Some Inequalities for n-Time Differentiable Mappings Using a Multi-Step Kernel with Applications

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Volume 18, Number 1 (2020), 50-62

URL:https://doi.org/10.28924/2291-8639 DOI:10.28924/2291-8639-18-2020-50

SOME INEQUALITIES FOR n-TIME DIFFERENTIABLE MAPPINGS USING A MULTI-STEP KERNEL WITH APPLICATIONS

SOFIAN OBEIDAT∗

Department of Basic Sciences, Deanship of Preparatory Year, University of Hail, Hail 2440, Saudi Arabia

∗_{Corresponding author:} _{[email protected]}

Abstract. In this paper, we develop a new multi-step kernel and use it to establish new Ostrowski’s type inequalities forn-time differentiable mappings, whosen-th derivatives satisfy convexity and quasi-convexity conditions. Applications of our findings to random variables and approximation of integrals are given.

1. Introduction

The classical Ostrowski inequality (1938, see [8]) is given as follows: ifx∈[x1, x2] then

g(x)− 1

x2−x1

x2

Z

x1 g(t)dt

≤

₁

4 +

(x−x1+x2 2 )

2

(x2−x1)2

(x2−x1)kg0k_∞, (1.1)

where g is a differentiable function defined on a finite interval [x1, x2], whose derivative is integrable and

bounded over [x1, x2].The constant 1/4 is the best possible. Whenx=x1+₂x2,Inequality1.1reduces to the

midpoint version

g(x1+x2 2 )−

1

x2−x1

x2

Z

x1 g(t)dt

≤(x2−x1)

4 kg

0_k ∞.

The importance of Ostrowski’s type inequalities is due to their applications in different aspects. For

generalizations and variants of Ostrowski’s type inequalities, we refer the reader to [2] and [4].

Received 2019-10-26; accepted 2019-11-22; published 2020-01-02.

2010Mathematics Subject Classification. Primary 26D15, 26D20; Secondary 26D99.

Key words and phrases. convex functions; H¨older inequality; quasi-convex function; Ostrowski inequality. c

2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

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Recently, several authors have derived Ostrowski’s type inequalities for n-time differentiable mappings

whose n-th derivatives satisfy different types of convexity conditions, see for example [?], [3], [5], [7], [10]

and [12].

In particular, Ozdemir and Yildiz, in [9], obtained the following three theorems forn-time differentiable

mappings .

Theorem 1.1. [9] Suppose thatnis a positive integer, g:J ⊆R→Risn-time differentiable mapping on

J◦ andx1,x2∈J◦ withx1< x2. If g(n)∈L1([x1, x2])and

g(n)

is convex on [x1, x2], then

Z x2

x1

g(x)dx−

n−1

X

k=0

1 + (−1)k(x2−x1)

k+1

2k+1₍_k_{+ 1)!} g

(k)x1+x2

2

(1.2)

≤ (x2−x1)

n+1

2n₍_n_{+ 1)!} "

g(n)(x1) +

g(n)(x2) 2

#

.

J◦ andx1,x2∈J◦ withx1< x2. If g(n)∈L1([x1, x2])and

g(n)

q

is convex on [x1, x2], whereq >1, then

Z x2

x1

g(x)dx−

n−1

X

k=0

1 + (−1)k(x2−x1)

k+1

2k+1₍_k_{+ 1)!} g

(k)x1+x2

2

(1.3)

≤ (x2−x1)

n+1

2n+1_n_!

₁

1 +pn

1p 



g(n)(x1) q

+ 3g(n)(x2) q

4

!1q

+3

g(n)(x1) q

+g(n)(x2) q

4

#

,

where 1_p+1_q = 1.

J◦ andx1,x2∈J◦ withx1< x2. If g(n)∈L1([x1, x2])andg(n) q

is convex on [x1, x2], whereq≥1, then

Z x2

x1

g(x)dx−

n−1

X

k=0

1 + (−1)k(x2−x1)k+1

2k+1₍_k_{+ 1)!} g

(k)x1+x2

2

(1.4)

≤ (x2−x1)

n+1

2n+1₍_n_{+ 1)!}

"

_n_{+ 1} 2n+ 4

g

(n)₍_x1₎ q

+ n+ 3 2n+ 4

g

(n)₍_x2₎

q1q

+

_n_{+ 3} 2n+ 4

g

(n)₍_x1₎

q

+ n+ 1 2n+ 4

g

(n)₍_x2₎

q1q#

.

In this paper, we generalize the inequalities obtained by Ozdemir and Yildiz in [9] for convex and quasi

convex functions using a multi-step kernel. Then we introduce some applications of our findings to random

variables and approximation of integrals. Throught this paper, _R denotes the set of all real numbers and

J ⊂Rdenotes an interval. The concepts of convex and quasi convex functions, which are well known in the

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Definition 1.1. [6] A functiong:J ⊂R→Ris said to be convex if the inequality

g(tx1+ (1−t)x2)≤tg(x1) + (1−t)g(x2),

holds for allx1,x2∈J andt∈[0,1].

Definition 1.2. [11] A functiong:J ⊂R→Ris said to be quasi convex if the inequality

g(tx1+ (1−t)x2)≤max{g(x1), g(x2)}

holds for allx1,x2∈J andt∈[0,1].

2. _{Main Results}

We start this section with the following two Lemmas.

Lemma 2.1. Suppose thatn andm are positive integers, g:J ⊆R→Ris n-time differentiable mapping

on J◦ and x1, x2 ∈ J◦ with x1 < x2. Suppose that g(n) ∈L1([x1, x2]). Then for every s1, s2 ∈ [0,1] the

identity

Z s2

s1

g(x2+t(x1−x2))dt (2.1)

= n−1

X

k=0

h

(s2−r)k+1g(k)₍_x2₊_s2₍_x1₋_x2₎₎₋₍_s1₋_r₎k+1

g(k)₍_x2₊_s1₍_x1₋_x2₎₎i

(x2−x1)−k(k+ 1)!

+(x2−x1) n

n!

Z s2

s1

(t−r)ng(n)(x2+t(x1−x2))dt,

holds for each r∈R.

Proof. Using integration by parts repeatedly, we get that

1

n! Z s2

s1

(t−r)ng(n)(x2+t(x1−x2))dt (2.2)

= −

n−1

X

k=0

(t−r)k+1g(k)(x2+t(x1−x2))

(x2−x1)n−k(k+ 1)! s2

s1

+ 1

(x2−x1)n Z s2

s1

g(x2+t(x1−x2))dt.

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Lemma 2.2. Suppose thatn andm are positive integers, g:J ⊆R→Ris n-time differentiable mapping

onJ◦ andx1,x2∈J◦ withx1< x2.If g(n) ∈L1([x1, x2])then

Z x2

x1

g(x)dx (2.3)

=

2m−1

X

l=1

n−1

X

k=0

h

1 + (−1)ki2−m(k+1)

(x2−x1)−k−1(k+ 1)!g

(k)(2l−1)x1+ (2m−2l+ 1)x2

2m

+ (x2−x1)

n+1Z 1

0

Pn,m(t)g(n)(x2+t(x1−x2))dt,

where

Pn,m(t) = 1

n!     

   

tn_, _if_t_∈

0,₂1m

t−21−mln, if t∈ 2l−1 2m ,

2l+1 2m

, l= 1,2,· · ·,2m−1−1

(t−1)n, if t∈ 1− 1 2m,1

.

Proof. Using Lemma2.1, we have

Z 21m

0

g(x2+t(x1−x2))dt (2.4)

= n−1

X

k=0

h

1 2m

k+1

g(k)x1+(2m−1)x2 2m

i

(x2−x1) −k

(k+ 1)!

+(x2−x1) n

n!

Z ₂1m

0

Z 1

(1− 1 2m)

g(x2+t(x1−x2))dt (2.5)

= n−1

X

k=0

− −1 2m

k+1

g(k)(2m−1)x1+x2 2m

(x2−x1)−k(k+ 1)!

+(x2−x1) n

n!

Z 1

(1− 1 2m)

and

Z 2₂lm+1

2l−1 2m

g(x2+t(x1−x2))dt (2.6)

= n−1

X

k=0

h

g(k)(2l+1)x1+(2m−2l−1)x2 2m

+ (−1)kg(k)(2l−1)x1+(2m−2l+1)x2 2m

i

2m(k+1)₍_x 2−x1)

−k

(k+ 1)!

+(x2−x1) n

n!

Z 2₂l+1m

2l−1 2m

forl= 1,2,· · ·,2m−1₋₁_. _{Combining (}_2.4_{), (}_2.5_{) and (}_2.6_{), the result follows.}

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Theorem 2.1. Suppose thatnandmare positive integers,g:J ⊆R→Risn-time differentiable mapping

onJ◦ andx1,x2∈J◦ withx1< x2.If g(n)∈L1([x1, x2])andg(n)

is convex on[x1, x2], then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1)−k−1(k+ 1)! (2.7)

×g(k)

₍₂_l₋₁₎_x1_{+ (2}m₋₂_l_{+ 1)}_x2 2m

≤ (x2−x1)

n+1

2mn₍_n_{+ 1)!} "

g(n)(x1) +

g(n)(x2) 2

#

.

Proof. Using convexity ofg(n)

on [x1, x2], we get that

Z 1

0

Pn,m(t)g(n)(x2+t(x1−x2))dt

(2.8)

≤ 1

n! Z ₂1m

0

tnh(1−t) g

(n)₍_x2₎ +t

g

(n)₍_x1₎ i

dt

+1

n!

2m−1₋₁

X

l=1

Z 2₂l+1m

2l−1 2m

t− l

2m−1

n

h (1−t)

g

(n)₍_x2₎ +t

g

(n)₍_x1₎ i

dt

+1

n! Z 1

1− 1 2m

(1−t)nh(1−t) g

(n)₍_x 2)

+t

g

(n)₍_x 1)

i

dt

= 1

2mn₍_n_{+ 1)!} "

g(n)(x1)

+

g(n)(x2)

2

#

.

Using Identity2.3 and Inequality2.8, the result follows.

Remark 2.1. In Theorem2.1,

(1) If m= 1, Inequality2.7 reduces to Inequality1.2.

(2) If m= 2, Inequality2.7 reduces to

Z x2

x1

g(x)dx−

n−1

X

k=0

1 + (−1)k2−2(k+1)

(x2−x1)−k−1(k+ 1)!

g(k)

₃_x1₊_x2 4

(2.9)

+g(k)

_x

1+ 3x2

4

≤ (x2−x1)

n+1

22n₍_n_{+ 1)!} "

g(n)(x1)

+

g(n)(x2)

2

#

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Z x2

x1

g(x)dx−

n−1

X

k=0

1 + (−1)k2−3(k+1)

(x2−x1)−k−1(k+ 1)!

g(k)

₇_x

1+x2

8

(2.10)

+g(k)

₅_x1_{+ 3}_x2 8

+g(k)

₃_x1_{+ 5}_x2 8

+g(k)

_x1_{+ 7}_x2 8

≤ (x2−x1)

n+1

23n₍_n_{+ 1)!} "

g(n)(x1) +

g(n)(x2) 2

#

.

on J◦ and x1, x2 ∈J◦ with x1 < x2. If g(n) ∈L1([x1, x2]) and

g(n)

q

is convex on [x1, x2], where q >1,

then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1)−k−1(k+ 1)! (2.11)

×g(k)

₍₂_l₋₁₎_x1_{+ (2}m₋₂_l_{+ 1)}_x2 2m

≤ (x2−x1)

n+1

2m(n+1)_n_!

₁

1 +pn

1p

×

" 1 2m+1

g

(n)₍_x1₎ q

+

1− 1

2m+1

g

(n)₍_x2₎

q1q

+

1− 1

2m+1

g

(n)₍_x1₎ q

+ 1 2m+1

g

(n)₍_x2₎

q1_q

+2

2m−1−1

X

l=1

_l

2m−1

g

(n)₍_x 1)

q

+

1− l

2m−1

g

(n)₍_x 2)

q1q 

,

where 1_p+1_q = 1.

Proof. Using Holder’s inequality, we get that

Z 1

0

(2.12)

≤ 1

n! Z ₂1m

0

tpndt

!1p _Z 1 2m

0

g

(n)₍_x2₊_t₍_x1₋_x2₎₎

q

dt

!1q

+ 1

n!

2m−1₋₁

X

l=1

Z 2₂l+1m

2l−1 2m

t− l

2m−1

pn

dt

!1p _Z 2l+1 2m

2l−1 2m

g

(n)₍_x2₊_t₍_x1₋_x2₎₎

q

dt

!1q

+ 1

n! Z 1

1− 1 2m

(1−t)pndt

!1p _Z 1

1− 1 2m

g

(n)₍_x

2+t(x1−x2))

q

dt

!1q

(7)

Using convexity ofg(n) q

on [x1, x2],we find that

Z 1

0

(2.13)

≤ 1

2m(n+1)_n_!

₁

1 +pn

1p

×

" 1 2m+1

g

(n)₍_x1₎ q

+

1− 1

2m+1

g

(n)₍_x2₎

q1q

+

1− 1

2m+1

g

(n)₍_x1₎ q

+ 1 2m+1

g

(n)₍_x1₎

q1q

+2

2m−1−1

X

l=1

_l

2m−1

g

(n)₍_x 1)

q

+

1− l

2m−1

g

(n)₍_x 2)

q1q 

.

(1) If m= 1, Inequality2.11 reduces to Inequality 1.3.

Z x2

x1

g(x)dx−

n−1

X

k=0





1 + (−1)k2−2(k+1)

(x2−x1)−k−1(k+ 1)!

g(k)

3x1+x2

4

(2.14)

+g(k)

_x1_{+ 3}_x2

4

≤ (x2−x1)

n+1

22(n+1)_n_!

₁

1 +pn

1p 

2

g(n)(x1)

q

+g(n)(x2)

q

2

!q1

+ 7

g(n)(x1)

q

+

g(n)(x2)

q

8

!1q +

g(n)(x1)

q

+ 7

g(n)(x2)

q

8

!1q

.

on J◦ and x1, x2 ∈J◦ with x1 < x2. If g(n) ∈L1([x1, x2]) and g(n) q

is convex on [x1, x2], where q≥1,

then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1)−k−1(k+ 1)! (2.15)

×g(k)

₍₂_l₋₁₎_x1_{+ (2}m₋₂_l_{+ 1)}_x2 2m

≤ (x2−x1)

n+1

2m(n+1)₍_n_{+ 1)!}

×

"

2−m₍_n_{+ 1)} (n+ 2)

g

(n)₍_x 1)

q

+

1−2

−m₍_n_{+ 1)}

(n+ 2)

g

(n)₍_x 2)

(8)

+2

2m−1₋₁

X

l=1

21−m

g

(n)₍_x1₎ q

+ 1−21−ml

g

(n)₍_x2₎

q1q

+

1−2

−m₍_n_{+ 1)}

(n+ 2)

g

(n)₍_x1₎

q

+2

−m₍_n_{+ 1)}

(n+ 2) g

(n)₍_x2₎

q1q#

.

Proof. Using convexity ofg(n) q

Z1 0

Pn,m(t)g(n)(x2+t(x1−x2))dt (2.16) ≤ 1 n!

Z ₂1m

0

tndt !1_p

Z ₂1m

0

tn g

(n)

(x2+t(x1−x2)) q

dt !1_q

+ 1

n!

Z1

1− 1 2m

(1−t)ndt !_p1

Z1

1− 1 2m

(1−t)n

g

(n)

(x2+t(x1−x2)) q

dt !1_q

+ 1

n!

2m−1−1 X

l=1 

 Z 2₂l+1m

2l−1 2m

t− l

2m−1 n dt !1_p

× Z 2₂l+1m

2l−1 2m

t− l

2m−1 n g

(n)

(x2+t(x1−x2)) q

dt !1_q



≤ 2

−m(n+1)

(n+ 1)!

2−m(n+ 1) (n+ 2)

g

(n)

(x1) q +

1−2

−m

(n+ 1) (n+ 2)

g

(n)

(x2)

q1_q

+2 −m(n+1)

(n+ 1)!

1−2

−m₍_n_{+ 1)}

(n+ 2)

g

(n)

(x1) q

+2

−m₍_n_{+ 1)}

(n+ 2)

g

(n)

(x2)

q1_q

+2

−m(n+1)+1

(n+ 1)!

2m−1₋₁ X

l=1

21−m

g

(n)

(x1) q

+ 1−21−ml g

(n)

(x2)

q1_q .

Using (2.3) and (2.16), the result follows.

(1) If m= 1, Inequality2.15 reduces to Inequality 1.4.

Z x2

x1

g(x)dx−

n−1 X k=0  

1 + (−1)k2−2(k+1)

(x2−x1) −k−1

(k+ 1)!

g(k)

₃_x

1+x2

4

(2.17)

+g(k)

_x1_{+ 3}_x2 4

≤ (x2−x1)

n+1

22(n+1)₍_n_{+ 1)!}

_n_{+ 1}

4n+ 8

g

(n)₍_x 1) q +

₃_n_{+ 7}

4n+ 8

g

(n)₍_x 2)

q1q

+2 (x2−x1) n+1

22(n+1)₍_n_{+ 1)!}

g(n)(x1) q

+g(n)(x2) q

2

!1q

+ (x2−x1) n+1

22(n+1)₍_n_{+ 1)!}

₃_n_{+ 7} 4n+ 8

g

(n)₍_x1₎ q

+

_n_{+ 1} 4n+ 8

g

(n)₍_x2₎

q1q

(9)

onJ◦ andx1,x2∈J◦ withx1< x2.If g(n)∈L1([x1, x2])andg(n)

is quasi-convex on[x1, x2], then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1)−k−1(k+ 1)! (2.18)

×g(k)

₍₂_l

−1)x1+ (2m−2l+ 1)x2

2m

≤ (x2−x1)

n+1

2mn₍_n_{+ 1)!} max n

g

(n)₍_x1₎ ,

g

(n)₍_x2₎ o

.

Proof. Using quasi-convexity of g(n)

Z 1

0

(2.19)

≤ max

g(n)(x1)

,

g(n)(x2)

n!

" Z 21m

0 tndt

+

2m−1−1

X

l=1

Z 22l+1m

2l−1 2m

t− l

2m−1

n

dt+ Z 1

1− 1 2m

(1−t)ndt





= 1

2mn₍_n_{+ 1)!}max n

g

(n)₍_x1₎ ,

g

(n)₍_x2₎ o

.

on J◦ and x1, x2 ∈ J◦ with x1 < x2. If g(n) ∈L1([x1, x2]) and g(n) q

is quasi-convex on [x1, x2], where

q >1,then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1) −k−1

(k+ 1)! (2.20)

×g(k)

₍₂_l₋₁₎_x1_{+ (2}m₋₂_l_{+ 1)}_x2 2m

≤ (x2−x1)

n+1

2mn_n_!

₁

1 +pn

p1 maxn

g

(n)₍_x 1)

q

, g

(n)₍_x 2)

qo1q

,

(10)

Proof. Using Holder’s inequality and quasi-convexity ofg(n) q

Z 1

0

(2.21)

≤ 2

−m(n+1)+1

n!

₁

1 +pn

p1 maxn

g

(n)₍_x 1)

q

, g

(n)₍_x 2)

qo1q

+ 2m−1−1 maxn

g

(n)₍_x 1)

q

, g

(n)₍_x 2)

qo1q

= 1

2mn_n_!

₁

1 +pn

1p maxn

g

(n)₍_x1₎ q

,

g

(n)₍_x2₎

qo1q

on J◦ and x1, x2 ∈J◦ with x1 < x2. If g(n) ∈ L1([x1, x2]) andg(n) q

is quasi-convex on [x1, x2], where

q≥1, then

Z x2

x1

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(x2−x1)−k−1(k+ 1)! (2.22)

×g(k)

₍₂_l₋₁₎_x1_{+ (2}m₋₂_l_{+ 1)}_x2 2m

≤ (x2−x1)

n+1

2mn₍_n_{+ 1)!}

maxn g

(n)₍_x 1)

q

, g

(n)₍_x 2)

qo1q

Proof. Using quasi-convexity ofg(n) q

Z 1

0

(2.23)

≤ 2

2mn+m₍_n_{+ 1)!}

maxn g

(n)₍_x1₎ q

,

g

(n)₍_x2₎

qo1q

+ 2

2mn+m₍_n_{+ 1)!} 2

m−1₋₁ maxn

g

(n)₍_x1₎ q

,

g

(n)₍_x2₎

qo1q

= 1

2mn₍_n_{+ 1)!}

maxn g

(n)₍_x 1)

q

, g

(n)₍_x 2)

qo1q

.

3. _{Some Applications}

We start this section with an application to approximation of an integral. Recall that a partitionDof a

finite interval [c, d], c < d,is a finite sequence of numbersc=c0< c1<· · ·< cn=d.

Proposition 3.1. Suppose that n and m are positive integers, g : J ⊆ R → R is n-time differentiable

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g(n)_∈_L1_([_{c, d}_])_and g(n)

is convex on [c, d], then

d Z

c

g(x)dx=A(g, D) +E(g, D), (3.1)

where

A(g, D) (3.2)

= n−1

X

j=0 2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(cj+1−cj)−k−1(k+ 1)!

×g(k)

₍₂_l₋₁₎_c

j+ (2m−2l+ 1)cj+1

2m

,

and

|E(g, D)| (3.3)

≤ 1

2mn+1₍_n_{+ 1)!}

n−1

X

j=0

(cj+1−cj) n+1

g

(n)₍_c

j) +

g

(n)₍_c

j+1)

.

Proof. For eachj= 0,1,· · ·, n−1,applying Theorem2.1over the interval [cj, cj+1],we have

cj+1

Z

cj

g(x)dx−

2m−1

X

l=1

n−1

X

k=0





1 + (−1)k2−m(k+1)

(cj+1−cj)− k−1

(k+ 1)! (3.4)

×g(k)

(2l−1)cj+ (2m−2l+ 1)cj+1

2m

≤ (cj+1−cj)

n+1

2mn₍_n_{+ 1)!} "

g(n)(cj) +

g(n)(cj+1)

2

#

.

Note that

E(g, D) = d Z

c

g(x)dx−A(g, D)

= n−1

X

j=0



  

 

cj+1

Z

cj

g(x)dx−

2m−1

X

l=1

n−1

X

k=0

1 + (−1)k2−m(k+1)

(cj+1−cj)

−k−1

(k+ 1)!

×g(k)

₍₂_l₋₁₎_c

j+ (2m−2l+ 1)cj+1

2m

.

Using the triangle inequality and Inequality3.4, we get that

|E(g, D)| ≤ 1

2mn+1₍_n_{+ 1)!}

n−1

X

j=0

(cj+1−cj) n+1

g

(n)₍_c

j) +

g

(n)₍_c

j+1)

.

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Proposition 3.2. Let X be a random variable taking its values in the finite interval[c, d], where0< c < d,

with a probability density function g : [c, d] → [0,1]. If g is n-time differentiable, g(n) ∈ L1([c, d]) and

g(n)

is convex on [c, d],then for any positive integerm,

E(X)−



d−

2m−1

X

l=1

n X

k=1





1 + (−1)k2−m(k+1)

(d−c)−k−1(k+ 1)!

(3.5)

×g(k−1)

₍₂_l₋₁₎_c_{+ (2}m₋₂_l_{+ 1)}_d 2m

− 2

−m+1

(d−c)−1

2m−1

X

l=1

Pr

X ≤ (2l−1)c+ (2

m₋₂_l_{+ 1)}_d

2m





≤ (d−c)

n+2

2m(n+1)₍_n_{+ 2)!}

" g(n)(c)

+

g(n)(d)

2

#

,

whereE(X)is the expectation of X.

Proof. Let G(x) = R_cxg(t)dt for x ∈ [c, d]. Using integration by parts and the facts that G0 = g and

G(d) = 1,we get that

E(X) =d−

Z d

c

G(t)dt.

SinceG(k+1)₌_g(k)_,₀_≤_k_≤_n_{, applying Theorem}_2.1_on_G_{, we have}

Z d

c

G(t)dt−

2m−1

X

l=1

n X

k=1





1 + (−1)k2−m(k+1)

(d−c)−k−1(k+ 1)! (3.6)

×g(k−1)

₍₂_l₋₁₎_c_{+ (2}m₋₂_l_{+ 1)}_d 2m

− 2

−m+1

(d−c)−1

2m−1

X

l=1

Pr

X ≤ (2l−1)c+ (2

m₋₂_l_{+ 1)}_d

2m

≤ (d−c)

n+2

2m(n+1)₍_n_{+ 2)!}

" g(n)(c)

+

g(n)(d)

2

#

.

Thus,

E(X)−



d−

2m−1

X

l=1

n X

k=1





1 + (−1)k2−m(k+1)

(d−c)−k−1(k+ 1)!

×g(k−1)

₍₂_l₋₁₎_c_{+ (2}m₋₂_l_{+ 1)}_d 2m

− 2

−m+1

(d−c)−1

2m−1

X

l=1

Pr

X ≤ (2l−1)c+ (2

m₋₂_l_{+ 1)}_d

2m





≤ (d−c)

n+2

2m(n+1)₍_n_{+ 2)!}

" g(n)(c)

+

g(n)(d)

2

#

.

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Remark 3.1. In Proposition3.2, if m= 1 then Identity3.5reduces to

E(X)−d+ (d−c) Pr

X ≤ c+d

2

(3.7)

+ n X

k=1

1 + (−1)k2−(k+1)

(d−c)−k−1(k+ 1)!

g(k−1)

_c₊_d

2

≤ (d−c)

n+2

2(n+1)₍_n_{+ 2)!}

" g(n)(c)

+

g(n)(d)

2

#

.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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