Received 23/09/11
BOUNDS FOR TWO MAPPINGS ASSOCIATED TO THE HERMITE-HADAMARD INEQUALITY S.S. DRAGOMIR 1;2 AND I. GOMM 1 Abstract. Some inequalities concerning two mappings associated to the celebrated Hermite-Hadamard integral inequality for convex function with applications for special means are given.
1. Introduction The Hermite-Hadamard integral inequality for convex functions f : [a; b] ! R (HH)
f
a+b 2
1 b
a
Z
b
f (a) + f (b) 2
f (x) dx
a
is well known in the literature and has many applications for special means. In order to provide various re…nements of this result, the …rst author introduced in 1991, see [2], the following associated mapping H : [0; 1] ! R de…ned by H (t) :=
1 b
a
Z
b
f
tx + (1
t)
a
a+b 2
dx;
for a given convex function f : [a; b] ! R: The following theorem collects some of the main properties of H (see also [2], [3], [4] and [6]): Theorem 1. With the above assumptions, we have that the function H: (i) is convex on [0; 1] ; (ii) has the bounds: a+b 2
inf H (t) = H (0) = f
t2[0;1]
and sup H (t) = H (1) = t2[0;1]
1 b
a
Z
b
f (x) dx;
a
(iii) increases monotonically on [0; 1] :
1991 Mathematics Subject Classi…cation. 26D15; 25D10. Key words and phrases. Convex functions, Hermite-Hadamard inequality, Special means. 1
2
DRAGOM IR & GOM M
(iv) The following inequalities hold: a+b 2
f
(1.1)
2 b Z
a 1
Z
a+3b 4
f (x) dx 3a+b 4
H (t) dt
0
" 1 f 2
a+b 2
1
+
b
a
Z
#
b
f (x) dx :
a
The corresponding double integral mapping in connection with the HermiteHadamard inequalities was considered …rst in [3] and is de…ned as F : [0; 1] ! R; F (t) :=
1 (b
2
a)
Z
b
a
Z
b
f (tx + (1
t) y) dxdy:
a
The following theorem provides some of the main results concerning this mapping [3] (see also [4]): Theorem 2. Let f : [a; b] ! R be as above. Then (i) F + 12 = F 12 t 2 [0; 1] ; (ii) F is convex on [0; 1] ; (iii) We have the bounds:
2 0; 12
for all
sup F (t) = F (0) = F (1) = t2[0;1]
and F (t) = F (1
1 b
a
Z
t) for all
b
f (x) dx
a
and inf F (t) = F
t2[0;1]
1 2
=
1 (b
2
a)
Z
a
b
Z
b
f
a
x+y 2
dxdy;
(iv) The following inequality holds: f
a+b 2
F
1 2
;
(v) F decreases monotonically on 0; 21 and increases monotonically on (vi) We have the inequality: H (t)
1 2; 1
;
F (t) for all t 2 [0; 1] :
For other related results, see for instance the research papers [1], [8], [9], [10], [12], [11], [13], [14], [15], the monograph online [7] and the references therein. Motivated by the above results we establish in this paper some new bounds involving these two mappings. Applications for special means are also provided.
HERM ITE-HADAM ARD INEQUALITY
3
2. The Results Theorem 3. Let f : [a; b] ! R be a convex function on the interval [a; b]. Then we have (2.1)
0
2 min ft; 1 tg " " # Z b 1 a+b 1 f (x) dx + f 2 b a a 2 Z b t a+b f (x) dx + (1 t) f b a a 2 2 max ft; 1 tg # " " Z b 1 a+b 1 f (x) dx + f 2 b a a 2
and (2.2)
0
2 min ft; 1 1 b
a
Z
a
tg
"
1 b
a
Z
2 b
a
Z
#
f (x) dx 3a+b 4
H (t)
2 b
a
Z
a+3b 4
F
#
F
1 2
#
a
#
f (x) dx 3a+b 4
1 2
b
f (x) dx
a+3b 4
b
f (x) dx "
2 max ft; 1
tg
F (t) 1
b
a
Z
b
f (x) dx
a
;
for any t 2 [0; 1] : Proof. Recall the following result obtained by the …rst author in [5] that provides a re…nement and a reverse for the weighted Jensen’s discrete inequality: !# " n n 1X 1X (2.3) (xi ) xi n min fpi g n i=1 n i=1 i2f1;:::;ng ! n n 1 X 1 X pi (xi ) pi xi Pn i=1 Pn i=1 " n !# n 1X 1X n max fpi g (xi ) xi ; n i=1 n i=1 i2f1;:::;ng
where : C ! R is a convex function de…ned on the convex subset C of the linear space X; fxi gi2f1;:::;ng are vectors in C and fpi gi2f1;:::;ng are nonnegative numbers Pn with Pn := i=1 pi > 0: For n = 2 we deduce from (2.3) that
(2.4)
2 min ft; 1
tg
t (x) + (1 2 max ft; 1 for any x; y 2 C and t 2 [0; 1] :
(x) + 2 t) (y)
tg
(y)
(x) + 2
x+y 2 (tx + (1 t) y) (y)
x+y 2
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DRAGOM IR & GOM M
On making use of the inequality (2.4) we can write for the convex function f : [a; b] ! R that !# " a+b f (x) + f a+b x + 2 2 (2.5) 2 min ft; 1 tg f 2 2 tf (x) + (1 2 max ft; 1
t) f "
tg
a+b 2 f (x) + f 2
f
tx + (1
a+b 2
t)
a+b 2 !#
x + a+b 2 2
f
for any x 2 [a; b] and t 2 [0; 1] : Integrating over x 2 [a; b] in (2.5) we get (2.6)
2 min ft; 1 tg " "Z ! # # Z b b x + a+b 1 a+b 2 f (x) dx + f dx (b a) f 2 a 2 2 a Z b a+b t (b a) H (t) (b a) f (x) dx + (1 t) f 2 a 2 max ft; 1 tg " "Z # Z ! # b b x + a+b 1 a+b 2 f (x) dx + f (b a) f dx 2 a 2 2 a
and since
Z
a
b
f
x + a+b 2 2
!
dx = 2
Z
a+3b 4
f (s) ds 3a+b 4
then from (2.6) we get (2.1). Now, if we write the inequality (2.4) for the convex function f and integrate over x and y on [a; b] ; we get (2.7)
2 min ft; 1 tg "Z Z # Z bZ b b b f (x) + f (y) x+y dxdy f dxdy 2 2 a a a a Z bZ b Z bZ b [tf (x) + (1 t) f (y)] dxdy f (tx + (1 t) y) dxdy a
a
a
2 max ft; 1 tg "Z Z b b f (x) + f (y) dxdy 2 a a
for any t 2 [0; 1] : Since Z Z
a
b
a
b
Z
a
Z
a
b
Z
a
b
Z
a
b
f
x+y 2
a)
Z
a
f (x) + f (y) dxdy = (b 2
b
[tf (x) + (1
t) f (y)] dxdy = (b
#
dxdy ;
b
a
a)
f (x) dx; Z
a
b
f (x) dx
HERM ITE-HADAM ARD INEQUALITY
and
Z
b
Z
5
b
x+y 1 dxdy = F 2 2 a a then we deduce from (2.7) the desired result (2.2). f
(b
2
a) ;
Corollary 1. With the above assumptions we have (2.8)
0
2 min ft; 1 tg " " # # Z b Z a+3b 4 1 a+b 2 1 f (x) dx + f f (x) dx 2 b a a 2 b a 3a+b 4 Z b a+b 1 1 f (x) dx + f [H (t) + H (1 t)] b a a 2 2 2 max ft; 1 tg # # " " Z b Z a+3b 4 1 a+b 2 1 f (x) dx + f f (x) dx ; 2 b a a 2 b a 3a+b 4
for any t 2 [0; 1] : Proof. Follows from the inequality (2.1) written for 1 t instead of t, by adding the obtained two inequalities and dividing the sum by 2: 3. Applications for Lp -means Let us consider the convex mapping f : (0; 1) ! R; f (x) = xp ; p 2 ( 1; 0) [ [1; 1) n f 1g and 0 < a < b: De…ne the mapping Z b 1 p (tx + (1 t) A (a; b)) dx; t 2 [0; 1] : Hp (t) := b a a
It is obvious that Hp (0) = Ap (a; b) ; Hp (1) = Lpp (a; b) where, we recall that A (a; b) = a+b 2 , Lpp (a; b) :=
1 bp+1 p+1 b
ap+1 ; p 2 ( 1; 0) [ [1; 1) n f 1g a
and for t 2 (0; 1) we have (3.1)
Hp (t) =
1 [tb + (1
t) A (a; b)]
= Lpp (ta + (1
[ta + (1
t) A (a; b) ; tb + (1
t) A (a; b)]
Z
tb+(1 t)A(a;b)
y p dy
ta+(1 t)A(a;b)
t) A (a; b)) :
The following proposition holds, via Theorem 1, applied for the convex function f (x) = xp : Proposition 1. With the above assumptions, we have for the function Hp : (i) is convex on [0; 1] ; (ii) has the bounds: inf Hp (t) = Ap (a; b) ; sup Hp (t) = Lpp (a; b) ;
t2[0;1]
(iii) increases monotonically on [0; 1] :
t2[0;1]
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DRAGOM IR & GOM M
(iv) The following inequalities hold Ap (a; b)
(3.2)
Lpp (A (a; A (a; b)) ; A (b; A (a; b))) Z 1 Hp (t) dt A Ap (a; b) ; Lpp (a; b) : 0
Now, on making use of Theorem 3 we can state the following result as well: Proposition 2. With the above assumptions, we have (3.3)
0
2 min ft; 1
tg
2 max ft; 1
tg
1 p 3a + b a + 3b L (a; b) + Ap (a; b) ; Lpp 2 p 4 4 p p tLp (a; b) + (1 t) A (a; b) Hp (t) 1 p L (a; b) + Ap (a; b) 2 p
3a + b a + 3b ; 4 4
Lpp
for any t 2 [0; 1] : Now, consider the function Fp (t) :=
1 (b
2
a)
Z
b
a
Z
b
p
(tx + (1
t) y) dxdy:
a
We observe that Fp (1) = Fp (0) = Lpp (a; b) and for t 2 (0; 1) we have ! Z b Z b 1 1 p (3.4) Fp (t) = (tx + (1 t) y) dx dy b a a b a a ! Z b Z tb+(1 t)y 1 1 sp ds dy = b a a [tb + (1 t) y] [ta + (1 t) y] ta+(1 t)y Z b 1 Lp (ta + (1 t) y; tb + (1 t) y) dy: = b a a p Utilising Theorem 2 we can state the following results: Proposition 3. We have the following properties: (i) Fp + 21 = Fp 12 for all 2 0; 21 and Fp (t) = Fp (1 t 2 [0; 1] ; (ii) Fp is convex on [0; 1] ; (iii) We have the bounds:
t) for all
sup Fp (t) = Fp (0) = Fp (1) = Lpp (a; b)
t2[0;1]
and inf F (t) = Fp
t2[0;1]
1 2
=
1 (b
2
a)
Z
a
b
Z
a
b
x+y 2
p
dxdy;
(iv) The following inequality holds: Ap (a; b) Fp 21 ; (v) Fp decreases monotonically on 0; 12 and increases monotonically on (va) We have the inequality: Hp (t)
Fp (t) for all t 2 [0; 1] :
1 2; 1
;
HERM ITE-HADAM ARD INEQUALITY
7
We can calculate the double integral 1 2
Fp as follows. Observe that for p 6= Z
2
(b
a)
Z
Z
b
b
a
a
p
x+y 2
dxdy
1 we have
b
p+1
b+y 2
p
x+y 2
a
and for p 6=
1
=
dx = 2
a+y p+1 2
p+1
2 we have Z
b
a
b+y 2
p+1
a+y 2
p+1
dy = 2
b+a p+2 2
bp+2
p+2
and Z
b
a
dy = 2
a+b p+2 2
ap+2
p+2
:
Then we get Z
Z
b
a
b
x+y 2
a
p
" 4 bp+2 dxdy = (p + 1) (p + 2)
p+2
2
b+a 2
p+2
2
b+a 2
p+2
+a
#
which gives that (3.5)
1 2
Fp
for p 6= 2; 1: The case p =
=
4 (b
2
a) (p + 1) (p + 2)
2 gives that Z b x+y 2 a
2
"
b
p+2
dx = 4
1 a+y
1 b+y
Z
1 a+y
1 b+y !
p+2
+a
#
;
and Z
a
b
Z
b
a
2
x+y 2
!
dx dy = 4
b
a
A (a; b) G (a; b)
= 4 ln where G (a; b) = Therefore (3.6)
p
dy
2
= 8 ln
A (a; b) G (a; b)
;
ab is the geometric mean of the positive numbers a and b.
F
2
1 2
=
8 (b
2
a)
ln
A (a; b) G (a; b)
:
Finally, on making use of the inequality (2.2) we can state that:
;
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DRAGOM IR & GOM M
Proposition 4. We have the inequalities: (3.7)
0
tg Lpp (a; b)
2 min ft; 1 Lpp (a; b)
Fp
1 2
Fp
1 2
Fp (t)
2 max ft; 1
tg Lpp (a; b)
;
for any t 2 [0; 1] : References [1] A.G. AZPEITIA, Convex functions and the Hadamard inequality. Rev. Colombiana Mat. 28 (1994), no. 1, 7–12. [2] S.S. DRAGOMIR, A mapping in connection to Hadamard’s inequalities, An. Öster. Akad. Wiss. Math.-Natur., (Wien), 128(1991), 17-20. MR 934:26032. ZBL No. 747:26015. [3] S.S. DRAGOMIR, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167(1992), 49-56. MR:934:26038, ZBL No. 758:26014. [4] S.S. DRAGOMIR, On Hadamard’s inequalities for convex functions, Mat. Balkanica, 6(1992), 215-222. MR: 934: 26033. [5] S.S. DRAGOMIR, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74(3)(2006), 471-476. ´ and J. SÁNDOR, On some re…nements of Hadamard’s [6] S.S. DRAGOMIR, D.S. MILO´SEVIC inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4(1993), 21-24. [7] S.S. DRAGOMIR and C.E.M. PEARCE, Selected Topics on HermiteHadamard Inequalities and Applications, RGMIA Monographs, 2000. [Online http://rgmia.org/monographs/hermite_hadamard.html]. [8] A. GUESSAB and G. SCHMEISSER, Sharp integral inequalities of the Hermite-Hadamard type. J. Approx. Theory 115 (2002), no. 2, 260–288. [9] E. KILIANTY and S.S. DRAGOMIR, Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space. Math. Inequal. Appl. 13 (2010), no. 1, 1–32. [10] M. MERKLE, Remarks on Ostrowski’s and Hadamard’s inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 113–117. [11] C. E. M. PEARCE and A. M. RUBINOV, P-functions, quasi-convex functions, and Hadamard type inequalities. J. Math. Anal. Appl. 240 (1999), no. 1, 92–104. ´ and A. VUKELIC, ´ Hadamard and Dragomir-Agarwal inequalities, the Euler µ [12] J. PE CARI C formulae and convex functions. Functional equations, inequalities and applications, 105–137, Kluwer Acad. Publ., Dordrecht, 2003. [13] G. TOADER, Superadditivity and Hermite-Hadamard’s inequalities. Studia Univ. Babe¸ sBolyai Math. 39 (1994), no. 2, 27–32. [14] G.-S. YANG and M.-C. HONG, A note on Hadamard’s inequality. Tamkang J. Math. 28 (1997), no. 1, 33–37. [15] G.-S. YANG and K.-L. TSENG, On certain integral inequalities related to Hermite-Hadamard inequalities. J. Math. Anal. Appl. 239 (1999), no. 1, 180–187. 1 Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia. E-mail address :
[email protected] URL: http://rgmia.org/dragomir 2 School
of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa.
