ON ONE OF H. ALZER’S PROBLEMS

(1)

Journal of Inequalities in Pure and Applied Mathematics

http://jipam.vu.edu.au/

Volume 7, Issue 2, Article 71, 2006

ON ONE OF H. ALZER’S PROBLEMS

YU-DONG WU X

INCHANG

H

IGH

S

CHOOL

X

INCHANG

, Z

HEJIANG

312500 P.R. C

HINA

.

[email protected]

Received 16 November, 2005; accepted 17 January, 2006 Communicated by B. Yang

A

BSTRACT

. In this short note, the author solves an inequality problem which was posed by H.

Alzer with difference substitution.

Key words and phrases: Problem, Inequality, Difference substitution.

2000 Mathematics Subject Classification. 26D15.

1. T HE P ROBLEM

In 1993, H. Alzer posed the following inequality problem in [1]. In 2004, Ji-Chang Kuang reposed the problem in his monograph [2].

Problem 1.1. Let a

₁

, . . . , a

_n

(n ∈ N

^∗

) be real numbers with a

_i

∈ (0,

¹₂

], then prove or disprove

(1.1)

n

Y

k=1

a

_k

1 − a

_k

≤

P

n k=1

a

ⁿ_k

P

n

k=1

(1 − a

_k

)

ⁿ

where n = 4 or n = 5.

In 1995, Michael Vowe pointed out that the inequality (1.1) holds when n ≤ 3 (n ∈ N

^∗

) and does not hold when n ≥ 6 (n ∈ N

^∗

) in [3].

2. S OLUTION OF T HE P ROBLEM

In this section, we show the reader the proof of the inequality (1.1) while n = 4.

ISSN (electronic): 1443-5756

The author would like to thank Dr. Jian Chen and Prof. Lu Yang for their enthusiastic help.

341-05

(2)

2 Y.-D. WU

Proof. We set a

_k

=

^b_u^k

(k = 1, 2, 3, 4), where u > 0 and 0 < a

_k

≤

¹₂

(k = 1, 2, 3, 4), then 0 < b

_k

≤

¹₂

u (k = 1, 2, 3, 4). So the inequality (1.1) is equivalent to the inequality as follows.

(2.1)

4

Y

k=1

b

_k

u − b

k

≤

P

4 k=1

b

⁴_k

P

4

k=1

(u − b

_k

)

⁴

.

Inequality (2.1) is equivalent to the following inequality.

(b

₃⁴

+ b

₄⁴

+ b

₂⁴

+ b

₁⁴

− 4 b

₁

b

₂

b

₃

b

₄

)u

³

+ (−b

₁

b

₃⁴

− b

₁⁵

+ 4 b

₁

b

₂

b

₃

b

₄²

− b

₂

b

₃⁴

− b

₄

b

₃⁴

(2.2)

− b

₄

b

₁⁴

− b

₃

b

₄⁴

− b

₄⁵

− b

₄

b

₂⁴

− b

₁

b

₄⁴

− b

₃⁵

− b

₂

b

₄⁴

− b

₁

b

₂⁴

− b

₂

b

₁⁴

− b

₃

b

₁⁴

+ 4 b

₁

b

₂

b

₃²

b

₄

− b

₃

b

₂⁴

− b

₂⁵

+ 4 b

₁²

b

₂

b

₃

b

₄

+ 4 b

₁

b

₂²

b

₃

b

₄

)u

²

+ (b

₂

b

₃⁵

+ b

₁

b

₃⁵

+ b

₁⁵

b

₃

+ b

₁

b

₄⁵

+ b

₁⁵

b

₄

+ b

₂⁵

b

₃

− 6 b

₁

b

₂

b

₃

b

₄³

− 6 b

₁

b

₂

b

₃³

b

₄

− 6 b

₁

b

₂³

b

₃

b

₄

− 6 b

₁³

b

₂

b

₃

b

₄

+ b

₂

b

₄

b

₁⁴

+ b

₃

b

₄

b

₂⁴

+ b

₃

b

₄

b

₁⁴

+ b

₁

b

₂

b

₄⁴

+ b

₁

b

₂

b

₃⁴

+ b

₁

b

₃

b

₄⁴

+ b

₁

b

₃

b

₂⁴

+ b

₁

b

₄

b

₃⁴

+ b

1

b

4

b

24

+ b

2

b

3

b

44

+ b

2

b

3

b

14

+ b

2

b

4

b

34

+ b

2

b

45

+ b

25

b

4

+ b

3

b

45

+ b

35

b

4

+ b

15

b

2

+ b

₁

b

₂⁵

)u − b

₂

b

₃

b

₄⁵

+ 3 b

₁

b

₂

b

₄

b

₃⁴

− b

₁

b

₃

b

₄⁵

− b

₁

b

₃⁵

b

₄

+ 3 b

₁

b

₃

b

₄

b

₂⁴

+ 3 b

₁

b

₂

b

₃

b

₄⁴

− b

₁⁵

b

₃

b

₄

− b

₂

b

₃⁵

b

₄

− b

₂⁵

b

₃

b

₄

− b

₁

b

₂⁵

b

₄

+ 3 b

₂

b

₃

b

₄

b

₁⁴

− b

₁⁵

b

₂

b

₃

− b

₁

b

₂

b

₄⁵

− b

₁

b

₂

b

₃⁵

− b

₁⁵

b

₂

b

₄

− b

₁

b

₂⁵

b

₃

≥ 0.

Inequality (2.2) is symmetrical for b

_k

(1 ≤ k ≤ 4, k ∈ N

^∗

), so there is no harm in supposing that b

₁

≤ b

₂

≤ b

₃

≤ b

₄

. Then we can set

(2.3)



 

 

 

 

b

₂

= b

₁

+ c

₁

; b

₃

= b

₁

+ c

₁

+ c

₂

; b

₄

= b

₁

+ c

₁

+ c

₂

+ c

₃

;

u = 2(b

₁

+ c

₁

+ c

₂

+ c

₃

) + c

₄

,

where c

i

≥ 0 (1 ≤ i ≤ 4, i ∈ N

^∗

).

The substitution (2.3) was called a difference substitution in [4] (see also [5]). Substituting (2.3) in (2.2), we obtain the result (3.1) (see Appendix). Since every monomial on the left of (3.1) is nonnegative, the last inequality obviously holds, then the inequality (1.1) holds when n = 4.

Thus, the proof of the inequality (1.1) (n = 4) is completed.

3. R EMARKS

Remark 3.1. In the same manner, we can also prove the inequality (1.1) holds when n = 5.

Remark 3.2. The operations in this paper were implemented using mathematics software Maple 9.0.

J. Inequal. Pure and Appl. Math., 7(2) Art. 71, 2006 http://jipam.vu.edu.au/

(3)

ONONE OFH. ALZER’SPROBLEMS 3

A PPENDIX

30b

₁

c

₄²

c

₁⁴

+ 10 b

₁

c

₁⁶

+ 120 c

₄

b

₁³

c

₁

c

₂

c

₃

+ 720 c

₄

b

₁

c

₁²

c

₂

c

₃²

+ 9 c

₄

c

₁⁶

(3.1)

+ 824 c

₄

b

₁

c

₁²

c

₂²

c

₃

+ 10 c

₄

c

₂⁶

+ 498 c

₄

b

₁

c

₁³

c

₂

c

₃

+ 13 b

₁²

c

₁⁵

+ 51 b

₁²

c

₃⁵

+ 44 b

₁²

c

₂⁵

+ 8 b

₁³

c

₁⁴

+ 40 b

₁³

c

₃⁴

+ 32 b

₁³

c

₂⁴

+ 120 c

₄²

b

₁²

c

₁

c

₂

c

₃

+ 10 b

₁⁴

c

₃³

+ 20 b

₁⁴

c

₁

c

₂

c

₃

+ 152 b

₁³

c

₁²

c

₂

c

₃

+ 208 b

₁³

c

₁

c

₂²

c

₃

+ 200 b

₁³

c

₁

c

₂

c

₃²

+ 346 b

12

c

13

c

2

c

3

+ 624 b

12

c

12

c

22

c

3

+ 600 b

12

c

12

c

2

c

32

+ 550 b

12

c

1

c

2

c

33

+ 556 b

₁²

c

₁

c

₂³

c

₃

+ 780 b

₁²

c

₁

c

₂²

c

₃²

+ 904 b

₁

c

₁

c

₂²

c

₃³

+ 318 b

₁

c

₁⁴

c

₂

c

₃

+ 696 b

1

c

13

c

22

c

3

+ 652 b

1

c

13

c

2

c

32

+ 756 b

1

c

12

c

2

c

33

+ 824 b

1

c

12

c

23

c

3

+ 1128 b

₁

c

₁²

c

₂²

c

₃²

+ 490 b

₁

c

₁

c

₂

c

₃⁴

+ 504 b

₁

c

₁

c

₂⁴

c

₃

+ 912 b

₁

c

₁

c

₂³

c

₃²

+ 2 b

14

c

13

+ 8 b

14

c

23

+ 74 b

12

c

14

c

2

+ 40 b

13

c

13

c

2

+ 8 b

14

c

12

c

2

+ 10 b

₁⁴

c

₁²

c

₃

+ 12 b

₁⁴

c

₁

c

₂²

+ 10 b

₁⁴

c

₁

c

₃²

+ 20 b

₁⁴

c

₂²

c

₃

+ 20 b

₁⁴

c

₂

c

₃²

+ 48 b

₁³

c

₁³

c

₃

+ 80 b

₁³

c

₁²

c

₂²

+ 80 b

₁³

c

₁²

c

₃²

+ 80 b

₁³

c

₁

c

₂³

+ 80 b

₁³

c

₁

c

₃³

+ 120 b

₁³

c

₂

c

₃³

+ 112 b

₁³

c

₂³

c

₃

+ 160 b

₁³

c

₂²

c

₃²

+ 158 b

₁²

c

₁

c

₂⁴

+ 165 b

₁²

c

₁

c

₃⁴

+ 83 b

₁²

c

₁⁴

c

₃

+ 178 b

₁²

c

₁³

c

₂²

+ 180 b

12

c

13

c

32

+ 232 b

12

c

12

c

23

+ 220 b

12

c

12

c

33

+ 210 b

12

c

2

c

34

+ 370 b

₁²

c

₂²

c

₃³

+ 194 b

₁²

c

₂⁴

c

₃

+ 360 b

₁²

c

₂³

c

₃²

+ 112 b

₁

c

₁

c

₃⁵

+ 116 b

1

c

1

c

25

+ 236 b

1

c

12

c

24

+ 210 b

1

c

12

c

34

+ 62 b

1

c

15

c

2

+ 64 b

1

c

15

c

3

+ 168 b

₁

c

₁⁴

c

₂²

+ 158 b

₁

c

₁⁴

c

₃²

+ 260 b

₁

c

₁³

c

₂³

+ 224 b

₁

c

₁³

c

₃³

+ 122 b

₁

c

₂

c

₃⁵

+ 3 c

₁⁷

+ 4 c

₂⁷

+ 356 b

₁

c

₂³

c

₃³

+ 124 b

₁

c

₂⁵

c

₃

+ 276 b

₁

c

₂⁴

c

₃²

+ 280 c

₁²

c

₂

c

₃⁴

+ 534 c

₁²

c

₂²

c

₃³

+ 24 b

₁

c

₂⁶

+ 72 c

₁²

c

₂⁵

+ 61 c

₁²

c

₃⁵

+ 20 c

₁⁶

c

₂

+ 26 c

₁

c

₂⁶

+ 59 c

₂²

c

₁⁵

+ 72 c

₂²

c

₃⁵

+ 110 c

₁³

c

₂⁴

+ 85 c

₁³

c

₃⁴

+ 19 c

₁⁶

c

₃

+ 50 c

₁⁵

c

₃²

+ 102 c

₁⁴

c

₂³

+ 78 c

₁⁴

c

₃³

+ 110 c

₂³

c

₃⁴

+ 104 c

₂⁴

c

₃³

+ 24 c

₂⁶

c

₃

+ 64 c

₂⁵

c

₃²

+ 22 b

₁

c

₃⁶

+ 24 c

₁

c

₃⁶

+ 26 c

₂

c

₃⁶

+ 107 c

₁⁵

c

₂

c

₃

+ 268 c

₁⁴

c

₂²

c

₃

+ 242 c

14

c

2

c

32

+ 133 c

1

c

2

c

35

+ 380 c

13

c

23

c

3

+ 508 c

13

c

22

c

32

+ 310 c

24

c

12

c

3

+ 552 c

₂³

c

₁²

c

₃²

+ 280 c

₂²

b

₁

c

₃⁴

+ 46 c

₄

b

₁²

c

₁⁴

+ 84 c

₄

b

₁²

c

₂⁴

+ 78 c

₄

b

₁²

c

₃⁴

+ 6 c

₄

b

₁⁴

c

₁²

+ 8 c

₄

b

₁⁴

c

₂²

+ 6 c

₄

b

₁⁴

c

₃²

+ 56 c

₄

c

₁

c

₂⁵

+ 42 c

₄

c

₁

c

₃⁵

+ 28 c

₄

c

₁³

b

₁³

+ 48 c

₄

c

₂

c

₁⁵

+ 45 c

₄

c

₂

c

₃⁵

+ 192 c

₄

c

₁

b

₁²

c

₃³

+ 326 c

₁³

c

₂

c

₃³

+ 305 c

₁

c

₂²

c

₃⁴

+ 386 c

₁

c

₂³

c

₃³

+ 134 c

₁

c

₂⁵

c

₃

+ 298 c

₁

c

₂⁴

c

₃²

+ 44 c

₄

b

₁³

c

₃³

+ 33 c

₄

b

₁

c

₁⁵

+ 52 c

₄

b

₁

c

₂⁵

+ 39 c

₄

b

₁

c

₃⁵

+ 168 c

₄

b

₁²

c

₁³

c

₂

+ 8 c

₄

b

₁⁴

c

₁

c

₂

+ 4 c

₄

b

₁⁴

c

₁

c

₃

+ 8 c

4

b

14

c

2

c

3

+ 72 c

4

b

13

c

12

c

2

+ 60 c

4

b

13

c

12

c

3

+ 88 c

4

b

13

c

1

c

22

+ 60 c

₄

b

₁³

c

₁

c

₃²

+ 104 c

₄

b

₁³

c

₂²

c

₃

+ 96 c

₄

b

₁³

c

₂

c

₃²

+ 150 c

₄

c

₁⁴

b

₁

c

₂

+ 152 c

₄

c

₁³

b

₁²

c

₃

+ 276 c

₄

c

₁²

b

₁²

c

₂²

+ 208 c

₄

c

₁²

b

₁²

c

₃²

+ 187 c

₄

c

₁⁴

c

₂

c

₃

+ 376 c

₄

c

₁³

c

₂²

c

₃

+ 320 c

₄

c

₁³

c

₂

c

₃²

+ 214 c

₄

c

₁

b

₁

c

₂⁴

+ 169 c

₄

c

₁

b

₁

c

₃⁴

+ 252 c

₄

c

₂

b

₁²

c

₃³

+ 306 c

₄

c

₂²

b

₁

c

₁³

+ 240 c

₄

c

₂³

b

₁²

c

₁

+ 264 c

₄

c

₂³

b

₁²

c

₃

+ 352 c

₄

c

₂²

b

₁²

c

₃²

+ 404 c

₄

c

₂³

c

₁²

c

₃

+ 516 c

₄

c

₂²

c

₁²

c

₃²

+ 182 c

₄

c

₂

b

₁

c

₃⁴

(4)

4 Y.-D. WU

+ 436 c

₄

c

₁²

b

₁²

c

₂

c

₃

+ 552 c

₄

c

₁

b

₁²

c

₂²

c

₃

+ 496 c

₄

c

₁

b

₁²

c

₂

c

₃²

+ 48 c

₄

c

₂³

b

₁³

+ 130 c

4

c

12

c

24

+ 91 c

4

c

12

c

34

+ 42 c

4

c

15

c

3

+ 115 c

4

c

14

c

22

+ 82 c

4

c

14

c

32

+ 160 c

₄

c

₁³

c

₂³

+ 104 c

₄

c

₁³

c

₃³

+ 105 c

₄

c

₂²

c

₃⁴

+ 130 c

₄

c

₂³

c

₃³

+ 46 c

₄

c

₂⁵

c

₃

+ 98 c

₄

c

₂⁴

c

₃²

+ 135 c

₄

b

₁

c

₁⁴

c

₃

+ 228 c

₄

b

₁

c

₁³

c

₃²

+ 352 c

₄

b

₁

c

₁²

c

₂³

+ 252 c

₄

b

₁

c

₁²

c

₃³

+ 338 c

₄

b

₁

c

₂²

c

₃³

+ 202 c

₄

b

₁

c

₂⁴

c

₃

+ 344 c

₄

b

₁

c

₂³

c

₃²

+ 196 c

₄

c

₁

c

₂

c

₃⁴

+ 364 c

₄

c

₁

c

₂²

c

₃³

+ 216 c

₄

c

₁

c

₂⁴

c

₃

+ 368 c

₄

c

₁

c

₂³

c

₃²

+ 338 c

₄

c

₁²

c

₂

c

₃³

+ 590 c

₄

b

₁

c

₁

c

₂

c

₃³

+ 668 c

₄

b

₁

c

₁

c

₂³

c

₃

+ 868 c

₄

b

₁

c

₁

c

₂²

c

₃²

+ 16 c

₄²

b

₁³

c

₁

c

₂

+ 8 c

₄²

b

₁³

c

₁

c

₃

+ 80 c

₄²

b

₁²

c

₁²

c

₂

+ 58 c

₄²

b

₁²

c

₁²

c

₃

+ 96 c

₄²

b

₁²

c

₁

c

₂²

+ 62 c

₄²

b

₁²

c

₁

c

₃²

+ 16 c

₄²

b

₁³

c

₂

c

₃

+ 96 c

₄²

b

₁²

c

₂²

c

₃

+ 88 c

₄²

b

₁²

c

₂

c

₃²

+ 12 c

₄²

b

₁³

c

₁²

+ 34 c

₄²

b

₁²

c

₁³

+ 16 c

₄²

b

₁³

c

₂²

+ 48 c

42

b

12

c

23

+ 12 c

42

b

13

c

32

+ 96 b

1

c

42

c

13

c

2

+ 148 b

1

c

42

c

12

c

22

+ 120 b

₁

c

₄²

c

₁

c

₂³

+ 72 b

₁

c

₄²

c

₁³

c

₃

+ 88 b

₁

c

₄²

c

₁²

c

₃²

+ 196 b

₁

c

₄²

c

₁²

c

₂

c

₃

+ 240 b

₁

c

₄²

c

₁

c

₂²

c

₃

+ 208 b

₁

c

₄²

c

₁

c

₂

c

₃²

+ 9 c

₄²

c

₁⁵

+ 8 c

₄²

c

₂⁵

+ 36 c

₄²

b

₁

c

₂⁴

+ 38 c

₄²

c

₁

c

₂⁴

+ 36 c

₄²

c

₁⁴

c

₂

+ 68 c

₄²

c

₁³

c

₂²

+ 72 c

₄²

c

₁²

c

₂³

+ 8 c

₃⁶

c

₄

+ 5 c

₄²

c

₃⁵

+ 4 c

₃⁷

+ 38 c

₄²

b

₁²

c

₃³

+ 22 c

₄²

b

₁

c

₃⁴

+ 92 c

₄²

c

₁³

c

₂

c

₃

+ 144 c

₄²

c

₁²

c

₂²

c

₃

+ 120 c

₄²

c

₁²

c

₂

c

₃²

+ 23 c

₄²

c

₁

c

₃⁴

+ 27 c

₄²

c

₁⁴

c

₃

+ 38 c

₄²

c

₁³

c

₃²

+ 42 c

₄²

c

₁²

c

₃³

+ 24 c

₄²

c

₂

c

₃⁴

+ 46 c

₄²

c

₂²

c

₃³

+ 26 c

₄²

c

₂⁴

c

₃

+ 44 c

₄²

c

₂³

c

₃²

+ 80 c

₄²

b

₁

c

₁

c

₃³

+ 84 c

₄²

b

₁

c

₂

c

₃³

+ 96 c

₄²

b

₁

c

₂³

c

₃

+ 120 c

₄²

b

₁

c

₂²

c

₃²

+ 88 c

₄²

c

₁

c

₂

c

₃³

+ 100 c

₄²

c

₁

c

₂³

c

₃

+ 126 c

₄²

c

₁

c

₂²

c

₃²

+ 3 c

₄³

c

₁⁴

+ 6 c

₄³

b

₁²

c

₁²

+ 8 c

₄³

b

₁

c

₁³

+ 2 c

₄³

c

₂⁴

+ c

₄³

c

₃⁴

+ 8 c

₄³

b

₁²

c

₁

c

₂

+ 4 c

₄³

b

₁²

c

₁

c

₃

+ 8 c

₄³

b

₁²

c

₂

c

₃

+ 12 c

₄³

c

₁²

c

₂

c

₃

+ 16 c

₄³

b

₁

c

₁²

c

₂

+ 8 c

₄³

b

₁

c

₁²

c

₃

+ 20 c

₄³

b

₁

c

₁

c

₂²

+ 12 c

₄³

b

₁

c

₁

c

₃²

+ 12 c

₄³

b

₁

c

₂²

c

₃

+ 12 c

₄³

b

₁

c

₂

c

₃²

+ 12 c

₄³

c

₁

c

₂²

c

₃

+ 12 c

₄³

c

₁

c

₂

c

₃²

+ 20 c

₄³

b

₁

c

₁

c

₂

c

₃

+ 8 c

₄³

b

₁²

c

₂²

+ 6 c

₄³

b

₁²

c

₃²

+ 8 c

₄³

b

₁

c

₂³

+ 4 c

₄³

b

₁

c

₃³

+ 8 c

₄³

c

₁³

c

₂

+ 4 c

₄³

c

₁³

c

₃

+ 12 c

₄³

c

₁²

c

₂²

+ 6 c

₄³

c

₁²

c

₃²

+ 8 c

₄³

c

₁

c

₂³

+ 4 c

₄³

c

₁

c

₃³

+ 4 c

₄³

c

₂

c

₃³

+ 4 c

₄³

c

₂³

c

₃

+ 6 c

₄³

c

₂²

c

₃²

≥ 0.

R EFERENCES

[1] HORST ALZER, Problem 10337, Amer. Math. Monthly, 100(8) (1993), 798.

[2] JI-CHANG KUANG. Applied Inequalities (in Chinese), Shandong Science and Technology Press, 3rd. Ed., 2004, p. 156.

[3] M. VOWE, An unsettled problem, Amer. Math. Monthly, 102(7) (1995), 659–660.

[4] L. YANG, Solving harder problems with lesser mathematics, Proceedings of the 10th Asian Tech- nology Conference in Mathematics, December 12-16, 2005, Cheong-Ju, South Korea.

[5] BAO-QIAN LIU, The generating operation and its application in the proof of the symmetrical in- equality in n variables, Journal of Guangdong Education Institute (in Chinese), 25(3) (2005), 10–14.