Perturbed Companions of Ostrowski type inequalities for 𝑁-times differentiable functions and applications Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 9 (27), No 2, 2020, pp. 45-57 45

DOI: 10.15393/j3.art.2020.7130

UDC 517.2, 517.38, 517.51

S. Erden

PERTURBED COMPANIONS OF OSTROWSKI TYPE INEQUALITIES FOR N-TIMES DIFFERENTIABLE FUNCTIONS AND APPLICATIONS

Abstract. We firstly examine some inequalities obtained by using sets of complex-valued functions for functions whose high order derivatives are restricted. We also give some approximations for the functions whose derivatives up to the order n — 1 (n > 1) are continuous and whose the nth derivatives are of bounded variation. So, the results provide extensions of those presented in earlier works.

Key words: Function of bounded variation, Perturbed Ostrowski type inequalities

2010 Mathematical Subject Classification: 26D15, 26A45, 26D10

1. Introduction. The inequality discovered by Ostrowski in 1938 has been studied by a large number of researchers due to its comprehensive application fields in numerical analysis and certain special means. This inequality [21], established by using mappings whose first derivatives are bounded, is stated in the following manner.

Theorem 1. Let f : [a,b] ^ R be a differentiable mapping on (a,b),

f' : (a,b) ^ R is bounded on (a,b), i.e. H/'ll^ := sup |/' (i)| < Then

te(a,b)

we have the inequality

f (*) - I f (t)dt < 1 + V" 2i ' (b - o) H/'llc

w .......r 1 (X - *?)

b — a

L4 (b - a)

for all x E [a,b]. The constant 1 is the best possible.

Over the years, interested researchers have studied it to provide novel refinements, improvements, and generalizations of the inequality (1). For

© Petrozavodsk State University, 2020

instance, some authors deduced new Ostrowski-type inequalities for differentiate, twice differentiable, or higher-order differentiable functions in [7], [8], [9], and [22] (see also references therein). On the other side, the perturbed method has been much used to generalise integral inequalities. For example, after Dragomir had published his paper [14] involving the perturbed inequality of the Ostrowski type established by utilizing absolutely continuous functions, some authors focused on perturbed integral inequalities for twice and higher order differentiable mappings in [5], [16], [17], [18], and [19]. What is more, some companion perturbed inequalities for various assumptions of the functions are refined by using three- and five-step quadratic kernels in [15], [23], and [24].

In particular, some mathematicians focus on the Ostrowski-type inequalities obtained by using mappings of bounded variation, as well as the other function species. In the reference [11], Dragomir introduced the following useful result for functions of bounded variation:

Theorem 2. Let f : [a, b] ^ R be a mapping of bounded variation on [a,b]. Then

f (t)dt - (b - a) f (x) < [1(b - a)+ x - ^ \\J (f) (2)

a

a

holds for all x e [a,b]. The constant 1 is the best possible.

Morever, Dragomir indicated the original generalisation of the Ostrowskitype results for functions that are of bounded variation in [10]. Afterwards, results pertaining to the inequality (1) for functions whose first derivatives are of bounded variation, are given in [1], [6], and [20]. Also, certain generalized outcomes for mappings that possess n-th derivatives of bounded variation, are established in [3] and [13]. In addition to all the results, some companion versions of perturbed results concerning Ostrowski's inequality for bounded-variation mappings are examined in [2], [4], and [12].

We also note that Dragomir established the following identity, so as to observe some perturbed outcomes of Ostrowski-type inequalities in [14].

Theorem 3. Let f : [a, b] ^ C be absolutely continuous on [a,b] and x e [a, b]. Then, for any complex numbers Ai(x) and X2(x), we have

x b

If. If,

1 (t - a) [f(t) - X1(x)] dt + -- (t - b) [f(t) - X2(x)] dt =

b — a I b — a

b

1 1 f = I{%) + _ a) [{b - xf \2(x) - (x - a)2 - J f'(t)dt

a

where the integrals in the left-hand side are taken in the Lebesgue sense.

The primary purpose of this work is to deduce original inequalities for functions, whose higher-order derivatives are limited. For this, some approximations are examined with the help of the identity obtained by utilizing higher-order differentiable mappings. So, the new companion results are derived, regarding inequality (1) for functions whose n-th derivatives are bounded and of bounded variation. Relations between these results and inequalities given in the earlier works are also examined.

2. The case when f(n) is bounded. Before we can establish the inequalities that will be given in this section, we should mention the following identity.

Lemma 1. Let f : [a, b] ^ C be an n-time differentiable function on (a,b). Then, for any complex numbers Xi(x), i = 1,2,3 and all x e [a, a+r], we have the identity

X ^

r (t - a) ri(n)

J n!

a

[f (n)(t) - dt+

a+b—x

+ J l - n [f ^ - W] dt+

+ j [f(n)(t) - Asfr)] dt =

a+b—x ^

= S(f : n,x) - R(n, x) + (-1)n j f (t)dt, (3)

a

where S(f : n,x) and R(n,x) are defined by

n— (-1)n+1 [f™(a + b - x) + (-1)kf(k\x)] b(J : n,x) = (u + 1)! x

(k + 1)!

(x - a)k+l + (-1)k{^ - x)

k=0

'a + b \k+11 ~2

and

R(n,x) = [X1(x) + (—1)*%0r)] ^^ + [1 + (—1)»] (a-f — x)n+\

Proof. Combining the resulting identities by using fundamental analysis operators, after applying integration by parts n times to the three integrals in the right-hand side of the equality (3), the required identity can be easily obtained. □

The expression S(f : n, x) (4) will be used throughout this paper. Furthermore, we define the sets of complex-valued mappings, for 7, r E C and an interval of real numbers [a, b],

U M(7, r) := {/ : [a, b] ^ C | K [(r — f(t)) (M — 7)] > 0}

for almost every t e [ a, b] and

Am (7, r) :=

[a, b] ^ C

( )-

7 + r

2

< 1|r-71}

for a. e. t e [a, b]

Also, we shall give the following lemma so as to prove the next inequality.

Lemma 2. [14] For any 7, r E C, 7 = r, the sets Um (7, r) and A[a,&] (7, r) are nonempty, convex, and closed convex sets and

Uw (7, r) = AM] (7, r).

Theorem 4. Let f : [a, b] ^ C be an n—time differentiable function on (a, b) and x e [a, . If there exists 7^, r E C with 7^ = r, i = 1, 2, 3, such that

f(n) e A[0,a] (71, ri) n a[x,a+b_x] (72, r2) n a[a+b_xm (73, r3), (5) then we have the inequality

S(/ :niX) - [1 + (-1)™] 72#±l2M(E+l - xv+1

2(n + 1)! V 2

7i(x) + (-1)n7a(x) + ri (x) + (-lm^r)

(x - a)

n+i

(n + 1)!

2

+ (—1)™/ f(t)dt

<

\ra+i

\ra i -

x — a) e2

2 (n +1)! (n +1)!V 2

-r (6)

where £i = |r^(x) — 7i(x)| , £2 = ^(x) — 72 (^)| , £3 = ^(x) — 73| .

Proof. If we take absolute value of both left- and right-hand side of (3) for

iO^ 72 (x) + r2(x^ 73(x) + r3(x) —, A2(X) = ---, A3(x) = —

7i(x

Ai(x) =

2

we get the inequality

2

2

S (f:n,x) — [1 + (—1)ra ]

72(x) + r2(x) fa + b 2(n + 1)! V^

(¥ — ')

ra+i

7i (x) + (—1)ra 73 (x) + ri (x) + (—1)ra r3 (x)

(x — a)

ra+i

+ (—1)ray f(t)dt

a

a-\-b—x

<

(t — of

n!

f(n)(t) —

(n + 1)! 7i (x) + ri (x)

2

dt+

+ I n

a + b

+

f(n)(t) — (b — t)ra

72 (x) + r2 (x)

2

dt+

n!

a+b—x

j(n)^) — 73 0*0 + r3 (x)

dt.

Utilizing condition (5), on account of the definition of A[a,fc] (7, r), we write the inequality

(t — a)r

n!

f{n)(t) —

7i(x) + ri(x)

dt < 1 |ri(x) —7i(x)i i dt =

2 n!

a

1 ( T — a)n+i = ^ lri(^) — 7i(x)|iF+V.

Similarly, the results of the other integrals can also be obtained. Thus, the proof is completed. □

Corollary 1. To get the following inequalities, we use:

2

X

ri

2

2

X

X

2

the Holder inequality

i i mn + pq < (ma + pa)" (n3 + q3)13

1 + 1

a ~ 3

X-Y

where m,n,p,q > 0 and a > 1 with a + 3 = 1; the identity

X + Y

max {X, Y} = + +

- the maximum property of max {an, bn} = (max {a, b})n for a,b > 0 and n e N in the left-hand side of inequality (6).

The obtained inequalities are

12(x) + T2(x) /a + b \n+1

Ax)/a + b y

s(f-.n,x) - [i + w 2{n + 1)l [— -x

7i(x) + (-1)n73(x)+rx(x) + (-lfr(x)i (x-a)n+1

+(-i)n h m

1

(x - a)n+1 + - x)

<

(n + 1)! 1

ünTT)!

, n+ln

( n+1)!

l + 3

ma^i —-—, e 2 >,

(x - a)(n+r)p + - x)

(n+1)p

I 2

l

p

forp > 1, p + 1 = 1,

p

k (n +1)!

b — a . 3 a + b + \x -

4

4

n+1 r^l + ^3 . 1

I 2 ' °2J

where £i = |Ti (x) - 71 (x) , £2 = (x) - 72 (x) , £3 = ^3 (x) - 73 (x) .

Remark 1. Let f and x be defined as in Theorem 4. If there exists ji, r e C with ji = ri, i = 1, 2, such that

f[n) e A

] (71, ri) n A[ x, a+ — x ] (72, r2) n A[a+b—x,b] (7U ri) ,

then we have

s(f:n,x) - [1 + (-1)n]

72(x) + r2(x) fa + b ^\n+1 2(n

-x)

2

1

\

[1 + (-1)ra][7i(^) + ri(:r)] (x — a)

n+1

2

<

+ (-1)n / f(t)dt

(n + 1)

(n + 1)!

" , Nn+1 fa + b \n+r ei(x — a) + e2[ —--xj

<

(7)

where e1 = |r1 (x) — 71 (x)| and e2 = |r2 (x) — 72 (x)| . Additionally, if there exists 7^ r1 G C with 71 = r1 such that /(n) G A[a,fe] (7^ r1), then we have the conclusion

S(f:n,x) + (—1)nJ f(t)dt— [1 + (—1)n] [71 (X) + r1(x)]

2( n

— ^ + (x — a)n+1

<

<

|r1(x) — 71(x)| (n +1)!

a + 6 \n+1

2

— xj +(x — a)n+1 . (8)

Remark 2. If we select x = a in inequality (6), we have

z (—1)n+1 i /"=> (6) + (—1)

E-

k=Q

( k + 1)!

'b - a\k+1i

b

72(x) + r2(x) ( — x)"+1 + (—1)-| f(t)dt

a

|r2 (x) — 72 (x)| fb — a \n+1

— '1 + <—1>nl 2(n + 1)! ^ — x

<

<

(n + 1)!

Ç) — a^ n+1

a + b

Remark 3. If we take x = —-— in the inequality (6), then one concludes

the inequality

2

g (— ^ ? ^ ( ^ r + (—1)n j fm —

k=Q

71 (x) + (—1)n73 (x) + r1 (x) + (—1)n r (x)

2

( — a)

n+1

2n+1 (n + 1) !

<

1

k

n+i

< |ri (x) - 7i (x)| + |r3 (x) - 73 (x)| ^b - a\

Also, should we use the condition of the result (7) in this inequality, then we can find a new inequality.

3 a + b

Remark 4. Substitution of x = —-— in (6) gives

n-i (-l)^1 [l + (-1)k] f(k) (*+*) + (-1)k (6 -a)

fc=Q

.fc+i

4 fc+1 ( k + 1)!

72 (x) + r2 (x) ( b-a\ n I

- [1 + (-1)] 2 (n +1)! \—J +(-1) J f(t)dt-

n+1

7i (x) + (-1)nj3 (x) + ri (x) + (-1)n r3 (x)

2

(b -a)

n+i

4n+i (n + 1) !

<

<

(n + 1)!

)

n+i

i + 3 -^--+ £ 2

where £x = |ri (x) - (x)| , £2 = |r2 (x) - 72 (x)| and £3 = ^3 (x) - 73 (x)| What is more, applying the condition of the result (8) to this inequality, a new inequality can be found.

In addition to these results, one can deduce some inequalities, taking n = 1 in inequality (6) or the other results related to (6); these inequalities were published by Dragomir [15]. Furthermore, if we take n = 2 in (6) or the other results connected to (6), then we obtain some inequalities presented in [23] that is published by Sarikaya et. al.

3. The case when /(ra) is of Bounded Variation. We begin with the definition of bounded-variation functions and the concept of total variation, which is used throughout this section.

Definition 1. Let P : a = x0 < x1 < ... < xn = b be any partition of [a, b] and let Af(xi) = f(x+) — f(xi); then f is said to be of bounded variation, if the sum

£ |Af(xi)l

=i

is bounded for all such partitions.

1

Definition 2. Let f be of bounded variation on [a, b], and A f( P)

n

denote the sum |A f(xi) | corresponding to the partition P of [a, b].

i=1

The number

b

V (/) := sup{£ A f ( P) : P e P([a, b])} ,

a

is called the total variation of f on [a, b]. Here P([a, b]) denotes the family of partitions of [ , ].

Now, a perturbed inequality of the Ostrowski type for functions whose high-order derivatives are of bounded variation, are established in the following theorem.

Theorem 5. Let f : I ^ C be an n time differentiable function on 1° and [a, b\ C 1°. If the n-th derivative /(n) is of bounded variation on [a, b], then we have

b n S ( f:n,x) + (-1)n[ f(t)dt- [f (n)(a) + (-1)n f(n)(b)} (-X—T"

)n+1

(n+iyr

- [1 + (-1)n]

/(n)(x) + /(n)(a + b -x) fa + b \n+1

2 (n + 1)!

-x)

<

<

( x - a)

n+1

t(n + 1)!

V(/(n))+ V ( /(n))

a a+b—x

1 fa + b

+

a+b—x

^ -x)n+1 V (/(n)) (9

+ (n +1)!^ 2 x

for any x e [a, Op].

Proof. Writing f(n)(a), (f(n)(x) + /(n)(a + b — x))/2, f(n)(b) instead of A1(x), A2(x), A3 (x) in equation (3) respectively, then taking modulus of this equality, we find that

S(f :n,x) + (-1)n I f(t)dt - |/n)(a) + (-1)n/(n)

( x - a)

n+1

(n +1)!

- [1 + (-1)nP

f(n)(x) + f(n)(a + b -x) fa + b \n+i

2 (n + 1)!

- x)

<

<

+

(-L-^\f{n)(t) - fH -a) \dt+ \ fn) -1) - fn) - b) \dt+

a+b—x

a+b—x

+ s n

a + b

!(n\t) -

a + b — x)

dt.

Noting that /(n) : 1° ^ C is of bounded variation on [a,x], we get

X

\ f(n)(t) — f(n)( a) \< V(/(n))

and observe that

X +1

r ( t- a)n dt = (x — a)n

J n\ (n + 1)! '

a

The other integrals are also examined by noting that /(n) : 1° ^ C is of bounded variation on [x,a + b — x] and [a + b — x, b]: we can find the result (9), which finishes the proof. □

Remark 5. Suppose that all assumptions of Theorem 5 hold. If we take x = a in the inequality given this theorem, we have

n-1 -—l)n+1 [f(k)-b) + -—1)kf(k)-a)] r (b — a^+1

— [1 + -—1)n]

a + b — x) fb — a \n+'

2-n

<

(^T' + -—irj f-t)dt

a

1 J (^ "x)"+I V (f <■>)

-n + 1)W 2

In addition, if we choose x = a+r, we get the midpoint inequality

n—I -—1)n+If(k) (a2) [1 + -—1)k

k=0

- k + 1)J

^b — a

-2T)

x

n

2

2

- [/(ra)(«) + (-1)" f(n)

Finally, should we take x

( - )

n+1

2n+i(n + 1)!

+ -1)n / f(t)dt

3 a + b 4

, we have

<

<

(6 -a)

n+1

2n+1 (n + 1)

V (/(n)).

E

k=Q

1 (-1)n+1 [1 + (-1)k] f(k) (^) + (-1)fc/(k) (^) (b - a)k+1

4 fc+1( fc + 1)!

b

+ (-1)n i f(t)dt- [f (n)(a) + (-1)n /(n)

(6 -a)

n+1

4n+1 (n + 1)!

- t1 + (-1)nj Z(n)( ^) + Z(n)() £ b_-_a ^n+1

2 (n + 1)!

<

<

t( ^^ )n+1 V (n)).

(n + 1)

Besides the results that are presented in this section, taking n =1 in the inequality (9) or the other results pertaining to (6), we obtain some inequalities given in [15] by Dragomir. What is more, should we take n = 2 in expression (6) or the other results interested in (6), we can find some inequalities presented in [23] by Sarikaya et. al.

References

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Received October 14, 2019.

In revised form, April 11, 2020.

Accepted April 11, 2020.

Published online April 18, 2020.

Bartin University, Faculty of Science, Department of Mathematics,

Bartin-TURKEY

E-mail: [email protected]