✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❚❍➚ ❍➬◆● ❍❖❆

▼❐❚ ❙➮ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❍⑨▼ ▲➬■
❱⑨ Ù◆● ❉Ö◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

◆●❯❨➍◆ ❚❍➚ ❍➬◆● ❍❖❆

▼❐❚ ❙➮ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❍⑨▼ ▲➬■
❱⑨ Ù◆● ❉Ö◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ sì ❝➜♣
▼➣ sè✿ ✽✹✻✵✶✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆

P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨


✹

✶✳✶✳✷

❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐ ❦❤↔ ✈✐ ✼

Ù♥❣ ❞ö♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷✳✶

Ù♥❣ ❞ö♥❣ tr♦♥❣ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✶✹

✶✳✷✳✷

Ù♥❣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣
❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

❈❤÷ì♥❣ ✷✳ ❍➔♠ ❧ç✐ s✉② rë♥❣ ✈➔ ù♥❣ ❞ö♥❣
✷✳✶

✷✳✷

✷✳✸

✷✶

❍➔♠ J ✲❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✶✳✶

❍➔♠ ❧ç✐ tr➯♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶


❑➳t ❧✉➟♥

✹✶

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✷


✶

❇↔♥❣ ❦þ ❤✐➺✉
R

t➟♣ sè t❤ü❝

Lp [a, b]

❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ ✤♦↕♥ [a, b]

Co

♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ C

A

tr✉♥❣ ❜➻♥❤ ❝ë♥❣

G

❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤✳ ▲✉➟♥ ✈➠♥ ❝ô♥❣ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ s✉②
rë♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❦❤↔ ✈✐ n✲❧➛♥✱ ❤➔♠

J ✲❧ç✐✱ ❤➔♠ s✲❧ç✐✱ ❤➔♠ s✲❧ã♠ tr♦♥❣ ❝→❝ ❝æ♥❣ tr➻♥❤ ❬✼❪✱ ❬✽❪ ❝æ♥❣ ❜è ♥➠♠ ✷✵✶✷
✈➔ ✷✵✶✼✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤
❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐ ♠ët
❜✐➳♥✱ ❤➔♠ ❧ç✐ ❦❤↔ ✈✐ ❜➟❝ ♥❤➜t✱ ❜➟❝ ❤❛✐✱ ❜➟❝ n ✈➔ ù♥❣ ❞ö♥❣ ✤→♥❤ ❣✐→ ♠ët
sè ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t➟♣ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣
❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ ❤➔♠ J ✲❧ç✐ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❧î♣
❤➔♠ J ✲❧ç✐✱ ❦❤→✐ ♥✐➺♠ ❤➔♠ s✲❧ç✐✱ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ s✲❧ç✐✱ ✈➼ ❞ö ✈➲ ❤➔♠ s✲❧ç✐✳
❚r➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ s✲❧ç✐✱ tr➻♥❤ ❜➔②




tt ự t tự ũ ởt số ự ử
tr tr t
ữủ t t rữớ ồ ồ ồ
r q tr ồ t tỹ rữớ
ồ ồ t ồ tốt t t ồ t
ự ữủ tọ ỏ t ỡ t t ổ
tr tr rữớ ồ ồ ồ

t t tọ ỏ t ỡ s s tợ P
ừ ữớ t t ữợ t t

ỡ ỳ ữớ t tr t sự tổ
s t tốt t tổ tổ õ t ồ t ự

♠å✐ x, y ∈ [a, b] ✈➔ λ ∈ [0, 1] t❤➻

f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y).
❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ ♥➳✉ ❤➔♠ (−f ) ❧➔ ❧ç✐✳

❍➺ q✉↔ ✶✳✶✳✷ ✭❬✶✶✱ ❍➺ q✉↔ ✷✳✶❪✮ ❍➔♠ f (x) ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr➯♥ ❦❤♦↔♥❣ ♠ð

❧➔ ❤➔♠ ❧ç✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ♥â ❦❤æ♥❣ ➙♠
tr➯♥ t♦➔♥ ❦❤♦↔♥❣ (a, b)✳
(a, b) ⊆ R

❘➜t ♥❤✐➲✉ ❜➜t ✤➥♥❣ t❤ù❝ q✉❛♥ trå♥❣ ✤÷ñ❝ t❤✐➳t ❧➟♣ tø ❧î♣ ❝→❝ ❤➔♠ ❧ç✐✳
▼ët tr♦♥❣ ♥❤ú♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ♥ê✐ t✐➳♥❣ ♥❤➜t ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕


✺

❍❛❞❛♠❛r❞ ✭❝á♥ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛❞❛♠❛r❞✮✳ ❇➜t ✤➥♥❣ t❤ù❝ ❦➨♣ ♥➔②
✤÷ñ❝ ♣❤→t ❜✐➸✉ tr♦♥❣ ✤à♥❤ ❧þ s❛✉✳

✣à♥❤ ❧þ ✶✳✶✳✸ ✭❬✸✱ ❚❤❡ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ■♥t❡❣r❛❧ ■♥❡q✉❛❧✐t②❪✮ ❈❤♦ f

♠ët ❤➔♠ ❧ç✐ tr➯♥ [a, b] ⊂ R✱ a = b✳ ❑❤✐ ✤â
1
a+b
f
≤
2
b−a



a

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ❤➔♠ f ❧ç✐ tr➯♥ ✤♦↕♥ [a, b]✱ ♥➯♥ ✈î✐ ♠å✐ λ ∈ [0, 1] t❛ ❝â
f λa + (1 − λ)b ≤ λf (a) + (1 − λ)f (b).
▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ t❤❡♦ λ tr➯♥ ✤♦↕♥ [0, 1]✱ t❛ ♥❤➟♥ ✤÷ñ❝
1

1

f λa + (1 − λ)b dλ ≤ f (a)
0

1

0

❱➻

1

0

1

(1 − λ)dλ =

λdλ =
0


a+b
=f
.
2

✭✶✳✸✮


✻

❚➼❝❤ ♣❤➙♥ ❤❛✐ ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② t❤❡♦ λ tr➯♥ ✤♦↕♥ [0, 1] t❛ ♥❤➟♥ ✤÷ñ❝

 1
1
a+b
1
≤  f (λa + (1 − λ)b)dλ + f ((1 − λ)a + λb)dλ
f
2
2
0

0
b

1
=
b−a

f (x)dx.

(b − a)2 .
24

✭✶✳✹✮

a

m 2
x ✈î✐ ♠å✐ x ∈ [a, b]✳ ❑❤✐ ✤â✱
2
f (x) = g (x) − m ≥ 0, ∀x ∈ (a, b)

❈❤ù♥❣ ♠✐♥❤✳ ✣➦t f (x) = g(x) −

❝❤ù♥❣ tä ❤➔♠ f ❧➔ ❧ç✐ tr➯♥ ❦❤♦↔♥❣ ♠ð (a, b)✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝
❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ f t❛ ♥❤➟♥ ✤÷ñ❝

g

a+b
2

m
−
2

a+b
2

2


m a2 + ab + b2
.
2
3

a
b

1
=
b−a
a

❉♦ ✤â✱

m a2 + ab + b2 m
−
2
3
2

a+b
2

2

b

1


a+b
.
2

a

t tự tữỡ ữỡ ợ
b

1
m
(b a)2
24
ba

a+b
.
2

g(x)dx g
a

ữ t tự tự t ừ ữủ ự
ự t tự tự ừ t ử ự
tữỡ tỹ ữ ợ t tự tự t
M 2
h(x) =
x g(x), x [a, b].
2



a



t tự rtr ỗ

ỵ Lp [a, b] ổ t p 1 p < tr
[a, b] f (x) Lp [a, b] t
b

|f (x)|p dx < .
a

t sỷ f : [a, b] R R tr
[a, b] ợ a < b f L1 [a, b] t
b

f (a) + f (b)
1

2
ba

b

1
f (t)dt =
ba

b−a
f (a) + f (b)
1
f (a) − f (b) ≥
−
8
2
b−a

✭✶✳✼✮

f (x)dx ≥ 0.
a

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤♦ ❤➔♠ ϕ ✭①❡♠ ❬✶✵❪✮✿
1
a+b
ϕ
2
2

b

ϕ(a) + ϕ(b)
1
+
≥
2
b−a


tr➯♥ [a, b] ✈➔ p > 1✳ ◆➳✉ |f | ❧➔ q✲❦❤↔ t➼❝❤ tr➯♥ [a, b]✱ tr♦♥❣ ✤â p1 + 1q = 1✱
t❤➻

✣à♥❤ ❧þ ✶✳✶✳✽ ✭❬✹✱ ✣à♥❤ ❧þ ✷✻❪✮

b

1
f (a) + f (b)
−
2
b−a

1
p

f (t)dt ≤
a



1 (b − a) 
2 (p + 1) p1

 1q

b
q

✭✶✳✽✮

a

a+b
dx
2

1
p

×

1
b−a

1
q

b

| f (x) |q dx
a

tr♦♥❣ ✤â✱
b
a

a+b
x−
f (x)dx = 2
2

a

1
p

(b − a)p
(p + 1)2p

=

1
p

p

b

1
b−a

×

1
b−a

1
p

| f (x) |q dx
a

C ◦ ✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ C ✱ a, b ∈ C ✱ ✈î✐ a < b ✈➔ f ∈ L1 [a, b]✳ ❑❤✐ ✤â✱

❇ê ✤➲ ✶✳✶✳✾ ✭❬✹✱ ❇ê ✤➲ ✸❪✮

b

a+b
2

f

b

1
f (x)dx =
b−a

1
−
b−a
a

tr♦♥❣ ✤â✱
p(x) =

✭✶✳✾✮

p(x)f (x)dx,
a


−

1
b−a

1
p

f (t)dt ≤
a



1 (b − a) 
2 (p + 1) p1

 1q

b

q
|f (t)| dt .

✭✶✳✶✵✮

a

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r✱ t❛ ❝â✿
1
b−a


.


✶✵

▼➦t ❦❤→❝✱
a+b
2

b
p

| p(x) | dx =
a

b

| x − a |p dx +

| x − b |p dx
a+b
2

a

(b − a)p+1
= p
.
2 (p + 1)

✭✶✳✶✶✮

(ii)

◆➳✉ t❤➯♠ ❣✐↔ t❤✐➳t m ≤ f

(x) ≤ M ✱ m, M
b

2

f (a) + f (b)
1
(b − a)
≤
−
m
12
2
b−a

❧➔ ❝→❝ ❤➡♥❣ sè✱ t❤➻

(b − a)2
f (t)dt ≤ M
.
12

a


(f (a) + f (b)) −
f (x)dx.
=
2
a

(ii) ❚❛ ❝â✿
m(x − a)(b − x) ≤ (x − a)(b − x)f (x) ≤ M (x − a)(b − x)

✭✶✳✶✷✮




ợ ồ x [a, b] õ

m
2

b
a

b

1
(x a)(b x)dx
2
M

2


(b a)3
(x a)(b x) =
.
6

ứ s r
ởt số t tự n

ờ ờ sỷ f : [a, b] R n

f (n) L1[a, b] t
n1

b

f (t)dt =
a

k=0

(x a)k+1 f (k) (a) + (1k )(b x)k+1 f (k) (b)
(k + 1)!

(x a)n+1
+ (1 )
n!
n+1
n (b x)
+ (1 )


: (C R) R

(x a)k+1 f (k) (a) + (1k )(b x)k+1 f (k) (b)
(k 1)!

k=0
n+1



(x a)
(n + 1)|f (n) (a)| + |f (n) (x)|
(n + 2)!
(b x)n+1
+
|f (n) (x)| + (n + 1)|f (n) (b)| ,
(n + 2)!

x [a, b]




✶✷

❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ ✶✳✶✳✶✷ ✈➔ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ trà t✉②➺t ✤è✐ t❛
❝â t❤➸ ✈✐➳t✿
n−1



f (t)dt −
a

(x − a)k+1 f (k) (a) + (−1k )(b − x)k+1 f (k) (b)
(k − 1)!

k=0
n+1

1
(x − a)
(1 − t)n t|f (n) (x)| + (1 − t)|f (n) (a)| dt
≤
n!
0
1
n+1
(b − x)
+
(1 − t)n t|f (n) (x)| + (1 − t)|f (n) (b)| dt
n!
0
n+1
(x − a)
(n + 1)|f (n) (a)| + |f (n) (x)|
=
(n + 2)!
(b − x)n+1
+

✶✸

❝❤å♥ n = 2✱ x = a +2 b ✈➔ f (x) = f (a + b − x) t❤➻
b
1
f (a) + f (b)
−
f (t)dt
2
b−a a
(b − a)2
a+b
)| + 3|f (b)|
≤
3|f (a)| + 2|f (
192
2
(b − a)2
≤
|f (a)| + |f (b)| .
48

✣à♥❤ ❧þ ✶✳✶✳✶✻ ✭①❡♠ ❬✼✱ ✣à♥❤ ❧þ ✷✳✷❪✮ ●✐↔ sû f : C ⊂ R → R ❧➔ ❤➔♠ ❦❤↔

✈✐ n✲❧➛♥✱ a, b ∈ C ✈➔ a < b✱ x ∈ [a, b]✳ ◆➳✉ f (n) ∈ L1[a, b] ✈➔ |f (n)|q ✱ n ≥ 1✱
❧ç✐ tr➯♥ [a, b] t❤➻
n−1

b



ð ✤➙② p1 + 1q = 1✳
❈❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✶✳✶✳✶✷ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❍☎♦❞❡r
t❛ ♥❤➟♥ ✤÷ñ❝
n−1

b

f (t)dt −
a

k=0

(x − a)k+1 f (k) (a) + (−1k )(b − x)k+1 f (k) (b)
(k − 1)!

(x − a)n+1
≤
n!

1
p

1

(1 − t)np dt
0

(b − x)n+1
+

❱➻ |f (n) |q ❧➔ ❤➔♠ ❧ç✐ tr➯♥ [a, b] ♥➯♥
n−1

b

f (t)dt −
a

k=0

(x − a)k+1 f (k) (a) + (−1k )(b − x)k+1 f (k) (b)
(k − 1)!

(x − a)n+1
≤
n!

1
np + 1

(b − x)n+1
+
n!
=

1
np + 1

1
p

+
n!
2

1
q

1
q

.

❚r♦♥❣ ✣à♥❤ ❧þ ✶✳✶✳✶✻✱ ♥➳✉ t❛ ❝❤å♥
✈➔ f (x) = f (a + b − x) t❤➻

❍➺ q✉↔ ✶✳✶✳✶✼ ✭①❡♠ ❬✼✱ ❇ê ✤➲ ✷✳✶❪✮
a+b
n = 2✱ x =
2

f (a) + f (b)
1
−
2
b−a

b

f (t)dt
a

×
+
.


2
2





✶✳✷

Ù♥❣ ❞ö♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞

✶✳✷✳✶

Ù♥❣ ❞ö♥❣ tr♦♥❣ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤

❚✐➸✉ ♠ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët ✈➔✐ ù♥❣ ❞ö♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕
❍❛❞❛♠❛r❞ ✤➸ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ s❛✉ ✤➙②✿
✭❛✮ ❚r✉♥❣ ❜➻♥❤ ❝ë♥❣✿

A = A(a, b) :=

a+b
,
2


✭❞✮ ❚r✉♥❣ ❜➻♥❤ ❧æ❣❛r✐t✿

b−a
, a = b;
ln b − ln a
L = L(a, b) :=
 a, a = b,



a, b > 0.

✭✶✳✶✽✮

a = b;

✭✶✳✶✾✮

✭❡✮ ❚r✉♥❣ ❜➻♥❤ p✲❧æ❣❛r✐t✿





bp+1 − ap+1
Lp = Lp (a, b) :=
(p + 1) (b − a)

 a, a = b,


♥➳✉ a = b✳

▼➺♥❤ ✤➲ ✶✳✷✳✷ ✭❬✸✱ ▼➺♥❤ ✤➲ ✶❪✮
[a, b] ⊂ (0, ∞) .

❑❤✐ ✤â✱

Lpp − tp
≥A−t
ptp−1

●✐↔ sû p ∈ (−∞, 0) ∪ [1, ∞) \ {−1} ✈➔
✈î✐ ♠å✐

t ∈ [a, b].

✭✶✳✷✵✮


✶✻

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t →♥❤ ①↕ f : [a, b] −→ [a, +∞)✱ f (x) = xp ✈î✐ p t❤ä❛
♠➣♥

p ∈ (−∞, 0) ∪ [1, ∞) \ {−1} ,
t❛ t❤✉ ✤÷ñ❝

1
b−a


L
Lp
G
L
A−H
L−H L−a
A−a
≥
,
≥
,
H
L
L
a
b−A
b−L
≥
.
L
b

▼➺♥❤ ✤➲ ✶✳✷✳✸ ✭❬✹✱ ▼➺♥❤ ✤➲ ✶✷❪

✭✶✳✷✶✮
✭✶✳✷✷✮
✭✶✳✷✸✮

❳➨t p > 1 ✈➔ [a, b] ⊂ [0, +∞)✳ ❑❤✐ ✤â✱



(b − a)

1
p

2(p + 1)

1
p

1
q

b

x(p−1)q dx

p
a

▼➦t ❦❤→❝✱
b

x
a

(p−1)q

bpq−q+1 − apq−q+1

1 (b − a)

2(p + 1) p

p

=

p(b − a)Lp (a, b) q
1

.

2(p + 1) p
❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✹✮ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

▼➺♥❤ ✤➲ ✶✳✷✳✹ ✭❬✹✱ ▼➺♥❤ ✤➲ ✶✸❪✮
−1

❈❤♦ p > 1 ✈➔ 0 < a < b✳ ❑❤✐ ✤â✱
(b − a)

−1

0 ≤ H (a, b) − L (a, b) ≤

1

L


−

▼➦t ❦❤→❝✱

a

dx
x2 q

✭✶✳✷✺✮

.
t❛ ❝â✿

1
x
1
q

.

b

x−2q dx = (b − a)Lpp (a, b),
a

✈î✐

−2q =


p−1

(a, b)

p−1
p

1
q

.

Ù♥❣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣
tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣

❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞
✤÷ñ❝ sû ❞ö♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❦➨♣✳


✶✽

❉÷î✐ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ö✳

❱➼ ❞ö ✶✳✷✳✺ ❈❤♦ 0 < a < b < +∞✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
b

b2 − a2
a+b+2
ln
2

a+b
ln
,
2
2

✈➔

f (a) + f (b) a ln(1 + a) + b ln(1 + b)
=
.
2
2
❉♦ ✤â✱ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐ f (x) t❛
♥❤➟♥ ✤÷ñ❝
b

f

a+b
2

1
≤
b−a

f (t)dt ≤

f (a) + f (b)
.

b

≤

x ln(1 + x)dx
a

≤

b−a
[a ln(1 + a) + b ln(1 + b)] .
2


✶✾

❱➼ ❞ö ✶✳✷✳✻ ❈❤♦ b > a > 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
(a + b)e

(a+b)2
4

2

2

2

2


=
e
2

a+b
2

✈➔

2

f (a) + f (b) aeb
= a2
2
be

b
2

xex dx =

1 b2
2
e − ea .
2

a

⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐ f (x) t❛ ♥❤➟♥ ✤÷ñ❝
2

2

2

≤ eb − ea ≤ aea + beb .

❱➼ ❞ö ✶✳✷✳✼ ❈❤♦ p, q > 0✱ f ❧➔ ❤➔♠ ❧ç✐ tr➯♥ C ✱ [a, b] ⊂ C ✱ v =
0≤y≤

pa + qb
✈➔
p+q

b−a
min(p, q)✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿
p+q
v+y

f

pa + qb
p+q

1
≤
2y

f (t)dt ≤
v−y



1
[f (u − v) + f (u + v)] .
2

✭✶✳✷✻✮

v−y

▼➦t ❦❤→❝ ✈➻ f ❧➔ ❤➔♠ ❧ç✐ tr➯♥ C ⊃ [a, b] ✈➔ v =

x2 < x3 ≤ b,
f (x2 ) ≤

pa + qb
♥➯♥ ✈î✐ a ≤ x1

f (t)dt ≤

1
[f (u − v) + f (u + v)]
2

v−y

≤

1 b−v
v−a
pf (a) + qf (b)
f (a) +
f (b) ≤
.
2 b−a
b−a
p+q

✭✶✳✷✼✮
✭✶✳✷✽✮




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