On inequalities related to some quasi-convex functions Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 4(22), No. 2, 2015, pp. 45-64

DOI: 10.15393/j3.art.2015.2869

45

UDC 517.54, 517.542

Z. Liu

ON INEQUALITIES RELATED TO SOME QUASI-CONVEX

FUNCTIONS

Abstract. Estimations of errors in inequalities related to some quasi-convex functions in literature are simplified. Two new general inequalities for functions whose n-th derivatives for any positive integer n in absolute values are quasi-convex have been established. Some special cases are discussed with applications in numerical integration and special means.

Key words: inequalities, quasi-convex function, Simpson type rule, numerical integration, .special means

2010 Mathematical Subject Classification: 26D15

1. Introduction. It is well known that a function f : [a, 6] ^ R is called quasi-convex on [a, 6] if

f (Ax +(1 - A) y) < max{f (x) ,f (y)}

for all x,y E [a, 6] and A E [0,1] (e.g., see [1] and [2]). Thus we see clearly that if f : [a, 6] ^ R is quasi-convex on [a, 6] then for any t E [a, 6] we have

f (t) < max{f (a) ,f (6)}.

It should be noticed that any convex function is a quasi-convex function and there exist quasi-convex functions which are neither convex nor continuous (e.g., see [3] and [4]).

Along this paper, we consider a real interval I C R, and denote that I° is the interior of I.

In [5-7], we see the following three inequalities for quasi-convex functions.

©Petrozavodsk State University, 2015

[MglHl

Theorem A. [7, Theorem 6] Let f : I C R ^ R be a differentiable mapping on I°, a, b G I° with a < b and f' G L1[a,b]. If |f'|q is quasi-convex on [a, b] and q > 1, then the following inequality holds:

f (a) + 4f (^) + f (b)] - — f f (t) dt < ^ (max{|f' (a) |q, |f' (b)|q})1 .

<

(1)

Theorem B. [5, Theorem 4] Let f : I C R ^ R be a twice differentiable

i P

mapping on I°, a,b G I° with a < b and f'' G L1 [a, b]. If |f ''| p-1 is quasi-convex on [a, b], for p > 1, then the following inequality holds:

- ^ f f (t) dt

<

<

(b-a)2 / VM M r(1+p)

< 8

r( 3 +P)

(2)

(max{|f'' (a) |q, |f'' (b) |q})1

where q =

p

p - 1

Theorem C. [6, Theorem 3] Let f'' : I C R ^ R be an absolutely continuous function on I° such that f''' G L1[a, b], where a,b G I° with a < b. If If'''^,q = p—1 is quasi-convex on [a, b], for some fixed p > 1, then the following inequality holds:

If (t) dt - ^ f (a)+ 4 Z^f) + f (b)

<

<

2 P (b — 48

(B (p +1, 2p +1))p I" (max{|f''' (a) |q, |f''' (a++b) |q})1 +

+ (max{|f''' (b) |q, |f''' |q})

1 n

q

(3)

It should be noticed that

(max{|f' (a) |q, |f' (b) |q})q = max{|f' (a) |, |f' (b) |},

which has been overlooked in the literature (see e.g., [3-13]). The inequalities (1), (2) and (3) have a uniform bound independent of q. Indeed, for any q > 0 and positive integer n, |f(n) |q is quasi-convex on [a, b] if

2

4

and only if |f(n) | is quasi-convex on [a, 6]. Thus, instead of Theorem A, Theorem B and Theorem C, we actually just have the following three theorems as:

Theorem 1. Let f : I C R ^ R be a differentiate mapping on I°, a, 6 G I° with a < 6 and f' G L1 [a, 6]. If |f is quasi-convex on [a, 6], then the following inequality holds:

f (a) + 4f (a+b) + f (&)] - ¿a J f (t) dt

max{|f' (a) If' (b) |}.

<

< 5(b-a) — 36

(4)

Theorem 2. Let f : I C R ^ R be a twice differentiable mapping on I°, a, 6 G I° with a < 6 and f'' G L1 [a, 6]. If |f'' | is quasi-convex on [a, 6], then the following inequality holds:

b 2 f <a) + f <6) thi ff (i) dt| * T max{lf" («) I, If" (6) !>• (5)

Theorem 3. Let f" : I C R ^ R be an absolutely continuous function on I° such that f''' G L1 [a, 6], where a, 6 G I° with a < 6. If |f"'I is quasi-convex on [a, 6], then the following inequality holds:

(b-a)4 — 1152

<

J f (t) dt - If (a) + 4( /Mfi) + f (b)

a

max{|f"' (a) |f''' (a+b) |} + maxf (a+b) |f''' (b) |}

(6)

In this work, we will derive two new general inequalities for functions whose nth derivatives for any positive integer n in absolute values are quasi-convex, which provide some generalizations of the above three inequalities and some other interesting inequalities as special cases. Some applications in numerical integration and to special means are also given.

2. The Results.

Lemma. (see [14]) Let f : [a, 6] ^ R be such that the (n — 1) th derivative

f (n-1) (n > 1) is absolutely continuous on [a, 6] and f(n) G L1[a, 6]. Then

we have the identity

J f (x) dx = 0f (a) + 2 (1 - 0) f (o+b) + 0f (b)

a

+ V [1 - (2k + 1) 0] (b- a)2k+1 f (2k) / a+b \ +

+

k=1

(2k + 1)!22k + (-1)n f Kn (x,0) f(n) (x) dx,

(7)

where 0 G [0,1] and

\TO— 1

f (x - a)n 0 (b - a) (x - a)n _ if x G [a a+b]

K (x 0) = ; n! 2(n - 1)! _ 1 " ' (x - b)n + 0 (b ~ a)(x ~ b)n , if x G (a+b,b].

n!

2 (n - 1)!

(8)

Theorem 4. Let f : [a, b] ^ R be such that the (n - 1) th derivative f(n-1) (n > 1) is absolutely continuous on [a, b] and f(n) G L1[a, b]. If |f (n)| is quasi-convex on [a, b], then we have

J f (x) dx - ^ [0f (a) + 2 (1 - 0) f (a+b) + 0f (b)]-

-

^ [1 - (2k + 1) 0] (b - a)

2k+1

_f (2k)

a+b

k=1 (2k + 1)!22k V 2

< I (n, 0) max{|f(n) (a) |, |f(n) (b) |},

<

(9)

where 0 G [0, 1] and

I (n, 0) =

[1 - (n + 1) 0 + 2nn0n+1] (b - a)

(n + 1)!2n [(n + 1) 0 - 1] (b - a)n+1 (n + 1)! 2n ,

n+1

n< 1,

n > 1.

(10)

Proof. From (7) of the Lemma, we have

J f (x) dx - ^ [0f (a) + 2(1 - 0) f ( ) + 0f (b)

a

V [1 - (2k + 1) 0] (b - a)2k+1 f(2fc) (a + b

i'OL I 1 M 02k f

(2k + 1)! 22k V 2

b b

J Kn (x, 0) f(n) (x) dx < / (x, 0) f(n) (x) | dx < (11)

Jb

b] |f(n) (x) |/|Kn (x,0) | dx <

a

< max{|f(n) (a) |, |f(n) (b) |} J |Kn (x, 0) | dx.

By elementary calculus, it is not difficult to get the following results:

\«+1

[1 - (n +1) 0 + 2nn0n+1 ] (b - a)n 1

, n < 1,

|Kn (x,0) | dx = ^ ++1)An

n > 1.

[(n +1) 0 - 1] (b - a)n+1 _ 1

(n + 1)!2n - e'

(12)

Consequently, the inequality (9) with (10) follows from (11) and (12). The proof is completed. □

Corollary 1. Let f : [a, b] ^ R be such that the (n - 1) th derivative f(n-1) (n — 1) is absolutely continuous on [a, b] and f(n) G L1[a, b]. If |f (n)| is quasi-convex on [a, b], then we get a midpoint type inequality

f / (x) dx - b—. / (a+b) - | jL+gk f <*> ( ^) <

(b - an+1

(b - a)n+1

((n+1))!2n max{|f(n) (a) |, |f(n) (b) |},

a trapezoid type inequality

J f (x) dx - ^ [f (a) + f (b)] + E (kk(b+ 1a!)22k-1 f(2k) ( < frV^ max{|f(n) (a) |, |f(n) (b) |},

b

2

a Simpson type inequality

b b — a f f (x) --—

n 6

f (a) + 4f ( —+— ) + f (b)

+

+ E

(k — 1) (b — a)

2k+1

3(2k + 1)! 22k-1

_f (2k)

a + b

<

where

< /(n, 3)max{|f(n) (a) |f(n) (b) |}

J(n, 3

316,

81 , +1 (n-2)(b-a)n+1 3(n+1)!2n

n = 1, n = 2,

n3

and an averaged midpoint-trapezoid type inequality

where

J f (x) dx —

b — a rf (a) + 2^a + b

+ f (b)

+ E

(2k — 1) (b — a)

2k+1

k=1 (2k + 1)!22k+1

_f (2k)

a + b

<

< /(n, 2)max{|f(n) (a) |, |f(n) (b) |},

J(n, ¿1 =

1

48

n = 1,

(n-1)(b-a)n+1 n> 2 (n+1)!2n+1 , n - 2-

+

Proof. Set e = 0,1,1,1 in (9) and (10). □ 32

Remark 1. For n = 1, we have

ff (x) dx —ef (a)+ 2(1 — e) f ( ^ ) + ef (b)

a

< 1 — 2e + 2e2 (b — a)2 max{|f' (a) |, |f' (b) |}.

2

5

2

If we take 6 = 0,1, -, - in (13), then we get a midpoint inequality

3 2

f (x) dx — (b — a) f

a+b

< max{|f(n) (a) |, |f(n) (b) |},

a trapezoid inequality b

f f (x) dx — b—- [f (a) + f (6)]

<

(b — a)2

max{|f' (a) |, |f' (b) |},

a Simpson inequality

b b — a / f (x) dx--—

a 6

5 (b — a)

<

36

f (a) + 4f -+— ^ + f (b)

>

- max{|f' (a) |, |f' (b) |}

<

(14)

which recapture the inequality (4), and an averaged midpoint-trapezoid inequality

b b — a If (x) dx--1—

f (a) + 2f ( ~"+"" ) + f (b)

<

< ^ max{|f' (a) |, |f' (b) |}.

Remark 2. For n = 2, we have

s f (x) dx —b-a [ef (a) + 2(1 — e) f + ef (b) a < J (2, e) max{|f'' (a) |, |f'' (b) |},

<

where

J (2,e) =

11

[1-3e+8g3](b-a)3 24 '

[30- 1](b-a)3 24 ,

n< 2, n - 2.

(15)

(16)

If we take 6 = 0,1,77, 7: in (15) and (16), then we get a midpoint inequality

3 2

f (x) dx — (b — a) f

a+b

<

(b — a)3

24T~

max{|f'' (a) |, |f'' (b) |},

b

2

b

2

a trapezoid inequality

f (x) dx - ^ [f (a) + f (b)]

<

(b - a)3 12

max{|f" (a) |, |f" (b) |},

which recapture the inequality (5), a Simpson inequality

f ( x ) dx -

b - a 6

f (a) + 4f

a+b 2

+ f (b)

< max{|f" (a) |, |f" (b) |}

<

(17)

and an averaged midpoint-trapezoid inequality

f ( x ) dx -

ba

f (a) + 2f

a+b 2

+ f (b)

< ^^^ max{|f" (a) |, |f" (b) |}.

Remark 3. For n = 3, we have

<

J f (x) dx - b-a 0f (a) + 2(1 - 0) f (a+^) + 0f (b)

(1 - 30) (b - a)3 „ /a + b

24

f '

<

< J (3, 0) max{|f(a) |, |f" (b) |},

(18)

where

J (3,0) =

11

[1-40+5404](b-a)4

192 „ [40-1](b-a)4 192 ,

n< 2, n > 2.

(19)

If we take 0 = 0,1,77, o in (18) and (19), then we get a midpoint type 32

inequality

b ,3

J f (x) dx - (b - a) f ( a+b ) - f " ( a+b )

<

(b - a)4 192

max{|f"' (a) |, |f"' (b) |},

b

2

a trapezoid type inequality

/ f (x) dx — ^ [f (a) + f (b)] + f'' (a+b)

<

(b — a)4 64

max{|f''' (a) |, |f''' (b) |},

a Simpson inequality

f ( x ) dx —

b — a 6

f (a) + 4f

a+b 2

+ f (b)

< ^^4 max{|f''' (a) |, |f''' (b) |}

<

(20)

and a midpoint-trapezoid type inequality

b b — a / f (x) dx--—

a4

f (a) + 2f (a) + f (b)

(b — a)

48"

2

+

/ "(a^

<

(b — a)4 192

max{|f''' (a) |, |f''' (b) |}.

Theorem 5. Let f : [a, 6] ^ R be such that the (n — 1) th derivative f(n-1) (n > 1) is absolutely continuous on [a, 6] and f(n) G L1[a, 6]. If |f (n)| is quasi-convex on [a, 6], then we have

sf (x) dx — ^[ef (a) + 2 (1 — e) f (*+*) + ef (b)]—

— ^ [1 — (2k + 1) e] (b — a) k=1

2k+1

<

J (n,e)

2

(2k + 1)!22k max j f(n) (a)

f(2k)

a+b

<

f

(n)

a+b

+

+ max{|f(n) |, |f(n) (b) |}

where e G [0,1] and J (n, e) is as in (10).

Proof. From (7) of the Lemma, we have

If (x) dx - ^ 0f (a) + 2(1 - 0) f + 0f (b)

a

_ sf [1 - (2k + 1) 0] (b - a)2k+1 (2fc) /a+b\

¿1 (2k + 1)! 22k f V 2 ;

b b

J Kn (x, 0) f(n) (x) dx < / |Kn (x, 0) f(n) (x) | dx =

aa

a + b b

= j |Kn (x,0) f(n) (x) | dx + J |Kn (x,0) f(n) (x) | dx <

< max

xe[a, a++b ]

+ max

xe[ a+b ,b]

a + b 2

a + b 2

x) | J |Kn (x, 0) | dx+

|Kn (x, 0) | dx <

a + b 2

a + b 2

< max{|f(n) (a) |, |f(n) (a+^) |} / |Kn (x, 0) | dx+

(22)

+ max{| f(n) (a+b) |,

|Kn (x, 0) | dx,

a+b 2

Observe that

b b J |Kn (x,0) | dx = J |Kn (x,0) | dx = 2 J |Kn (x,0) | dx = ,

a a+b a

2

the inequality (21) follows from (22). The proof is completed. □

Corollary 2. Let f : [a, b] ^ R be such that the (n - 1) th derivative f(n-1) (n — 1) is absolutely continuous on [a, b] and f(n) G L1[a, b]. If |f (n)| is quasi-convex on [a, b], then we get a midpoint type inequality

/ f (x) dx - ^ f (a+b) - i:

(b - a)2k+1 f (2k^ a + b\ k=i (2k + 1)! 22kf V 2 ;

) (a) , f (n)f > +

(b - a)

n+1

(n + 1)! 2n+1

max

<

+ max{|f(n) (a+^) |, |f(n) (b) |}

a trapezoid type inequality

— k (b — a)

2k+1

J f (x) dx — 4=2[f (a) + f (b)] + E (2fc+ !)!22t-1 f<2k>

" < (n^ ^maxf (a)f ^a±*Ï

max { f(n) (a)

f

a+b

+

+ max{|f(n) (a+^) |, |f(n) (b) |} a Simpson type inequality

f (x) dx -

b — a 6

f (a) + 4f

a

+b

+ ^ (k — 1) (b — a)2k+1 (2k) + k=1 3 (2k + 1)! 22k-1 f

2 ;

a+b

+ f (b)

+

<

J(n 3)

max

f(n) (a) , f

(n)

2

a+b

<

+

+ max{|f(n) |, |f(n) (b) |}

where

Jln, 3

36,

81 , +1 (n-2)(b-a)n+1 3(n+1)!2n

n = 1, n = 2,

n 3,

and an averaged midpoint-trapezoid type inequality

f (x) dx -

ba

f (a) + 2f

a+b

+ f (b)

+

+

-T (2k — 1) (b — a)

2k+1

k=1 (2k + 1)! 22k+1

f(2k)

<

J(n, 2)

a+b

<

max{|f(n) (a)|,|f(n) (a+ |} +

+ max{|f(n) |, |f(n) (b) |}

where

n, 2 ' =

48, +1

(n-1)(b-a)n+1 (n+1)!2n+1

n = 1, n > 2.

<

2

2

5

2

2

1

Proof. Set 0 = 0,1,1,1 in (21) and (10). □ 32

Remark 4. For n = 1, we have

b b — a r

J f (x) dx - a 0f (a)+ 2(1 - 0) f

1 - 20 + 202 )2 < -(b - a)

a+b

+ 0f (b)

<

8

+ max

max{|f' (a) |, |f'(^1} +

(23)

f' №) , |f' (b) |

If we take 0 = 0,1, -, - in (23), then we get a midpoint inequality 32

J f (x) dx - (b - a) f ( a+b )

<

<

(b - a)2 8

max {|f' (a) |, |f'(^ |} +

+ max

f' (a+b:

, | f ' (b) |

a trapezoid inequality

If (x) dx - ^ [f (a) + f (b)]

<

<

(b - a)2 8

max

' (a) |,

f'

a+b

+ max

f' (a?) , |f' (b) |

+

a Simpson inequality

Jb b a r J f (x) dx - ^ f (a) + 4f

a+b

+ f (b)

<

5 (b - a)2

72

ma^ |f' (a) |, f

a+b

+ max

f' (a+b

, | f ' (b) |

2

and an averaged midpoint-trapezoid inequality

b b — a If (x) dx--4—

a

f (a) + 2f ) + f (b)

<

(b — a)

2

16 + max

max1 |f'(a) |,

f'

, /a + b

f' (a+b

+

, If' (b) I

<

Remark 5. For n = 2, we have

/f (x) dx — b-a 0f (a)+ 2(1 — 0) f + 0f (b)

<

I (2,0) 2

max {|f'' (a) |,

f'

a+b

+

+ max

f (a+b:

, If'' (b) I

<

(24)

where I (2, 0) is as expressed in (16).

If we take 0 = 0,1, -, - in (24) and (16), then we get a midpoint

3 2

inequality

If (x) dx — (b — a) f (a+b)

<

<

(b — a)

3

48 + max

max {If'' (a) I,

f'

a+b

f (a+b:

, If'' (b) I

+

a trapezoid inequality

If (x) dx — ^ [f (a) + f (b)]

<

(b — a)3 8

max If'' (a) I, f

+ max

f'' №

a+b

, If'' (b) I

a Simpson inequality

Jb b - a

f f (x) dx -

6

f (a) + 4f ( —) + f (b)

<

(b - a)

3

162 + max

max

(a) |,

f

a+b

f'' m , |f'' (b) |

<

+

and an averaged midpoint-trapezoid inequality

Jb b - a

If (x) dx —t~

f (a) + 2f (a) + f (b)

<

(b - a)2

96 + max

max

{|f'' (a) |,

f

a+b

f (a+b:

, |f '' (b) |

<

+

Remark 6. For n = 3, we have

Jf (x) dx - ^ 0f (a) + 2(1 - 0) f (a+b) + 0f (b)

(1-30)(b-a)3 f '' ( a+b N 24 f V 2 ,

+ max

< I(3,0)

< 2

max{|f''' (a) |, f''' (o+b)

f''' (a+b)

+

(25)

, |f''' (b) |

where I (3, 0) is as expressed in (19).

If we take 0 = 0,1, ^, ^ in (25) and (19), then we get a midpoint type inequality

32

f (x) dx - (b - a) f

a + b\ (b - a)

2

(b - a)4

384 + max

max |f ''' (a) |, f

24

a+b

f„I a + b

f''' (a+b

2

<

, |f ''' (b) |

b

3

a trapezoid type inequality

J f (x) dx — ^ [f (a) + f (b)] + f '' (a+b)

<

<

(b — a)

4

128 + max

max {|f ''' (a) |,

f'

a+b

+

f ''' /a+bN

, If ''' (b) |

a Simpson inequality

b b — a / f (x) dx —

6

f (a) + 4f ( —) + f (b)

<

(b — a)

4

<

1152 + max

max |f ''' (a) |,

a+b

f

f''' , If''' (b) |

+

which recapture the inequality (6) and an averaged midpoint-trapezoid type inequality

(b — a)3

48"

f (x) dx —

f^ a +b

ba

f (a) + 2f

a+b

+ f (b)

2

<

(b — a)4 384

max

(a) I, f

+

'" [ a + b

+

+ max

f''' , If''' (b) |

2

3. Applications in numerical integration. We restrict further considerations to the Simpson quadrature rule.

Theorem 6. Let n = (xo = a < x1 < ■ ■ ■ < xn = 6} be a given subdivision of the interval [a, 6] such that h = xi+1 — Xj = h = ^^ and let the assumptions of Theorem 1 hold. Then we have

n1

If (t) dt — |E f (xi ) + 4f

i=0

2

+ f (xi+1)

<

< 5 (b36na)2 max{|f' (a) |, |f' (b) |}.

Proof. From the inequality (14) in Remark 1 we obtain

Xi+1

I f (t) dt - h f (xi) + 4f + f (xi+1)

<

<

<

5h 36

max{|f' (xi) |, |f (xi+1) |}<

5 (b - a)2 36n2

max{|f' (a) |, |f' (b) |}.

By summing (27) over i from 0 to n - 1, we get

n — 1

i=0

Xi+1

E / f (t) dt - h f (xi) + 4f

xi + Xi+1 2

+ f (xi+1)

<

<

5 (b - a)2 36n

max{|f' (a) |, |f' (b) |}.

(27)

(28)

Consequently, the inequality (26) follows from (28). □

Theorem 7. Let n = {x0 = a < x1 < ■ ■ ■ < xn = b} be a given subdivision of the interval [a, b] such that h = xi+1 - xi = h = and let the assumptions of Theorem 2 hold. Then we have

b n—1r /

f f (t) dt - h E f (xi) + 4f (i^) + f (xi+1)

i=0

<

(b - a)3 81n2

max{|f" (a) |, |f" (b) |}.

<

(29)

Proof. From the inequality (17) in Remark 2 we obtain

Xi+1

I f (t) dt - h f (xj ) + 4f

h3

xi +Xi + 1 2

+ f (xi+1)

< -max{|f" (xj) |, f (xj+1) |}<

< ^bin^ max{|f" (a) |, |f" (b) |}. By summing (30) over i from 0 to n - 1, we get

<

(30)

n1

i=0

Xi+1

E I f (t) dt - h f (xj) + 4f

Xi + Xi+1 2

+ f (xi+1)

<

(b - a)3 81n2

max{|f" (a) |, |f" (b) |}.

<

(31)

Consequently, the inequality (29) follows from (31). □

Theorem 8. Let n = (x0 = a < x1 < ■ ■ ■ < xn = 6} be a given subdivision of the interval [a, 6] such that h = xi+1 — xj = h = ^^ and let the assumptions of Theorem 3 hold. Then we have

b n—1r /

/ f (t) dt — h £ f (xi) + 4f ( Hi+Xii1) + f (xi+i)

i=0

\4

< ^|3r- max{If''' (a) I, If''' (b) I}.

<

(32)

Proof. From the inequality (20) in Remark 3 we obtain

Xi+1

I f (t) dt — h f (xi) + 4f (+ f (xi+i) h4

<

max{If''' (xi) I, If''' (xi+i) I}<

<

576 (b — a)4

576n4

max{If''' (a) I, If''' (b) I}.

<

(33)

By summing (33) over i from 0 to n — 1, we get

n — 1

i=0

Xi+1

E / f (t) dt — h f (xi) + 4f

+ f (xi+i)

<

(b — a)4 576n3

max{If''' (a) I, If''' (b) I}.

<

(34)

Consequently, the inequality (32) follows from (34). □

4. Applications to special means. We now consider the applications of the Simpson inequalities (14), (17) and (20) to the following special means:

a + 6

(1) The arithmetic mean: A (a, 6) := —-—, a, 6 > 0.

(2) The Geometric mean: G (a, 6) := v/a6, a, 6 > 0.

2a6

(3) The harmonic mean: H (a, 6) :=--, a, 6 > 0.

a+b

(4) The logarithmic mean: L (a, b) : =

b — a ln b ln a

, a = b, a, b > 0.

1 ( bb \ 1/(b-a)

(5) The identric mean: I (a, b) := - I — ) , a = b, a, b > 0.

e \aa/

&P+1 - ap+1

_ (p + 1) (b - a) .

, a = b, a, b >

(6) The p-logarithmic mean: Lp (a, b) = > 0, p = -1, 0.

Using the Simpson inequalities (14), (17) and (20), some new inequalities are derived for the above means.

Proposition 1. Let a, b G R, 0 <a<b and n G N, n > 3. Then we have

12

^ A (an ,bn) + 2 An (a, b) - Ln (a, b)

<

5n (b - a) b

36

n1

12

1A (an,bn) + 2An (a, b) - Ln (a, b)

<

n (n - 1) (b - a) b 81

2n2

and

12

1A (an ,bn) + 2 An (a, b) - Ln (a, b)

<

n (n - 1) (n - 2) (b - a) b

576

3 n- 3

Proof. The assertion follows from applying the inequalities (14), (17) and (20) to the mapping f (x) = xn,x G [a, b] and n G N which implies that |f/ (x) | = nxn-1, |f" (x) | = n (n - 1) xn-2 and |f"' (x) | = = n (n - 1) (n - 2) xn-3 are quasi-convex on [a, b]. □

Proposition 2. Let a, b G R, 0 < a < b. Then we have

and

12

3H-1 (a,b) +3 A-1 (a, b) - L-1 (a, b) 12

3H-1 (a,b) + 3 A-1 (a, b) - L-1 (a, b)

12

3H-1 (a,b) + 3 A-1 (a, b) - L-1 (a,b)

<

<

<

5 (b - a) 36a2

(b - a)2 81a3

(b - a)3 288a4 .

Proof. The assertion follows from applying the inequalities (14), (17) and (20) to the mapping f (x) = —, x G [a, b] which implies that |f/ (x) | = —^,

2 x 6 x |f" (x) | = —3 and |f(x) | = —4 are quasi-convex on [a, b]. □

x3

x4

Proposition 3. Let a, 6 G R, 0 < a < 6. Then we have

1

ln G (a, b) + 2 ln A (a, b) — ln I (a, b)

<

5 (b — a) 36a

and

ln G (a, b) + 2 ln A (a, b) — ln I (a, b)

ln G (a, b) + 2 ln A (a, b) — ln I (a, b)

<

<

(b — a)2 81a2

(b — a)3 288a3

3

1

3

1

3

Proof. The assertion follows from applying the inequalities (14), (17) and

(10) to the mapping f (x) = ln x, x G [a, 6] which implies that |f' (x) | = —,

x

12

|f" (x) | = and |f"' (x) | = —^ are quasi-convex on [a, 6]. □

Acknowledgment. The author would like to thank the referees for their helpful comments and suggestions.

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Received September 5, 2015.

In revised form, November 11, 2015.

University of Science and Technology Liaoning

Institute of Applied Mathematics, School of Science

Anshan 114051, Liaoning, China;

Room 2004, 4 Lane 123

Guiping Road, Shanghai 200233, China

E-mail: lewzheng@163.net