings, to [15] fors-Godunova–Levin mappings, to [16] for harmonically convex mappings, to [17] for preinvex mappings, to [18] for MTm -preinvex mappings, to [19] forh-convex mappings, to [20] forr-convex mappings, and see the references cited therein.

In 2012, Mubeen and Habibullah introduced the following class of fractional integrals.

Deﬁnition 1.1([21]) Letf ∈L1_[_a_,_b_{], then the Riemann–Liouville}_k_{-fractional integrals} kJ

μ

a+f(x) andkJ μ

b–f(x) of orderμ> 0 are given as

kJ_aμ+f(x) = 1

kk(μ)

x

a

(x–t)μk–1_f₍_t_{) d}_t ₍₀≤_a_<_x_<_b₎

and

kJbμ–f(x) = 1

kk(μ)

b

x

(t–x)μk–1_f₍_t_{) d}_t ₍₀≤_a_<_x_<_b_),

respectively, wherek> 0 andk(μ) is thek-gamma function deﬁned byk(μ) =

∞

0 t μ–1_×

e–tkk dt. Furthermore,_k(_μ+k) =_μ_k(_μ) and_kJ0

a+f(x) =kJb0–f(x) =f(x).

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the properties of Riemann–Liouvillek-fractional integrals are very dissimilar to those of classical Riemann–Liouville fractional integrals. Due to this, the Riemann–Liouville k -fractional integrals have aroused many researchers’ interest. Properties and estimations for the integral inequality related to this operator can be sought out in [22–27] and the references cited therein.

The main purpose of the current paper is to establish some new bounds on Hermite– Hadamard’s and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized (m,h)-preinvex. To do this, the authors derive a generalk -fractional integral identity along with multi parameters for twice diﬀerentiable mappings. By using this integral identity, the authors derive some new inequalities of Simpson and Hermite–Hadamard type for these mappings.

To end this section, we restate some special functions and deﬁnitions as follows. Let us consider the following special functions:

(1) The beta function:

β(x,y) =(x)(y)

(x+y) =

1

0

tx–1(1 –t)y–1dt, x,y> 0;

(2) The incomplete beta function:

β(a;x,y) =

a

0

tx–1(1 –t)y–1dt, 0 <a< 1,x,y> 0;

(3) The hypergeometric function:

2F1(a,b;c;z) = 1

β(b,c–b)

1

0

tb–1(1 –t)c–b–1(1 –zt)–adt, c>b> 0,|z|< 1.

Deﬁnition 1.2([28]) A functionf : [0,∞)→Ris nameds-convex in the second sense along withs∈(0, 1] if

f(αx+βy)≤αsf(x) +βsf(y)

holds for allx,y∈[0,∞) andα,β≥0 along withα+β= 1.

Deﬁnition 1.3([29]) A functionf :A⊆R→Ris calleds-Godunova–Levin function of the second kind along withs∈[0, 1] if

ftx+ (1 –t)y≤f(x) ts +

f(y) (1 –t)s

holds for allx,y∈Aandt∈(0, 1).

Deﬁnition 1.4([30]) A functionf :A⊆R→Ris namedtgs-convex onAif f is non-negative and

ftx+ (1 –t)y≤t(1 –t)f(x) +f(y)

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Deﬁnition 1.5([31]) A functionf:A⊆R→Ris calledMT-convex iff is non-negative and

ftx+ (1 –t)y≤

√

t

2√1 –tf(x) +

√ 1 –t

2√t f(y)

holds for allx,y∈Aandt∈(0, 1).

Deﬁnition 1.6([32]) A setA⊆Rn_{is called}_m_{-invex with respect to the mapping}_η_:_A_×

A×(0, 1]→Rn_{for some ﬁxed}_m_∈_{(0, 1] if}_mx₊_λη₍_y_,_x_,_m₎_∈_A_{holds for all}_x_,_y_∈_A_and

λ∈[0, 1].

Deﬁnition 1.7([33]) LetA⊆Rbe an openm-invex subset with respect toη:A×A×

(0, 1]→R, and leth1,h2: [0, 1]→R0. A functionf :A→Ris said to be generalized (m,h1,h2)-preinvex if

fmx+tη(y,x,m)≤mh1(t)f(x) +h2(t)f(y) (1.2)

is valid for allx,y∈Aandt∈[0, 1]. If inequality (1.2) reverses, thenf is said to be gener-alized (m,h1,h2)-preincave onA.

Clearly, if we takeh1(t) =h(1 –t),h2(t) =h(t) in Deﬁnition 1.7, thenf becomes general-ized (m,h)-preinvex functions as follows.

Deﬁnition 1.8 LetA⊆Rbe an openm-invex subset with respect toη:A×A×(0, 1]→

R, and leth: [0, 1]→R0. A functionf :A→Ris called generalized (m,h)-preinvex if

fmx+tη(y,x,m)≤h(t)f(y) +mh(1 –t)f(x) (1.3)

is valid for allx,y∈Aandt∈[0, 1].

Remark1.1 Let us discuss some special cases of Deﬁnition 1.8 as follows:

(i) choosingh(t) = 1, we obtain the deﬁnition of generalized(m,P)-preinvex functions; (ii) choosingh(t) =ts_for_s_∈_{(0, 1], we obtain the deﬁnition of generalized}

(m,s)-Breckner-preinvex functions;

(iii) choosingh(t) =t–s_for_s_∈_{(0, 1), we obtain the deﬁnition of generalized} (m,s)-Godunova–Levin–Dragomir-preinvex functions;

(iv) choosingh(t) =t(1 –t), we obtain the deﬁnition of generalized(m,tgs)-preinvex functions;

(v) choosingh(t) = √t

2√1–t, we obtain the deﬁnition of generalizedm-MT-preinvex functions.

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2 Main results

In order to derive our main results, we need the subsequent identity.

Lemma 2.1 Let A⊆Rbe an open m-invex subset with respect toη:A×A×(0, 1]→R\{0}

for some ﬁxed m∈(0, 1],and let a,b∈A,a<b withη(b,a,m) > 0.Assume that f :A→Ris a twice diﬀerentiable function on A such that fis integrable on[ma,ma+η(b,a,m)].Then the following identity for Riemann–Liouville k-fractional integrals along with x∈[a,b],

λ∈[0, 1],μ> 0,and k> 0exists:

If,η(μ,k;x,λ,m,a,b)

= η

μ

k+2₍_x_,_a_,_m₎

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ–tμk

fma+tη(x,a,m)dt

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ–tμk

fmb+tη(x,b,m)dt, (2.1)

where

:= 1 –λ

η(b,a,m) η

μ

k₍_x_,_a_,_m₎_f_ma₊_η₍_x_,_a_,_m₎

+ (–1)μk_η

μ

k(_x,_b,_m)_f_mb+_η(_x,_b,_m)

+ λ

η(b,a,m) η

μ

k(x,a,m)f(ma) + (–1)

μ

k_η

μ

k(x,b,m)f(mb)

+ 1

μ

k+1

–λ

η(b,a,m) (–1)

μ

k+1_η

μ

k+1₍_x_,_b_,_m₎_f_mb₊_η₍_x_,_b_,_m₎

–ημk+1₍_x_,_a_,_m₎_f_ma₊_η₍_x_,_a_,_m₎

–k(μ+k)

η(b,a,m) kJ μ

(ma+η(x,a,m))–f(ma) +kJ₍μ_mb₊η(x,b,m))+f(mb)

andkis the k-gamma function.

Proof By integration by parts and replacing the variable, we can state

1

0

t

μ

k + 1

λ–tμk

fma+tη(x,a,m)dt

=t

μ

k + 1

λ–tμk

f(ma+tη(x,a,m))

η(x,a,m)

1

0

– μ k + 1

η(x,a,m)

1

0

λ–tμk_f_ma+_t_η(_x,_a,_m)d_t

=

μ

k + 1

λ– 1

f(ma+η(x,a,m))

η(x,a,m)

– μ k + 1

η(x,a,m)

λ–tμkf(ma+tη(x,a,m))

η(x,a,m)

1

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+

1

0

μ

kt μ

k–1f(ma+tη(x,a,m))

η(x,a,m) dt

=

μ

k + 1

λ– 1

f(ma+η(x,a,m))

η(x,a,m)

+ μ k+ 1

η2₍_x_,_a_,_m₎

(1 –λ)fma+η(x,a,m)+λf(ma)

– μ k

ημk₍_x_,_a_,_m₎

ma+η(x,a,m)

ma

(u–ma)μk–1_f₍_u_{) d}_u

=

μ

k + 1

λ– 1

f(ma+η(x,a,m))

η(x,a,m)

+ μ k+ 1

η2₍_x_,_a_,_m₎

(1 –λ)fma+η(x,a,m)+λf(ma)

– k(μ+k)

ημk₍_x_,_a_,_m₎k

J₍μ_ma₊_η₍_x_,_a_,_m₎₎–f(ma)

. (2.2)

Similarly, we get

1

0

t

μ

k + 1

λ–tμk

fmb+tη(x,b,m)dt

=

μ

k + 1

λ– 1

f(mb+η(x,b,m))

η(x,b,m)

+ μ k+ 1

η2₍_x_,_b_,_m₎

(1 –λ)fmb+η(x,b,m)+λf(mb)

– k(μ+k)

ημk₍_x_,_b_,_m₎k

J₍μ_ma₊_η₍_x_,_b_,_m₎₎–f(mb)

. (2.3)

Multiplying both sides of (2.2) and (2.3) by η

μ

k+2(x,a,m) (μ_k+1)η(b,a,m)and

(–1)

μ

k+2η

μ

k+2(x,b,m)

(μ_k+1)η(b,a,m) , respectively, and adding the resulting identities together, we obtain the desired result.

Remark2.1 In Lemma 2.1, if we putk= 1 andη(b,a,m) =b–maalong withm= 1, then we get the following identity:

(1 –λ)

(x–a)μ_{+ (}_b_–_x₎μ

b–a

f(x) +λ

(x–a)μ_f₍_a_{) + (}_b_–_x₎μ_f₍_b₎

b–a

+

1

μ+ 1–λ

(b–x)μ+1_{– (}_x_–_a₎μ+1

b–a

f(x) –(μ+ 1)

b–a J

μ

x–f(a) +J_xμ+f(b)

= (x–a) μ+2

(μ+ 1)(b–a)

1

0

t (μ+ 1)λ–tμftx+ (1 –t)adt

+ (b–x) μ+2

(μ+ 1)(b–a)

1

0

t (μ+ 1)λ–tμ_f_tx_{+ (1 –}_t₎_b_d_t_,

which is proved by İşcan in [34]. Further, if we putμ= 1,λ=1₂, andx=aorx=b, then the above identity recaptures Lemma 1 in [35].

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Theorem 2.1 Let A⊆Rbe an open m-invex subset with respect toη:A×A×(0, 1]→R\ {0}for some ﬁxed m∈(0, 1],and let a,b∈A,a<b withη(b,a,m) > 0.Assume that f :A→

Ris a twice diﬀerentiable function on A such that fis integrable on[ma,ma+η(b,a,m)].If

|f|q_{for q}_≥₁_{is a generalized}₍_m_,_h₎_{-preinvex function with respect to}_η_{and h}_{: [0, 1]}_→_R 0,

then the following inequality for k-fractional integrals with x∈[a,b],λ∈[0, 1],μ> 0,k> 0

exists:

≤C1–

1

q

0 (k,μ,λ)

η

μ

k+2₍_x_,_a_,_m₎

(μ_k+ 1)η(b,a,m)

f(x)qC1(k,μ,λ;h)

+mf(a)qC2(k,μ,λ;h)

1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

f(x)qC1(k,μ,λ;h)

+mf(b)qC2(k,μ,λ;h)

1

q

, (2.4)

where

C0(k,μ,λ) =

1

0

t

μ

k + 1

λ–tμkdt

=

⎧ ⎪ ⎨ ⎪ ⎩ μ

k[(

μ

k+1)λ]

1+ 2_μk

μ

k+2

–(

μ

k+1)λ

2 + 1

μ

k+2

, 0≤λ≤ μ1

k+1

, (μ_k+1)λ

2 – 1

μ

k+2,

1

μ

k+1<

λ≤1,

(2.5)

C1(k,μ,λ;h) =

1

0

t

μ

k + 1

λ–tμkh(t) dt, 0≤_λ≤1, (2.6)

and

C2(k,μ,λ;h) =

1

0

t

μ

k + 1

λ–tμkh(1 –t) dt, 0≤_λ≤1. (2.7)

Proof Applying Lemma 2.1 and the power mean inequality, we have

≤ η

μ

k+2₍_x_,_a_,_m₎

(μ_k + 1)η(b,a,m)

₀1t

μ

k + 1

λ–tμk_f_ma₊_t_η₍_x_,_a_,_m₎_d_t

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ–tμkfmb+t_η(x,b,m)dt

≤ η

μ

k+2(x,a,m)

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ–tμk_d_t

1–1_q

× 1 0

t

μ

k + 1

λ–tμk_f_ma₊_t_η₍_x_,_a_,_m₎q_d_t

1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ–tμkdt

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× 1 0

t

μ

k + 1

λ–tμk_f_mb₊_t_η₍_x_,_b_,_m₎q_d_t

1

q

=C1–

1

q

0 (k,μ,λ)

× η

μ

k+2₍_x_,_a_,_m₎

(μ_k + 1)η(b,a,m)

₀1t

μ

k + 1

1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

1

0

t

μ

k + 1

λ

–tμk_f_mb₊_t_η₍_x_,_b_,_m₎q_d_t

1

q

. (2.8)

Since|f|qis generalized (m,h)-preinvex on [ma,ma+η(b,a,m)], we get

1

0

t

μ

k + 1

≤

1

0

t

μ

k + 1

λ–tμk_h₍_t₎_f₍_x₎q₊_mh_{(1 –}_t₎_f₍_a₎q_d_t

=f(x)qC1(k,μ,λ;h) +mf(a) q

C2(k,μ,λ;h) (2.9)

and

1

0

t

μ

k + 1

λ–tμk_f_mb₊_t_η₍_x_,_b_,_m₎q_d_t

≤

1

0

t

μ

k + 1

λ–tμk_h(_t)_f(_x)q+_mh(1 –_t)_f(_b)qd_t

=f(x)qC1(k,μ,λ;h) +mf(b) q

C2(k,μ,λ;h), (2.10)

whereC0(k,μ,λ),C1(k,μ,λ;h), andC2(k,μ,λ;h) are deﬁned by (2.5)–(2.7), respectively. Hence, if we use (2.9) and (2.10) in (2.8), we can get the desired result. This completes the

proof.

Let us point out some special cases of Theorem 2.1.

I. Ifh(t) =ts_{in Theorem 2.1, then we have the following results.}

Corollary 2.1 In Theorem2.1,if |f|q _{for q}_≥₁ _{is generalized}₍_m_,_s₎_{-Breckner-preinvex}

functions,then,for s∈(0, 1]and m∈(0, 1],we have

≤C1–

1

q

0 (k,μ,λ)

× η

μ

k+2(x,a,m)

(μ_k + 1)η(b,a,m)

f(x)qT1(k,μ,λ;s) +mf(a) q

T2(k,μ,λ;s)

1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

f(x)qT1(k,μ,λ;s) +mf(b)qT2(k,μ,λ;s)

1

q

(9)

where we use the fact that

T1(k,μ,λ;s) =

⎧ ⎪ ⎨ ⎪ ⎩

2μ

k[(

μ

k+1)λ]

1+k(s_μ+2)

(s+2)(μ_k+s+2) – (μ_k+1)λ

s+2 + 1

μ

k+s+2

, 0≤λ≤ μ1

k+1

, (μ_k+1)λ

s+2 – 1

μ

k+s+2,

1

μ

k+1<

λ≤1,

T2(k,μ,λ;s) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎣

β(μ_k+ 2,s+ 1) – (μ_k + 1)λβ(2,s+ 1)

+ 2(μ_k+ 1)λβ([(μ_k+ 1)λ]kμ_{; 2,}_s_{+ 1)}

– 2β([(μ_k + 1)λ]kμ_;μ

k + 2,s+ 1)

⎤ ⎥ ⎥

⎦, 0≤λ≤ μ1

k+1

,

(μ_k + 1)λβ(2,s+ 1) –β(μ_k + 2,s+ 1), 1

μ

k+1

<λ≤1,

and C0(k,μ,λ)is deﬁned by(2.5).

Corollary 2.2 In Theorem2.1,if the mappingη(b,a,m) =b–ma along with m= 1,taking x=a+₂b,then for s∈(0, 1],we have the following inequality for s-convex functions:

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b

2 ,λ, 1,a,b

=(1 –λ)f

a+b

2

+λ

f(a) +f(b) 2

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ

(a+₂b)–f(a) +kJ μ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k+ 1)C

1–1_q 0 (k,μ,λ)

f

a+b

2

qT1(k,μ,λ;s) +f(a)qT2(k,μ,λ;s)

1

q

+f

a+b

2

qT1(k,μ,λ;s) +f(b) q

T2(k,μ,λ;s)

1

q

.

Remark2.2 In Corollary 2.2, (i) takingλ=1₂, we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b 2 ,

1 2, 1,a,b

=1 4

f(a) + 2f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 2

×f

a+b

2

qT1

k,μ,1 2;s

+f(a)qT2

k,μ,1 2;s

1

q

+f

a+b

2

qT1

k,μ,1 2;s

+f(b)qT2

k,μ,1 2;s

1

q

(10)

(ii) takingλ=1₃, we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b 2 ,

1 3, 1,a,b

=1 6

f(a) + 4f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 3

×f

a+b

2

qT1

k,μ,1 3;s

+f(a)qT2

k,μ,1 3;s

1

q

+f

a+b

2

qT1

k,μ,1 3;s

+f(b)qT2

k,μ,1 3;s

1

q

.

Specially, if we putk=μ=s= 1, then we obtain

1₆f(a) + 4f

a+b

2

+f(b)

– 1

b–a

b

a

f(x) dx

≤(b–a)2 162

59 96

f

a+b

2

q+37

96f

₍_a₎q 1

q

+

59 96

f

a+b

2

q+37

96f

₍_b₎q 1

q

≤(b–a)2 162

59|f(a)|q_{+ 133}_|_f₍_b₎_|q 192

1

q

+

_59|

f(b)|q+ 133|f(a)|q

192

1

q

. (2.11)

It is noted that the result of the ﬁrst inequality in (2.11) is proved by İşcan in [34], which is better than the result presented by Sarikaya et al. in [36, Theorem 6].

Remark2.3 In Corollary 2.2, if we takek= 1 andλ= 0, 1, we have the results (f ) and (h) in [34, Corollary 2.3], respectively. Further, if we takeμ= 1, we have the results (g) and (i) in [34, Corollary 2.3], respectively.

II. Ifh(t) =t–s_{in Theorem 2.1, then we have the following results.}

Corollary 2.3 In Theorem2.1,if|f|q_{for q}_≥₁_{is generalized}₍_m_,_s₎_{-Godunova–Levin–}

Dragomir-preinvex functions,then,for s∈(0, 1)and m∈(0, 1],we have

≤C1–

1

q

(11)

× η

μ

k+2₍_x_,_a_,_m₎

(μ_k + 1)η(b,a,m)

f(x)qU1(k,μ,λ;s) +mf(a)qU2(k,μ,λ;s)

1

q

+(–1)

μ

k+2_η

μ

k+2(x,b,m)

(μ_k + 1)η(b,a,m)

f(x)qU1(k,μ,λ;s) +mf(b) q

U2(k,μ,λ;s)

1

q

,

U1(k,μ,λ;s) =

⎧ ⎪ ⎨ ⎪ ⎩

2μ

k [(

μ

k+1)λ]

1+k(2–_μs)

(2–s)(μ_k–s+2) – (μ_k+1)λ

2–s + 1

μ

k–s+2

, 0≤λ≤ μ1

k+1

, (μ_k+1)λ

2–s – 1

μ

k–s+2,

1

μ

k+1<

λ≤1,

U2(k,μ,λ;s) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎣

β(μ_k + 2, 1 –s) – (μ_k + 1)λβ(2, 1 –s)

+ 2(μ_k + 1)λβ([(μ_k + 1)λ]μk_{; 2, 1 –}_s₎

– 2β([(μ_k + 1)λ]μk_;μ

k + 2, 1 –s)

⎤ ⎥ ⎥

⎦, 0≤λ≤ μ1

k+1,

(μ_k + 1)λβ(2, 1 –s) –β(μ_k + 2, 1 –s), μ1

k+1

<λ≤1,

Corollary 2.4 In Theorem2.1,if the mappingη(b,a,m) =b–ma together with m= 1, tak-ing x=a+₂b,for s∈(0, 1),we have the following inequality for s-Godunova–Levin functions:

(1 –λ)f

a+b

2

+λ

f(a) +f(b) 2

–2

μ

k–1_k(_μ+_k)

(b–a)μk

kJ₍μa+b

2 )–

f(a) +kJ₍μa+b

2 )+

f(b)

≤ (b–a)2 8(μ_k+ 1)C

1–1q

0 (k,μ,λ)

f

a+b

2

qU1(k,μ,λ;s) +f(a) q

U2(k,μ,λ;s)

1

q

+f

a+b

2

qU1(k,μ,λ;s) +f(b) q

U2(k,μ,λ;s)

1

q

.

Remark2.4 In Corollary 2.4, (a) ifλ=1₃, then we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b 2 ,

1 3, 1,a,b

=1 6

f(a) + 4f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 3

×f

a+b

2

qU1

k,μ,1 3;s

+f(a)qU2

k,μ,1 3;s

1

q

+f

a+b

2

qU1

k,μ,1 3;s

+f(b)qU2

k,μ,1 3;s

1

q

(12)

Specially, if we putk= 1 =μ, then we obtain a Simpson-type inequality:

If

1, 1;a+b 2 ,

1 3, 1,a,b

=1 6

f(a) + 4f

a+b

2

+f(b)

– 1

b–a

b

a

f(x) dx

≤(b–a)2 16

8 81

1–1_q

f

a+b

2

qU1

1, 1,1 3;s

+f(a)qU2

1, 1,1 3;s

1

q

+f

a+b

2

qU1

1, 1,1 3;s

+f(b)qU2

1, 1,1 3;s

1

q

;

(b) ifλ=1₂, then we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b 2 ,

1 2, 1,a,b

=1 4

f(a) + 2f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 2

×f

a+b

2

qU1

k,μ,1 2;s

+f(a)qU2

k,μ,1 2;s

1

q

+f

a+b

2

qU1

k,μ,1 2;s

+f(b)qU2

k,μ,1 2;s

1

q

.

Specially, if we putk= 1 =μ, then we obtain an averaged midpoint-trapezoid-type inequality:

If

1, 1;a+b 2 ,

1 2, 1,a,b

=1 4

f(a) + 2f

a+b

2

+f(b)

– 1

b–a

b

a

f(x) dx

≤(b–a)2 16

1 6

1–1_q

f

a+b

2

qU1

1, 1,1 2;s

+f(a)qU2

1, 1,1 2;s

1

q

+f

a+b

2

qU1

1, 1,1 2;s

+f(b)qU2

1, 1,1 2;s

1

q

;

(c) ifλ= 0, then we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b

2 , 0, 1,a,b

=f

a+b

2

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ (a+2b)–

f(a) +kJμ (a+2b)+

(13)

≤ (b–a)2 8(μ_k + 1)

1 μ k+ 2

1–1q|_f₍a+b

2 )|q μ k –s+ 2

+f(a)qβ

μ

k + 2, 1 –s

1

q

+

_|

f(a+₂b)|q μ k –s+ 2

+f(b)qβ

μ

k + 2, 1 –s

1

q

.

Specially, if we putk= 1 =μ, then we obtain a midpoint-type inequality:

If

1, 1;a+b

2 , 0, 1,a,b

=f

a+b

2

– 1

b–a

b

a

f(x) dx

≤(b–a)2 16

1 3

1–1_q_|_f₍a+b 2 )|q 3 –s +f

₍_a₎q

β(3, 1 –s)

1

q

+

_|

f(a+₂b)|q 3 –s +f

₍_b₎q

β(3, 1 –s)

1

q

;

(d) ifλ= 1, then we obtain

2

μ

k–1

(b–a)μk–1 If

μ,k;a+b

2 , 1, 1,a,b

=f(a) +f(b)

2 –

2μk–1_k(_μ+k)

(b–a)μk

kJ₍μa+b

2 )–

f(a) +kJ₍μa+b

2 )+

f(b)

≤ (b–a)2 8(μ_k + 1)

μ k(

μ k + 3) 2(μ_k+ 2)

1–_q1

×

μ

k( μ

k –s+ 3)|f( a+b

2 )|q (2 –s)(μ_k –s+ 2)

+f(a)q

μ

k + 1

β(2, 1 –s) –β

μ

k + 2, 1 –s

1

q

+

μ k(

μ

k –s+ 3)|f( a+b

2 )|q (2 –s)(μ_k –s+ 2)

+f(b)q

μ

k + 1

β(2, 1 –s) –β

μ

k + 2, 1 –s

1

q

.

Specially, if we putk= 1 =μ, then we obtain a trapezoid-type inequality:

If

1, 1;a+b

2 , 1, 1,a,b

=f(a) +f(b)

2 –

1

b–a

b

a

f(x) dx

≤(b–a)2 16

2 3

1–1q_{(4 –}_s)|_f₍a+b

2 )| q

(2 –s)(3 –s) +f

₍_a₎q

2β(2, 1 –s) –β(3, 1 –s) 1

q

+

(4 –s)|f(a+₂b)|q (2 –s)(3 –s) +f

₍_b₎q

2β(2, 1 –s) –β(3, 1 –s) 1

q

(14)

III. Ifh(t) =t(1 –t) in Theorem 2.1, then we have the following results.

Corollary 2.5 In Theorem2.1,if|f|q_{for q}_≥₁_{is generalized}₍_m_,_tgs₎_{-preinvex functions}_,

then,for m∈(0, 1],we have

≤C1–

1

q

0 (k,μ,λ)θ 1

q₍_k_,_μ_,_λ₎ η

μ

k+2₍_x_,_a_,_m₎

(μ_k + 1)η(b,a,m)

f(x)q+mf(a)q 1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

f(x)q+mf(b)q1q

,

θ(k,μ,λ) =

⎧ ⎪ ⎨ ⎪ ⎩

2μ

k[(

μ

k+1)λ]

3k

μ+1

3(μ_k+3) –

μ

k[(

μ

k+1)λ]

4k

μ+1

2(μ_k+4) – (μ_k+1)λ

12 + 1

(μ_k+3)(μ_k+4), 0≤λ≤ 1

μ

k+1,

(μ_k+1)λ 12 –

1 (μ_k+3)(μ_k+4),

1

μ

k+1

<λ≤1

Corollary 2.6 In Theorem2.1,if the mappingη(b,a,m) =b–ma along with m= 1,taking x=a+₂b,we get the following inequality for tgs-convex functions:

(1 –λ)f

a+b

2

+λ

f(a) +f(b) 2

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ (a+2b)–

f(a) +kJμ (a+2b)+

f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q

0 (k,μ,λ)θ 1

q₍_k_,_μ_,_λ₎

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

1₆f(a) + 4f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 3

θ1q

k,μ,1 3

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

Specially, if we putk= 1 =μ, then we obtain a Simpson-type inequality:

1₆f(a) + 4f

a+b

2

+f(b)

– 1

b–a

b

a

(15)

≤(b–a)2 162

23 160

1

q

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

;

(b) ifλ=1₂, then we obtain

1₄f(a) + 2f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ (a+2b)–

f(a) +kJμ (a+2b)+

f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 2

θ1q

k,μ,1 2

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

Specially, if we putk= 1 =μ, then we derive an averaged midpoint-trapezoid-type inequality:

1₄f(a) + 2f

a+b

2

+f(b)

– 1

b–a

b

a

f(x) dx

≤(b–a)2 96

1 5

1

q

f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

;

(c) ifλ= 0, then we obtain

f

a+b

2

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)

1 μ k + 2

1–1q ₁

(μ_k + 3)(μ_k + 4)

1

q

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

Specially, if we putk= 1 =μ, then we derive a midpoint-type inequality:

f

a+b

2

– 1

b–a

b

a

f(x) dx

≤(b–a)2 48

3 20

1

q

f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

(16)

(d) ifλ= 1, then we obtain

f(a) +₂f(b)–2

μ

k–1_k(_μ+k)

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k + 1)

μ k(

μ k + 3) 2(μ_k+ 2)

1–_q1μ k(

μ2 k2 +8

μ k + 19) 12(μ_k + 3)(μ_k + 4)

1

q

×f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

Specially, if we putk= 1 =μ, then we obtain a trapezoid-type inequality:

f(a) +₂f(b)– 1

b–a

b

a

f(x) dx

≤(b–a)2 24

7 40

1

q

f

a+b

2

q+f(a)q

1

q

+f

a+b

2

q+f(b)q

1

q

.

IV. Ifh(t) =

√

t

2√1–t in Theorem 2.1, then we have the following results.

Corollary 2.7 In Theorem2.1,if|f|qfor q≥1is generalized m-MT -preinvex functions,

then,for m∈(0, 1],we have

≤C1–

1

q

0 (k,μ,λ)

η

μ

k+2₍_x_,_a_,_m₎

(μ_k+ 1)η(b,a,m)

f(x)q1(k,μ,λ) +mf(a)q2(k,μ,λ)

1

q

+(–1)

μ

k+2_η

μ

k+2₍_x_,_b_,_m₎

(μ_k + 1)η(b,a,m)

f(x)q1(k,μ,λ) +mf(b) q

2(k,μ,λ)

1

q

,

1(k,μ,λ)

=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡ ⎣12β(

μ k +

5 2,

1 2) –

3λπ(μ_k+1) 16 + (

μ

k+ 1)λβ([( μ k+ 1)λ]

k

μ_;5

2, 1 2) –β([(μ_k + 1)λ]μk_;μ

k + 5 2,

1 2)

⎤

⎦, 0≤λ≤ μ1

k+1,

3λπ(μ_k+1) 16 –

1 2β(

μ k +

5 2,

1 2),

1

μ

k+1

<λ≤1,

2(k,μ,λ)

=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡ ⎣12β(

μ k +

3 2,

3 2) –

λπ(μ_k+1) 16 + (

μ

k + 1)λβ([( μ k + 1)λ]

k

μ_;3

2, 3 2) –β([(μ_k+ 1)λ]kμ_;μ

k + 3 2,

3 2)

⎤

⎦, 0≤λ≤ μ1

k+1,

λπ(μ_k+1) 16 –

1 2β(

μ k +

3 2,

3 2),

1

μ

k+1<

λ≤1,

(17)

Corollary 2.8 In Theorem2.1, if the mappingη(b,a,m) =b–ma together with m= 1,

taking x=a+₂b,we get the following inequality for MT -convex functions:

(1 –λ)f

a+b

2

+λ

f(a) +f(b) 2

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk k Jμ

(a+₂b)+f(b)

≤ (b–a)2 8(μ_k+ 1)C

1–1_q 0 (k,μ,λ)

f

a+b

2

q1(k,μ,λ) +f(a)q2(k,μ,λ)

1

q

+f

a+b

2

q1(k,μ,λ) +f(b) q

2(k,μ,λ)

1

q

.

1₆f(a) + 4f

a+b

2

+f(b)

–2

μ

k–1_k₍_μ₊_k₎

(b–a)μk

kJμ (a+2b)–

f(a) +kJμ (a+2b)+

f(b)

≤ (b–a)2 8(μ_k + 1)C

1–1_q 0

k,μ,1 3

f

a+b

2

q1

k,μ,1 3

+f(a)q2

k,μ,1 3

1

q

+f

a+b

2

q1

k,μ,1 3

+f(b)q2

k,μ,1 3

1

q

.

Specially, if we putk= 1 =μ, then we derive a Simpson-type inequality:

1₆f(a) + 4f

a+b

2

+f(b)

– 1

b–a

b

a

f(x) dx

≤(b–a)2 16

8 81

1–1_q

f

a+b

2

q1

1, 1,1 3

+f(a)q2

1, 1,1 3

1

q

+f

a+b

2

q1

1, 1,1 3

+f(b)q2

1, 1,1 3

1

q

,

where

1

1, 1,1 3

=71 √

2 648 +

1 8arcsin

√ 3 3 –

π

32

and

2

1, 1,1 3

=19 √

2 648 +

1 12arcsin

1 3+

1

16arctan2 √