Journal of Engineering Technology and Applied Sciences, 2017
Received 3 April 2017, Accepted 29 April 2017
Published online 10 June 2017
e-ISSN: 2548-0391
Vol. 2, No. 1, 27-31
doi: will be added
APPLICATIONS OF WEIGHTED ARITHMETIC
GEOMETRIC MEANS INEQUALITY TO FUNCTIONAL
INEQUALITIES
Halil İbrahim Çelik
Department of Mathematics, Faculty of Arts and Sciences,Marmara University, İstanbul, Turkey
[email protected]
Abstract
In this paper, a functional inequality is proven and the result is illustrated with some elementary
functions.
Keywords: Arithmetic-geometric means inequality, Young inequality, extremum values, functional
inequalities, elementary functions, monotone sequences.
1. Introduction
Let f be a positive real function defined on a subset E of the real line ℝ .The function ℎ: 𝐸𝐸 →
ℝ, ℎ(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) is defined by ℎ(𝑥𝑥) = 𝑒𝑒 𝑓𝑓(𝑥𝑥)ln𝑓𝑓(𝑥𝑥) where the function lnx is the natural
logarithm function. If f and g are two such functions defined on the set E, then it is a natural
problem to compare the values 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) 𝑔𝑔(𝑥𝑥) for each 𝑥𝑥 ∈ 𝐸𝐸. Since the arithmetic and
geometric means inequality is a very important elementary tool for making estimates or
approximation it is expected that the inequality play a role for the solution of these types of
problems.
Let 𝑊𝑊 = �𝜆𝜆 = �𝜆𝜆1, 𝜆𝜆2, . . . , 𝜆𝜆𝑛𝑛 � ∈ ℝ𝑛𝑛 : 𝜆𝜆𝑗𝑗 ∈ [0,1] 𝑓𝑓𝑓𝑓𝑓𝑓 𝑗𝑗 = 1,2, . . . , 𝑛𝑛, ∑𝑛𝑛𝑗𝑗=1 𝜆𝜆𝑗𝑗 = 1� be the set of
weight vectors and 𝑃𝑃 = �𝑥𝑥 = �𝑥𝑥1, 𝑥𝑥2, ⋯ 𝑥𝑥𝑛𝑛 � ∈ ℝ𝑛𝑛 : 𝑓𝑓𝑓𝑓𝑓𝑓 𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛, 𝑥𝑥𝑗𝑗 > 0� be the set of
positive vectors.
Definition 1.1. Let 𝜆𝜆 ∈ 𝑊𝑊and 𝑥𝑥 ∈ 𝑃𝑃. The real number
𝐴𝐴𝑛𝑛 (𝑥𝑥, 𝜆𝜆) = 𝜆𝜆1 𝑥𝑥1 + 𝜆𝜆2 𝑥𝑥2 + ⋯ + 𝜆𝜆𝑛𝑛 𝑥𝑥𝑛𝑛
is called the weighted arithmetic mean of n positive real numbers 𝑥𝑥1 , 𝑥𝑥2 , ⋯ , 𝑥𝑥𝑛𝑛 of weight 𝜆𝜆 and
the number
𝜆𝜆
𝜆𝜆
𝜆𝜆
𝐺𝐺𝑛𝑛 (𝑥𝑥, 𝜆𝜆) = 𝑥𝑥1 1 . 𝑥𝑥2 2 ⋯ 𝑥𝑥𝑛𝑛 𝑛𝑛
is called the weighted geometric mean of n positive real numbers 𝑥𝑥1 , 𝑥𝑥2 , ⋯ , 𝑥𝑥𝑛𝑛 of weight 𝜆𝜆.
For the weight vector
𝐴𝐴𝑛𝑛 (𝑥𝑥, 𝜆𝜆) = 𝐴𝐴𝑛𝑛 =
1 1
1
𝜆𝜆 = �𝑛𝑛 , 𝑛𝑛 , ⋯ , 𝑛𝑛�
1
(𝑥𝑥 + 𝑥𝑥2 + ⋯ + 𝑥𝑥𝑛𝑛 )
𝑛𝑛 1
is the arithmetic mean of the 𝑛𝑛 positive real numbers 𝑥𝑥1 , 𝑥𝑥2 , ⋯ , 𝑥𝑥𝑛𝑛 and
1
𝐺𝐺𝑛𝑛 (𝑥𝑥, 𝜆𝜆) = 𝐺𝐺𝑛𝑛 = (𝑥𝑥1 . 𝑥𝑥2 ⋯ 𝑥𝑥𝑛𝑛 )𝑛𝑛 = 𝑛𝑛�𝑥𝑥1 . 𝑥𝑥2 ⋯ 𝑥𝑥𝑛𝑛
is the geometric mean of the 𝑛𝑛 positive real numbers 𝑥𝑥1 , 𝑥𝑥2 , ⋯ , 𝑥𝑥𝑛𝑛 .
We have the following inequality between arithmetic and geometric means.
Theorem 1.2 (Arithmetic and geometric means inequality [2, 5]). 𝐺𝐺𝑛𝑛 (𝑥𝑥, 𝜆𝜆) ≤ 𝐴𝐴𝑛𝑛 (𝑥𝑥, 𝜆𝜆)for all
positive real numbers 𝑥𝑥1 , 𝑥𝑥2 , ⋯ , 𝑥𝑥𝑛𝑛 and for all 𝜆𝜆 ∈ 𝑊𝑊and the inequality is strict unless 𝑥𝑥1 =
𝑥𝑥2 = ⋯ = 𝑥𝑥𝑛𝑛 .
Arithmetic-geometric means inequalities has numerous applications in mathematics and other
areas. We note the following example which shows that certain type of extremum value
problems can be resolved easily by using this inequality.
Example 1.3. We will find the minimum value of the function
sin𝑥𝑥
sin𝑦𝑦
3
sin𝑧𝑧
𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = sin𝑦𝑦 + � sin𝑧𝑧 + �sin𝑥𝑥 over the set 𝛺𝛺 = {(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∈ ℝ3 : 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 ∈ ℝ}.
Solution. From the arithmetic and geometric means inequality for every (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∈ 𝛺𝛺 we have
𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) =
=
sin𝑥𝑥
sin𝑦𝑦 3 sin𝑧𝑧
+�
+�
sin𝑦𝑦
sin𝑧𝑧
sin𝑥𝑥
6 1 1 sin𝑥𝑥 sin𝑦𝑦 sin𝑧𝑧
sin𝑥𝑥 1 sin𝑦𝑦 1 sin𝑦𝑦 1 3 sin𝑧𝑧 1 3 sin𝑧𝑧 1 3 sin𝑧𝑧
+ �
+ �
+ �
+ �
+ �
≥ 6�
sin𝑦𝑦 2 sin𝑧𝑧 2 sin𝑧𝑧 3 sin𝑥𝑥 3 sin𝑥𝑥 3 sin𝑥𝑥
4 27 sin𝑦𝑦 sin𝑧𝑧 sin𝑧𝑧
= 22⁄3 31⁄2
Since the equality occurs when
3
3
sin𝑥𝑥
1
sin𝑦𝑦
1 3 sin𝑧𝑧
= 2 � sin𝑧𝑧 = 3 �sin𝑥𝑥 or equivalently when sin𝑦𝑦 =
sin𝑦𝑦
√3√2and sin𝑧𝑧 = 2 √3sin𝑥𝑥 by the definition of minimum value it follows that
𝑚𝑚𝑚𝑚𝑚𝑚{𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧): (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∈ 𝛺𝛺} = 22⁄3 31⁄2 .
For positive real numbers a and b and for λ ∈ (0,1) the weighted arithmetic-geometric means
inequality is the inequality
28
1
𝑎𝑎 𝜆𝜆 𝑏𝑏1−𝜆𝜆 ≤ 𝜆𝜆𝜆𝜆 + (1 − 𝜆𝜆)𝑏𝑏.
1
For 𝑝𝑝 > 1if we take 𝜆𝜆 = 𝑝𝑝 , 1 − 𝜆𝜆 = 𝑞𝑞 and replace 𝑎𝑎with 𝑎𝑎𝑝𝑝 and 𝑏𝑏 with 𝑏𝑏 𝑞𝑞 we get the Young’s
inequality 𝑎𝑎𝑎𝑎 ≤
𝑎𝑎𝑝𝑝
𝑝𝑝
+
𝑏𝑏 𝑞𝑞
𝑞𝑞
.
This inequality is one of the most important inequality in mathematics because the famous
Cauchy, Hölder and Minkowski inequalities follow from it.
2. The functional inequality and applications
Theorem 2.1. Let 𝑓𝑓, 𝑔𝑔 be positive real valued functions defined on a subset 𝐸𝐸of the real line
ℝ . Then 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥) 𝑔𝑔(𝑥𝑥) if 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥) and 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) ≥ 1 for each 𝑥𝑥 ∈ 𝐸𝐸 . The
inequality is strict if 𝑓𝑓(𝑥𝑥) < 𝑔𝑔(𝑥𝑥) and 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) ≥ 1and the equality occurs if and only if
𝑓𝑓(𝑥𝑥)
ln(𝑔𝑔(𝑥𝑥))
=
for 𝑥𝑥 ∈ 𝐸𝐸.
𝑔𝑔(𝑥𝑥)
ln(𝑓𝑓(𝑥𝑥))
𝑓𝑓(𝑥𝑥)
Proof. Let 𝑥𝑥 ∈ 𝐸𝐸be arbitrary, 𝑎𝑎 = 𝑓𝑓(𝑥𝑥), 𝑏𝑏 = 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥)and 𝜆𝜆 = 𝑔𝑔(𝑥𝑥). Then 𝑎𝑎 > 0, 𝑏𝑏 > 0and
𝜆𝜆 ∈ (0,1] . Since 1 − 𝜆𝜆 ≥ 0 and 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) ≥ 1 by the weighted arithmetic and geometric
means inequality
𝑓𝑓(𝑥𝑥)
𝑔𝑔(𝑥𝑥)
𝑓𝑓(𝑥𝑥)
𝑓𝑓(𝑥𝑥)
𝑔𝑔(𝑥𝑥)
𝑓𝑓(𝑥𝑥)
(𝑓𝑓(𝑥𝑥)
1−
𝑓𝑓(𝑥𝑥)
𝑔𝑔(𝑥𝑥)
≤
+ 𝑔𝑔(𝑥𝑥))
𝑓𝑓(𝑥𝑥)
≤ 𝑔𝑔(𝑥𝑥) 𝑓𝑓(𝑥𝑥) + �1 − 𝑔𝑔(𝑥𝑥)� (𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥)) = 𝑔𝑔(𝑥𝑥) .
𝑓𝑓(𝑥𝑥)
It follows that 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥) 𝑓𝑓(𝑥𝑥) and the inequality is strict when 𝑓𝑓(𝑥𝑥) < 𝑔𝑔(𝑥𝑥) and 𝑓𝑓(𝑥𝑥) +
𝑔𝑔(𝑥𝑥) ≥ 1.
Remark 2.2. 1) In the proof of the Theorem 2.1 we cannot directly use the properties of the
convex function 𝜑𝜑: (0, ∞) → ℝ, 𝜑𝜑(𝑥𝑥) = 𝑥𝑥 𝑥𝑥 = 𝑒𝑒 𝑥𝑥ln𝑥𝑥 since this function has the global
1
minimum value at the point 𝑥𝑥 = 𝑒𝑒. But since the function 𝜑𝜑(𝑥𝑥) = 𝑥𝑥 𝑥𝑥 is monotone increasing
1
1
on�𝑒𝑒 , ∞�the inequality is trivial when 𝑒𝑒 ≤ 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥).
2) Theorem 2.1 holds true for positive real valued function defined on arbitrary sets not just on
the subsets of the real line.
By mathematical induction we get the following result.
Corollary 2.3. For 𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛 let 𝑓𝑓𝑗𝑗 , 𝑔𝑔𝑗𝑗 be positive real valued functions defined on a subset
𝐸𝐸 of the real line ℝ. Then ∏𝑛𝑛𝑗𝑗=1 𝑓𝑓𝑗𝑗 (𝑥𝑥) 𝑓𝑓𝑗𝑗(𝑥𝑥) ≤ ∏𝑛𝑛𝑗𝑗=1 𝑔𝑔𝑗𝑗 (𝑥𝑥) 𝑔𝑔𝑗𝑗(𝑥𝑥) 𝑖𝑖𝑖𝑖𝑓𝑓𝑗𝑗 (𝑥𝑥) ≤ 𝑔𝑔𝑗𝑗 (𝑥𝑥)and 𝑓𝑓𝑗𝑗 (𝑥𝑥) +
𝑔𝑔𝑗𝑗 (𝑥𝑥) ≥ 1 for each 𝑥𝑥 ∈ 𝐸𝐸 and 𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛 The inequality is strict if 𝑓𝑓𝑗𝑗 (𝑥𝑥) < 𝑔𝑔𝑗𝑗 (𝑥𝑥) and
𝑓𝑓𝑗𝑗 (𝑥𝑥) + 𝑔𝑔𝑗𝑗 (𝑥𝑥) ≥ 1and the equality occurs if and only if
1,2, ⋯ , 𝑛𝑛.
For the polynomial functions we have the following result.
29
𝑓𝑓𝑗𝑗 (𝑥𝑥)
𝑔𝑔𝑗𝑗
=
(𝑥𝑥)
ln𝑔𝑔𝑗𝑗 (𝑥𝑥)
ln𝑓𝑓𝑗𝑗 (𝑥𝑥)
for 𝑥𝑥 ∈ 𝐸𝐸and 𝑗𝑗 =
Corollary 2.4. Let 𝑝𝑝 and 𝑞𝑞 be real polynomial functions such that 𝑚𝑚 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑛𝑛. If
0 ≤ 𝑝𝑝(0) ≤ 𝑞𝑞(0) , 0 ≤ 𝑝𝑝(𝑗𝑗) (0) ≤ 𝑞𝑞 (𝑗𝑗) (0) for 𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛 and 𝑝𝑝(0) + 𝑞𝑞(0) ≥ 1 , where
𝑝𝑝(𝑗𝑗) (0) and are 𝑞𝑞 (𝑗𝑗) (0) the 𝑗𝑗 𝑡𝑡ℎ derivatives of 𝑝𝑝 and 𝑞𝑞 at 𝑥𝑥 = 0 respectively, then 𝑝𝑝(𝑥𝑥)𝑝𝑝(𝑥𝑥) ≤
𝑞𝑞(𝑥𝑥)𝑞𝑞(𝑥𝑥) for all 𝑥𝑥 ∈ [0, ∞).
Proof. From the hypothesis we have 𝑝𝑝(𝑥𝑥) ≤ 𝑞𝑞(𝑥𝑥) and 𝑝𝑝(𝑥𝑥) + 𝑞𝑞(𝑥𝑥) ≥ 𝑝𝑝(0) + 𝑞𝑞(0) ≥ 1 for all
𝑥𝑥 ∈ [0, ∞). So the inequality follows from the Theorem 2.1.
The following result is the problem 32 given in [4].
𝜋𝜋
Proposition 2.5. For all 𝑥𝑥 ∈ (0, 4 ) (sin𝑥𝑥)sin𝑥𝑥 < (cos𝑥𝑥)cos𝑥𝑥 .
𝜋𝜋
𝜋𝜋
Proof. Since sin𝑥𝑥 < cos𝑥𝑥 and sin𝑥𝑥 + cos𝑥𝑥 = √2cos � 4 − 𝑥𝑥� > √2cos 4 = 1 for all 𝑥𝑥 ∈
𝜋𝜋
�0, 4 � the inequality follows from the Theorem 2.1.
A similar reasoning gives the inequality (cos𝑥𝑥)cos𝑥𝑥 < (sin𝑥𝑥)sin𝑥𝑥 for all real numbers 𝑥𝑥 ∈
𝜋𝜋 𝜋𝜋
�4 , 2 �.Since the functions cosx and sinx are 2π-periodic functions these inequalities hold true
𝜋𝜋
𝜋𝜋 𝜋𝜋
on2n𝜋𝜋 translations of �0, 4 � and �4 , 2 � for each 𝑛𝑛 ∈ ℤ.
𝜋𝜋
Proposition 2.6. For all 𝑥𝑥 ∈ �0, 4 � (tan𝑥𝑥)tan𝑥𝑥 < (cot𝑥𝑥)cot𝑥𝑥
2
𝜋𝜋
Proof. Since tan𝑥𝑥 + cot𝑥𝑥 = sin2x ≥ 2 > 1 and tan𝑥𝑥 < cot𝑥𝑥 for all 𝑥𝑥 ∈ �0, 4 � the inequality
follows from the Theorem 2.1.
𝜋𝜋 𝜋𝜋
A similar argument gives the inequality (cot𝑥𝑥)cot𝑥𝑥 < (tan𝑥𝑥)tan𝑥𝑥 for all 𝑥𝑥 ∈ �4 , 2 � and since
the functions tan𝑥𝑥 and cot𝑥𝑥 are 𝜋𝜋 -periodic functions these inequalities are true for 𝑛𝑛𝑛𝑛
𝜋𝜋
𝜋𝜋 𝜋𝜋
translations of �0, 4 � and �4 , 2 � for each 𝑛𝑛 ∈ ℤ.
Proposition 2.7. For all 𝑥𝑥 ∈ (0, ∞) (sinh𝑥𝑥)sinh𝑥𝑥 < (cosh𝑥𝑥)cosh𝑥𝑥 .
Proof. Since
1
1
sinh𝑥𝑥 = 2 (𝑒𝑒 𝑥𝑥 − 𝑒𝑒 −𝑥𝑥 ) < 2 (𝑒𝑒 𝑥𝑥 + 𝑒𝑒 −𝑥𝑥 ) = cosh𝑥𝑥 and sinh𝑥𝑥 + cosh𝑥𝑥 = 𝑒𝑒 𝑥𝑥 > 1 for all
(0, ∞) the inequality follows from the Theorem 2.1.
𝑥𝑥 ∈
For the inverse hyperbolic functions cosh−1 𝑥𝑥 = ln�𝑥𝑥 + √𝑥𝑥 2 − 1� and sinh−1 𝑥𝑥 = ln�𝑥𝑥 +
√𝑥𝑥 2 + 1� we have the following result: By the first derivative test the function 𝜑𝜑(𝑥𝑥) =
�𝑥𝑥 + √𝑥𝑥 2 + 1��𝑥𝑥 + √𝑥𝑥 2 − 1� is strictly monotone increasing on[1, ∞) . Since 𝜑𝜑(1) = 1 +
√2 < 𝑒𝑒 by the intermediate value theorem for a continuous function there is a unique point 𝑐𝑐𝑒𝑒 ∈
(1, ∞)such that 𝜑𝜑(𝑐𝑐𝑒𝑒 ) = 𝑒𝑒.
−1 𝑥𝑥
Proposition 2.8. For all 𝑥𝑥 ∈ [𝑐𝑐𝑒𝑒 , ∞) (cosh−1 𝑥𝑥)cosh
−1 𝑥𝑥
< (sinh−1 𝑥𝑥)sinh
.
Proof. Since cosh−1 𝑥𝑥 = ln�𝑥𝑥 + √𝑥𝑥 2 − 1� < ln�𝑥𝑥 + √𝑥𝑥 2 + 1� = sinh−1 𝑥𝑥and
30
sinh−1 𝑥𝑥 + cosh−1 𝑥𝑥 = ln �𝑥𝑥 + �𝑥𝑥 2 + 1� + ln �𝑥𝑥 + �𝑥𝑥 2 − 1�
= ln �𝑥𝑥 + �𝑥𝑥 2 + 1� �𝑥𝑥 + �𝑥𝑥 2 − 1� ≥ ln �𝑐𝑐𝑒𝑒 + �𝑐𝑐𝑒𝑒2 + 1� �𝑐𝑐𝑒𝑒 + �𝑐𝑐𝑒𝑒2 − 1�
= ln𝑒𝑒 = 1
for all 𝑥𝑥 ∈ [𝑐𝑐𝑒𝑒 , ∞) the inequality follows from the Theorem 2.1.
Theorem 2.1 can also be applied to show that certain sequences of real numbers are monotone
increasing.
�1−
1
Example 2.9. The sequence ��1 − 𝑛𝑛−1�
1
�
𝑛𝑛−1
∞
�
𝑛𝑛=3
1
is monotone increasing.
1
Solution. Let 3 ≤ 𝑛𝑛 ∈ ℕ be arbitrary. Since 1 − 𝑛𝑛−1 < 1 − 𝑛𝑛
1
1
2n−1
1 − 𝑛𝑛−1 + 1 − 𝑛𝑛 = 2 − 𝑛𝑛(𝑛𝑛−1) ≥ 2 −
1
�1 − 𝑛𝑛−1�
�1−
1
�
𝑛𝑛−1
< �1 −
2n−1
1
1 �1−𝑛𝑛�
�
.
𝑛𝑛
1
Therefore the sequence ��1 − 𝑛𝑛−1�
�1−
2n
1
�
𝑛𝑛−1
and
> 1by Theorem 2.1 we have
∞
�
is monotone increasing.
𝑛𝑛=3
We end the paper with the following question.
Question. Can the hypothesis 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) ≥ 1weakened and is there a best constant smaller
than 1for the functional inequality 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) < 𝑔𝑔(𝑥𝑥) 𝑔𝑔(𝑥𝑥) to hold?
References
[1]
[2]
[3]
[4]
[5]
G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd. ed., Cambridge, New York
1959.
N. D. Kazarinoff, Analytic Inequalities, Holt, Rinehart and Winston, New York, 1961.
I. J. Maddox, Elements of Functional Analysis, Second ed., Cambridge Univ. Press, 1988.
Thomas.J. Mildford, Olympiad Inequalites, 2006, http://www.unl.edu/amc
W. Rudin, Real and Complex Analysis , McGraw-Hill Book Company, New York, 1987.
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