Topics in Inequalities - Theorems and Techniques Hojoo ... - Index of

Since it is homogeneous, we may normalize to u + v + w = 3. We are now required to show that
where G(t) = t x y , where t > 0. Since
x
y
G(u) + G(v) + G(w)
3
G(u) + G(v) + G(w)
3
≥ 1,
Similarly, we may deduce that M is monotone increasing on (−∞, 0).
≥ 1, we find that G is convex. Jensen’s inequality shows that
( )
u + v + w
≥ G
= G(1) = 1.
3
We’ve learned that the convexity of f(x) = x λ (λ ≥ 1) implies the monotonicity of the power means.
Now, we shall show that the convexity of x ln x also implies the power mean inequality.
Second Proof of the Monotonicity. Write f(x) = M (a,b,c) (x). We use the increasing function theorem. By
the lemma 3, it’s enough to show that f ′ (x) ≥ 0 for all x ≠ 0. Let x ∈ R − {0}. We compute
f ′ (x)
f(x) = d
dx (ln f(x)) = − 1 ( a x
x 2 ln + b x + c x )
+ 1 1
3 (ax ln a + b x ln b + c x ln c)
3 x
1
3 (ax + b x + c x )
or
x 2 f ′ (x)
f(x)
( a x + b x + c x
= − ln
3
)
+ ax ln a x + b x ln b x + c x ln c x
a x + b x + c x .
To establish f ′ (x) ≥ 0, we now need to establish that
( a
a x ln a x + b x ln b x + c x ln c x ≥ (a x + b x + c x x + b x + c x )
) ln
.
3
Let us introduce a function f : (0, ∞) −→ R by f(t) = t ln t, where t > 0. After the substitution p = a x ,
q = a y , r = a z , it becomes
( ) p + q + r
f(p) + f(q) + f(r) ≥ 3f
.
3
Since f is convex on (0, ∞), it follows immediately from Jensen’s inequality.
As a corollary, we obtain the RMS-AM-GM-HM inequality for three variables.
Corollary 4.2.1. For all positive real numbers a, b, and c, we have
√
a2 + b 2 + c 2
≥ a + b + c ≥ 3√ 3
abc ≥
3
3
1
a + 1 b + 1 .
c
Proof. The Power Mean inequality states that M (a,b,c) (2) ≥ M (a,b,c) (1) ≥ M (a,b,c) (0) ≥ M (a,b,c) (−1).
Using the convexity of x ln x or the convexity of x λ
the power means for n positive real numbers.
(λ ≥ 1), we can also establish the monotonicity of
Theorem 4.2.2. (Power Mean inequality) Let x 1 , · · · , x n > 0. The power mean of order r is defined by
( ) 1
M (x1,··· ,x n)(0) = n√ x
r r r
x 1 · · · x n , M (x1,··· ,x n)(r) = 1 + · · · + x n
(r ≠ 0).
n
Then, M (x1,··· ,x n) : R −→ R is continuous and monotone increasing.
We conclude that
Corollary 4.2.2. (Geometric Mean as a Limit) Let x 1 , · · · , x n > 0. Then,
(
n√ r ) 1
r
x1
r
+ · · · + x n x1 · · · x n = lim
.
r→0 n
Theorem 4.2.3. (RMS-AM-GM-HM inequality) For all x 1 , · · · , x n > 0, we have
√
x12 + · · · + x n
2
n
≥ x 1 + · · · + x n
n
≥ n√ n
x 1 · · · x n ≥
1
x 1
+ · · · + 1 .
x n
46