Download Two Inequalities for Differentiable Mappings and

Appl. Math. Lett. Vol. 11, No. 5, pp. 91-95, 1998
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T w o Inequalities for Differentiable M a p p i n g s
and Applications to Special Means of Real
N u m b e r s and to Trapezoidal Formula
S. S. D R A G O M I R
Department of Mathematics, University of Transkei
Private Bag X1, Unitra, Umtata, 5100 South Africa
R. P. A G A R W A L
Department of Mathematics, National University of Singapore
10 Kent Ridge Crescent, Singapore 119260
(Received and accepted July 1997)
A b s t r a c t - - I n this paper, we shall offer two inequalities for differentiable convex mappings which
are connected with the celebrated Hermite-Hadamard's integral inequality holding for convex functions. Some natural applications to special means of real numbers are given. Finally, some error
estimates for the trapezoidal formula are also addressed. © 1998 Elsevier Science Ltd. All rights
reserved.
Keywords--Herrnite-Hadamard inequality,Convex functions,Integralinequalities,Trapezoidal
formula, Special means, Numerical integration.
i.
INTRODUCTION
Let f : I c R --* R be a convex function on the interval I of real numbers and a, b E I with a < b.
The inequality
f ( _a~ . _b) < ~ -1a lab f(x) cl.T < f ( a ) +2f ( b )
(1.1)
is well k n o w n in t h e l i t e r a t u r e as H e r m i t e - H a d a m a r d ' s i n e q u a l i t y for convex f u n c t i o n s [1].
T h e a i m o f t h i s p a p e r is t o e s t a b l i s h s o m e results c o n n e c t e d w i t h t h e right p a r t of (1.1) as well
as t o a p p l y t h e m for s o m e e l e m e n t a r y inequalities for real n u m b e r s a n d in n u m e r i c a l i n t e g r a t i o n .
F o r several recent r e s u l t s c o n c e r n i n g H e r m i t e - H a d a m a r d ' s i n e q u a l i t y (1.1), see [2-6] w h e r e
f u r t h e r references a r e listed.
2. MAIN
RESULTS
W e begin with the following lemma.
LEMMA 2.1. L e t f : I ° C_ R -~ R be a differentiable
f E L[a, b], then the following e q u a / i t y holcls:
f(a)'i-2 f(b)
b -1 a
~abf(x) dx =
mapping on I °, a,b E I ° with a < b. If
b - a fol (1 - 2t)f(ta + (1 - t)b) dt.
T
91
(2.1)
92
PROOF.
S.S. DRAGOMIR AND R. P. AGARWAL
It SUffiCes to note that
1
I ----
~0
(1 -- 2t)f'(ta
+ (1 -- t)b)dt
_- f(ta+a (_l-t)b)g
(1_201 x + 2
/(a) +f(b)
=
b-a
2
- b-a
fo 1 f(ta+(1-t)b)dt
o
1 [b
b-a
a-b
J,~ f ( x ) d x .
REMARK 2.1. On using the change of the variable x = ta + (1 - t)b, t E [0, 1], equality (2.1) can
be written as
b -I faa b f(x) dx = b -1a
f(a) +
2 f(b)
lab( x
a 2- b ) f'(x) dx.
(2.2)
Some applicationsof identity (2.2)connected with Hermite-Hadamard's integralinequality for
convex functions have been presented in [7]. Here we shall offer some more, which are very
interesting.
THEOREM 2.2. Let f : I ° C_ R --* R be a differentiable mapping on I °, a, b E I ° with a < b. If
If'l is convex on [a, b], then the following inequality holds:
[ f ( a ) +2f ( b )
(b - a) (l/'(a)l + l/'(b)l)
b -1 a lab f(x) dx I <
(2.3)
PROOF. Using L e m m a 2.1,it followsthat
l y(a) + l(b)
b
2
1
a
f ( x ) dx =
<
2 Jo
b-a ~ 1
2
(1 - 2t)fCta + (1 - t)b) dt
l1 - 2tl
If'(ta +
(1 -
t)b)l dt
< b -2 a' fol I1 - 2tl [tl.f'(a)l + (1 - t)lY'(b)l] dt
= (b - a) (If(a)12
+ If(b)l) jo[1I1 - 2tl t dt
= (b - a) (I/'(a)l + If'(b)t)
8
where we have used the fact t h a t
/ol[1-2t[(1-t)dt= /oI 11-2tltdt= r
/112 ( 2 t - 1 ) t d t = i1.
(1-2tltdt+
J0
Another similar result is embodied in the following theorem.
THEOREM 2.3. Let f : I ° C_ R --, R be a differentiable mapping on l °, a, b 6 1 ° with a < b, a n d
let p > 1. If the new mapping [ftlP/(P--1) /.9 convex on [a, hi, then the £ollowing inequality holds:
2
b -a
f ( x ) d,T < 2 ( p + i ) 1 / p
2
.
(2.4)
Trapezoidal Formula
93
PROOF. Using L e m m a 2.1 and HSlder's integral inequality, we find
f(a) q- f(b)
1/b
-
b
<
f(x) dx
a
-
b- - a fol [1
2
- 2t[
[I'(ta +
(1 - t)b)[
(2.5)
dt
1
1/p
1
1/q
b
a
(fo
1
1
2
t
"
dt)
(fo
I]'(ta+(1-t)b)Iq
dr)
,
<- 2
where 1/p + 1/q = 1.
Using the convexity of
[f'(ta +
(1 -
If'l q,
we have
t)b)l q dt <
-
]01
[t[f'(a)[ q +
(1 -
t)lf(b)I q] dt = ]f'(a)lq + [f'(b)lq
2
(2.6)
Further, since
It -
2tl p
dt
(1 - 2t) p dt
=
+
-
1) p dt = 2
JO
(1 - 2t) p dt = _ _ 1
p + 1'
JO
(2.7)
a combination of (2.5)-(2.7) immediately gives the required inequality (2.4).
3. APPLICATIONS
TO
In the literature, the following means for
+2 / 3 '
A(a,f~) =
SPECIAL
positive
MEANS
real numbers a, f~, a ~ / 3 are well known:
arithmetic mean,
G(~,/3) = v f ~ ,
geometric mean,
f~-a
L(a,/3) = In/3 - In a '
logarithmic mean,
1 ( f ~ ) 1/(~-~)
I(a,/3) = e ~-~
,
identric mean,
[ ~,+, _ ~ . + 1 1
'"
generalized log-mean,
p~ -1,0.
There are several results connecting these means, e.g., see [8] for some new relations; however,
very few results are known for arbitrary real numbers. For this, it is clear t h a t we can extend
some of the above means as follows:
a+~
A(a,/3) =
2
--
L(a,/3)
'
a,/3 e R,
f~--O~
=
In
I~'l - In lal'
r z - + l - a"+l
L,(~,f~) =
L(~Vi)~--;)J
~' z • R\{0},
11/.
'
hEN,
n>l,
a,/3eR,
a<f~.
Now we shall use the results of Section 2 to prove the following new inequalities connecting
the above means for arbitrary real numbers.
PROPOSITION 3.1. Let a, b E R, a < b and n E N, n > 2.
Then,
the
following
IA(a",b ~) - L,(a,b)l < nCb~
4 a) A (laln-i, Ibl"-l).
inequa//ty
holds:
94
S.S. D R A G O M I R A N D R. P. A G A R W A L
PROOF. The proof is immediate from Theorem 2.2 applied for f ( x ) = x n, x E
R.
PROPOSITION 3.2. Let a, b E R, a < b, and n E N, n > 2. Then, for all p > 1, the following
inequality holds:
n(b-a)
[A(an, bn) - Ln(a,b)[ < 2(~'1)-~/"
[AOa[(n_l)p/(p_l)[b[(n_l)p/(p_l))](p-l)/p
"
PROOF. The proof is immediate from Theorem 2.3 applied for f ( x ) = x n, x e R.
PROPOSITION 3.3. Let a, b E R, a < b, and 0 • [a, b]. Then, the £ollowing inequa/ity holds:
A(a-l,b -1)
L-l(a,b) < ( b - a ) A o a r 2 , [ b [ - 2 )
PROOF. The proof is obvious from Theorem 2.2 applied for f ( x ) = 1/x, x E [a, b].
PROPOSITION 3.4. Let a, b E R, a < b, and 0 ~ [a, b]. Then, t'or p > 1, the/'ollowing inequa//ty
holds:
]A(a_lb_l)-~-l(a,b)[ ~ (b-a)[AOai_2p/(p_l),ibF2p/(p_l))](p-l)/p"
¥1T/.
-
PROOF. The proof is obvious from Theorem 2.3 applied for f ( x ) = 1/x, x E [a, b].
4. A P P L I C A T I O N S
TO TRAPEZOIDAL
FORMULA
Let d be a division of the interval [a, b], i.e., d : a = x0 < xl < ... < xn-1 < xn = b, and
consider the trapezoidal formula
n--1
T(f,d) = Z
i=0
/(xi) + .f(Xi+l)(xi+ 1 - - Xi).
2
It is well known that if the mapping f : [a, b] --+ R is twice differentiable on (a, b) and M =
maxtE(a,b ) [f'l(x)] < oo, then
ab f ( x ) dx = T ( f , d) + E ( f , d),
(4.1)
where the approximation error E ( f , d) of the integral f : f ( x ) d x by the trapezoidal formula
T ( f , d) satisfies
Mn-1
]E(f,d)l < - ~ E ( x i + l - x,) 3.
(4.2)
i=0
It is clear that if the mapping f is not twice differentiable or the second derivative is not
bounded on (a, b), then (4.2) cannot be applied. In recent papers [9-11], Dragomir and Wang
have shown that the remainder term E ( f , d) can be estimated in terms of the first derivative only.
These estimates have a wider range of applications. Here, we shall propose some new estimates
of the remainder term E ( f , d) which supplement, in a sense, those established in [9-11].
PROPOSITION 4.1. Let f be a differentiable mapping on I °, a, b E I ° with a < b. If[f'[ is convex
on [a,b], then in (4.1), for every division d of [a,b] ,the following holds:
ln-1
]E(f,d)l _< ~ ~--:.(xi+l - xi) 2 (I/'(x~)[ + [/'(xi+l)l)
i=0
< max{lf'(a)l,lf'(b)l} ~"~.(x,+1 - x,) 2.
-
4
i-----0
Trapezoidal Formula
PROOF. Applying Theorem 2.2 on the subinterval [27i,27i+1]
weget
I f(27i)
4- f(274-1'-1) (27i-I-1 -- 27i) -- / ~ ' + ' f(27) dx
2
I~x,
Summing over i from 0 to
triangle inequality, that
n -
<
-
95
(i =
O,..., rt
--
1) of the division d,
(X,+l - x,) 2 (If'(27,)l + If'(x,+l)l)
8
1 and taking into account that If'l is convex, we deduce, by the
n -1
T ( f , d)
-
~a b f(27) d27 ~-- 8 Z ( 2 7 " { - 1 - 27i)2 (I.f/(27i)[ 4- [ft(27i.{_l)[ )
i----0
n--1
< max{If'(a)['
4
[f'(b)[}
~-~(27,+1
- 27,)2.
i----0
PROPOSITION 4.2. Let f be a differentiable mapping on I °, a, b q I ° with a < b, and let p > 1.
If lf'l p/(v-1) is convex on [a,b], then in (4.1) for every division d of [a,b], the following holds:
1
Z (xi't'l _ z i ) 2 [f,(z~)lp/(v_l)
IE(f,d)l _< 2(p+ 1)1/p i--o
..f,(x~+l)lp/(p_l) '
(p--1)/p
rt--1
< max{If'(a)l, I f @ l } ~-'~(27,+1 - 27~)2.
--
2(p 4- 1) 1/v
i=0
PROOF. The proof uses Theorem 2.3 and is similar to that of Proposition 4.1.
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Academic Press, New York, (1991).
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Approx. 19, 29-34 (1990).
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(1992).
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