Journal of Inequalities in Pure and
Applied Mathematics
NOTE ON CERTAIN INEQUALITIES FOR MEANS IN TWO
VARIABLES
volume 6, issue 2, article 43,
2005.
TIBERIU TRIF
Universitatea Babeş-Bolyai
Facultatea de Matematică şi Informatică
Str. Kogălniceanu 1
3400 Cluj-Napoca, Romania
Received 13 October, 2004;
accepted 16 March, 2005.
Communicated by: S.S. Dragomir
EMail: ttrif@math.ubbcluj.ro
Abstract
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Victoria University
ISSN (electronic): 1443-5756
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Abstract
Given the positive real numbers x and y, let A(x, y), G(x, y), and I(x, y) denote
their arithmetic mean, geometric mean, and identric mean, respectively. It is
proved that for p ≥ 2, the double inequality
αAp (x, y) + (1 − α)Gp (x, y) < I p (x, y) < βAp (x, y) + (1 − β)Gp (x, y)
Note on Certain Inequalities for
Means in Two Variables
p
holds true for all positive real numbers x 6= y if and only if α ≤ 2e and β ≥ 32 .
This result complements a similar one established by H. Alzer and S.-L. Qiu
[Inequalities for means in two variables, Arch. Math. (Basel) 80 (2003), 201–
215].
Tiberiu Trif
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2000 Mathematics Subject Classification: Primary: 26E60, 26D07.
Key words: Arithmetic mean, Geometric mean, Identric mean.
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1
Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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1.
Introduction and Main Result
The means in two variables are special and they have found a number of applications (see, for instance, [1, 5] and the references therein). In this note we
focus on certain inequalities involving the arithmetic mean, the geometric mean,
and the identric mean of two positive real numbers x and y. Recall that these
√
means are defined by A(x, y) = x+y
, G(x, y) = xy, and
2
1
1 xx x−y
I(x, y) =
e yy
I(x, x) = x,
if x 6= y,
respectively. It is well-known that
(1.1)
G(x, y) < I(x, y) < A(x, y)
for all positive real numbers x 6= y. On the other hand, J. Sándor [6] proved
that
(1.2)
2
1
A(x, y) + G(x, y) < I(x, y)
3
3
for all positive real numbers x 6= y. Note that inequality (1.2) is a refinement
of the first inequality in (1.1). Also, (1.2) is sharp in the sense that 23 cannot be
replaced by any greater constant. An interesting counterpart of (1.2) has been
recently obtained by H. Alzer and S.-L. Qiu [1, Theorem 1]. Their result reads
as follows:
Note on Certain Inequalities for
Means in Two Variables
Tiberiu Trif
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Theorem 1.1. The double inequality
αA(x, y) + (1 − α)G(x, y) < I(x, y) < βA(x, y) + (1 − β)G(x, y)
holds true for all positive real numbers x 6= y, if and only if α ≤
2
3
and β ≥ 2e .
Another counterpart of (1.2) has been obtained by J. Sándor and T. Trif [8,
Theorem 2.5]. More precisely, they proved that
(1.3)
2
1
I 2 (x, y) < A2 (x, y) + G2 (x, y)
3
3
for all positive real numbers x 6= y. We note that (1.3) is a refinement of the
second inequality in (1.1). Moreover, (1.3) is the best possible inequality of the
type
(1.4)
I 2 (x, y) < βA2 (x, y) + (1 − β)G2 (x, y)
in the sense that (1.4) holds true for all positive real numbers x 6= y if and only
if β ≥ 23 .
It should be mentioned that (1.3) was derived in [8] as a consequence of
certain power series expansions discovered by J. Sándor [7]. We present here
an alternative proof of (1.3), based on the Gauss quadrature formula with two
knots (see [2, pp. 343–344] or [3, p. 36])
Z 1
1
1
1
1
1
1 (4)
1
+ √ + f
− √ +
f (ξ), 0 < ξ < 1.
f (t)dt = f
2
2 2 3
2
2 2 3
4320
0
Note on Certain Inequalities for
Means in Two Variables
Tiberiu Trif
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Choosing f (t) = log(tx + (1 − t)y) and taking into account that
Z 1
f (t)dt = log I(x, y),
0
we get
2 2
1 2
(x − y)4
1
.
A (x, y) + G (x, y) −
log I(x, y) = log
2
3
3
720(ξx + (1 − ξ)y)4
Consequently, it holds that
4 !
2 2
A (x, y) + 13 G2 (x, y)
1
x−y
< 3
exp
360 max(x, y)
I 2 (x, y)
4 !
x−y
1
.
< exp
360 min(x, y)
This inequality yields (1.3) and estimates the sharpness of (1.3). However, we
note that the double inequality (2.33) in [8] provides better bounds for the ratio
2 2
1 2
A (x, y) + G (x, y)
I 2 (x, y).
3
3
The next theorem is the main result of this note and it is motivated by (1.3)
and Theorem 1.1.
Theorem 1.2. Given the real number p ≥ 2, the double inequality
p
p
p
p
Note on Certain Inequalities for
Means in Two Variables
Tiberiu Trif
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p
(1.5) αA (x, y) + (1 − α)G (x, y) < I (x, y) < βA (x, y) + (1 − β)G (x, y)
p
holds true for all positive real numbers x 6= y if and only if α ≤ 2e and
β ≥ 23 .
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2.
Proof of Theorem 1.2
p
Proof. In order to prove that the first inequality in (1.5) holds true for α = 2e ,
we will use the ingenious method of E.B. Leach and M.C. Sholander [4] (see
also [1]). More precisely, we show that
p p
2
2
p
t −t
+ 1−
(2.1)
A e ,e
Gp et , e−t
e
e
< I p et , e−t ,
for all t > 0.
It is easily seen that (2.1) is equivalent to fp (t) < 0 for all t > 0, where fp :
(0, ∞) → R is the function defined by
fp (t) = (2 cosh t)p + ep − 2p − exp(pt coth t).
We have
p−1
4p sinh t(2 cosh t)
− p(sinh(2t) − 2t) exp(pt coth t)
.
2 sinh2 t
By means of the logarithmic mean of two variables,
x−y
L(x, y) =
log x − log y
L(x, x) = x,
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fp0 (t) = p
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if x 6= y,
the derivative fp0 may be expressed as
(2.2)
Tiberiu Trif
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fp0 (t) =
Note on Certain Inequalities for
Means in Two Variables
L(u(t), v(t))
g(t),
2 sinh2 t
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where
u(t) = 4 sinh3 t(2 cosh t)p−1 ,
v(t) = (sinh(2t) − 2t) exp(pt coth t),
g(t) = log u(t) − log v(t)
= (p + 1) log 2 + 3 log(sinh t) + (p − 1) log(cosh t)
− log(sinh(2t) − 2t) − pt coth t.
Note on Certain Inequalities for
Means in Two Variables
We have
3 cosh t (p − 1) sinh t 2 cosh(2t) − 2 p cosh t
pt
+
−
−
+
sinh t
cosh t
sinh(2t) − 2t
sinh t
sinh2 t
3 cosh2 t − sinh2 t − p(cosh2 t − sinh2 t)
pt
2 cosh(2t) − 2
=
+
2 −
sinh t cosh t
sinh(2t) − 2t
sinh t
cosh(2t) + 2 − p
pt
2 cosh(2t) − 2
=
+
,
2 −
sinh t cosh t
sinh(2t) − 2t
sinh t
g 0 (t) =
hence
(2.3)
g 0 (t) = g1 (t) + g2 (t),
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cosh(2t)
2t
2 cosh(2t) − 2
+
,
g1 (t) =
2 −
sinh t cosh t sinh t
sinh(2t) − 2t
(p − 2)t
p−2
g2 (t) =
−
.
2
sinh t cosh t
sinh t
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But
p−2
(t cosh t − sinh t)
sinh2 t cosh t
∞ X
p−2
1
1
=
t2k+1 .
−
2
(2k)!
(2k
+
1)!
sinh t cosh t k=1
g2 (t) =
Taking into account that p ≥ 2, we deduce that
(2.4)
g2 (t) ≥ 0
for all t > 0.
Further, let h : (0, ∞) → R be the function defined by
h(t) = sinh2 t cosh t(sinh(2t) − 2t)g1 (t).
Tiberiu Trif
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Then we have
h(t) = 2t sinh t + sinh t sinh 2t − 4t2
1
1
= 2t sinh t + cosh(3t) − cosh t − 4t2
2
2
∞ X
2
32k − 1 2k
=
+
t .
(2k
−
1)!
2(2k)!
k=2
Therefore h(t) > 0 for t > 0, hence
(2.5)
Note on Certain Inequalities for
Means in Two Variables
g1 (t) > 0
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for all t > 0.
J. Ineq. Pure and Appl. Math. 6(2) Art. 43, 2005
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By (2.3), (2.4), and (2.5) we conclude that g 0 (t) > 0 for t > 0, hence g is
increasing on (0, ∞). Taking into account that
sinh3 t
lim g(t) = (p + 1) log 2 + log lim
t→∞
t→∞ cosh t(sinh(2t) − 2t)
+ p lim (log(cosh t) − t) + p lim t(1 − coth t)
t→∞
= (p + 1) log 2 + log
t→∞
1
1
+ p log
2
2
Note on Certain Inequalities for
Means in Two Variables
= 0,
it follows that g(t) < 0 for all t > 0. By virtue of (2.2), we deduce that
fp0 (t) < 0 for all t > 0, hence fp is decreasing on (0, ∞). Since lim fp (t) = 0,
t&0
Tiberiu Trif
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we conclude that fp (t) < 0 for all t > 0. This proves the validity of (2.1).
q
Now let x 6= y be two arbitrary positive real numbers. Letting t = log xy
√ p
in (2.1) and multiplying the obtained inequality by
xy , we obtain
p
p 2
2
p
A (x, y) + 1 −
Gp (x, y) < I p (x, y).
e
e
p
Consequently, the first inequality in (1.5) holds true for α = 2e .
Let us prove now that the second inequality in (1.5) holds true for β = 23 .
Indeed, taking into account (1.3) as well as the convexity of the function t ∈
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p
(0, ∞) 7→ t 2 ∈ (0, ∞) (recall that p ≥ 2), we get
p
I p (x, y) = I 2 (x, y) 2
p2
2 2
1 2
< A (x, y) + G (x, y)
3
3
p2 1 2
p
2 2
≤
A (x, y) + G (x, y) 2
3
3
2 p
1 p
= A (x, y) + G (x, y).
3
3
Conversely, suppose that (1.5) holds true for all positive real numbers x 6= y.
Then we have
I p (x, y) − Gp (x, y)
< β.
α< p
A (x, y) − Gp (x, y)
Tiberiu Trif
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The limits
I p (x, 1) − Gp (x, 1)
lim p
=
x→0 A (x, 1) − Gp (x, 1)
p
yield α ≤ 2e and β ≥ 23 .
Note on Certain Inequalities for
Means in Two Variables
p
2
e
and
I p (x, 1) − Gp (x, 1)
2
lim p
=
x→1 A (x, 1) − Gp (x, 1)
3
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References
[1] H. ALZER AND S.-L. QIU, Inequalities for means in two variables, Arch.
Math. (Basel), 80 (2003), 201–215.
[2] P.J. DAVIS, Interpolation and Approximation, Blaisdell, New York, 1963.
[3] P.J. DAVIS AND P. RABINOWITZ, Numerical Integration, Blaisdell,
Massachusetts–Toronto–London, 1967.
[4] E.B. LEACH AND M.C. SHOLANDER, Extended mean values II, J. Math.
Anal. Appl., 92 (1983), 207–223.
Note on Certain Inequalities for
Means in Two Variables
Tiberiu Trif
[5] G. LORENZEN, Why means in two arguments are special, Elem. Math., 49
(1994), 32–37.
[6] J. SÁNDOR, A note on some inequalities for means, Arch. Math. (Basel)
56 (1991), 471–473.
[7] J. SÁNDOR, On certain identities for means, Studia Univ. Babeş-Bolyai,
Ser. Math., 38(4) (1993), 7–14.
[8] J. SÁNDOR AND T. TRIF, Some new inequalities for means of two arguments, Int. J. Math. Math. Sci., 25 (2001), 525–532.
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