(Luận văn thạc sĩ) Một số bất đẳng thức về hàm lồi và ứng dụng

✣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❨

❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽


✐✐✐

▼ö❝ ❧ö❝
❇↔♥❣ ❦þ ❤✐➺✉

✶

▼ð ✤➛✉

✷

❈❤÷ì♥❣ ✶✳ ❍➔♠ ❧ç✐ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞

✹

✶✳✶

✶✳✷

❍➔♠ ❧ç✐ ♠ët ❜✐➳♥ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳

✹

✶✳✶✳✶

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

✹

✶✳✶✳✷

❇➜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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✶✳✷

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

❍➔♠ s✲❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✷✳✶

✣à♥❤ ♥❣❤➽❛✳ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✷✳✷

❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ s✲❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ s✲❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✷✳✸✳✶

❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✷✳✸✳✷

▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ♠î✐ ✤÷ñ❝ t❤✐➳t ❧➟♣ tø ❜➜t ✤➥♥❣
t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✷✳✸✳✸

▼ët sè ù♥❣ ❞ö♥❣ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✹✵


✐✈

❑➳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

tr✉♥❣ ❜➻♥❤ ♥❤➙♥

H

tr✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛

L

tr✉♥❣ ❜➻♥❤ ❧æ❣❛r✐t

Lp

tr✉♥❣ ❜➻♥❤ p✲❧æ❣❛r✐t


✷

▼ð ✤➛✉
❍➔♠ ❧ç✐ ✈➔ t➟♣ ❧ç✐ ✤➣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ tø ❧➙✉ ❜ð✐ ❍☎♦❧❞❡r✱ ❏❡♥s❡♥✱
▼✐♥❦♦✇s❦✐✳ ✣➦❝ ❜✐➺t ✈î✐ ♥❤ú♥❣ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❋❡♥❝❤❡❧✱ ▼♦r❡❛✉✱ ❘♦❝❦✲
❛❢❡❧❧❛r ✈➔♦ ❝→❝ t❤➟♣ ♥✐➯♥ ✶✾✻✵ ✈➔ ✶✾✼✵ ✤➣ ✤÷❛ ❣✐↔✐ t➼❝❤ ❧ç✐ trð t❤➔♥❤ ♠ët
tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ♣❤→t tr✐➸♥ ♥❤➜t ❝õ❛ t♦→♥ ❤å❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♠ët sè
❤➔♠ ❦❤æ♥❣ ❧ç✐ t❤❡♦ ♥❣❤➽❛ ✤➛② ✤õ ♥❤÷♥❣ ❝ô♥❣ ❝❤✐❛ s➫ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t
♥➔♦ ✤â ❝õ❛ ❤➔♠ ❧ç✐✳ ❈❤ó♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✭❣❡♥❡r❛❧✐③❡❞
❝♦♥✈❡① ❢✉♥❝t✐♦♥✮✳ ✳ ✳
▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ❧ç✐✱
❤➔♠ ❧ç✐ ♠ët ❜✐➳♥✱ ❤➔♠ ❧ç✐ ♥❤✐➲✉ ❜✐➳♥✱ ❤➔♠ J ✲❧ç✐✱ ❤➔♠ s✲❧ç✐✱ ❜➜t ✤➥♥❣ t❤ù❝
❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐✱ ❤➔♠ ❧ç✐ ❦❤↔ ✈✐✱ ❤➔♠ s✲❧ç✐ ✈➔ ù♥❣ ❞ö♥❣
tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ t♦→♥ ♣❤ê t❤æ♥❣✱ ✤→♥❤ ❣✐→

❝→❝ ❣✐→ 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

t ỳ ổ ừ
ổ ụ ỷ ỳ ớ ỡ t t tợ tt ỳ ữớ
t ỳ ữớ s ợ tổ ỳ õ
tr tổ tỹ

t


ỗ


✹

❈❤÷ì♥❣ ✶

❍➔♠ ❧ç✐ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝
❍❡r♠✐t❡✕❍❛❞❛♠❛r❞
❈❤÷ì♥❣ ♥➔② ❣✐î✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ✈➲ ❤➔♠ ❧ç✐❀ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣
t❤ù❝ ❞↕♥❣ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐✱ ❤➔♠ ❧ç✐ ❦❤↔ ✈✐ ✈➔ ù♥❣ ❞ö♥❣
✤→♥❤ ❣✐→ ♠ët sè ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t➟♣
❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣
✤÷ñ❝ tê♥❣ ❤ñ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✸❪✱ ❬✹❪✱ ❬✼❪✱ ❬✽❪ ✈➔ ❬✶✵❪✳

✶✳✶
✶✳✶✳✶

❍➔♠ ❧ç✐ ♠ët ❜✐➳♥ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞
❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ❍➔♠ f : [a, b] ⊂ R → R ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ç✐ ♥➳✉ ✈î✐

♠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

b

f (x)dx ≤
a

f (a) + f (b)
.
2

❧➔

✭✶✳✶✮

❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮ ❝â t❤➸ ✈✐➳t ❧↕✐ ❞÷î✐ ❞↕♥❣✿
b

(b − a)f

a+b
2

≤

f (x)dx ≤ (b − a)

f (a) + f (b)
.
2

✭✶✳✷✮

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

(1 − λ)dλ.

λdλ + f (b)

1
2

0

✈➔ ❜➡♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥ x = λa + (1 − λ)b, s✉② r❛
1

b

1
f λa + (1 − λ)b dλ =
b−a
0

f (x)dx.
a

❑➳t ❤ñ♣ ✈î✐ ✭✶✳✸✮ t❛ ♥❤➟♥ ✤÷ñ❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ❝õ❛ ✭✶✳✶✮✳
❈ô♥❣ ❞♦ t➼♥❤ ❧ç✐ ❝õ❛ ❤➔♠ f ✱
1
f (λa + (1 − λ)b) + f ((1 − λ)a + λb)
2
λa + (1 − λ)b + (1 − λ)a + λb
≥f
2

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.

a

❇➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ❝õ❛ ✭✶✳✶✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

◆➳✉ g : [a, b] → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr➯♥
(t) ≤ M ✈î✐ ♠å✐ x ∈ [a, b]✱ m✱ M ❧➔ ❤➡♥❣ sè ①→❝ ✤à♥❤✱

❍➺ q✉↔ ✶✳✶✳✹ ✭①❡♠ ❬✸❪✮
[a, b] ⊆ R

t❤➻

✈➔ m ≤ g

b

1
m
(b − a)2 ≤
24
b−a

g(x)dx − g

a+b
2

≤

M

(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

=f

a+b
2
b

1
=
b−a

g(x) −

m 2
x dx
2

a
b

1
=
b−a

g(x)dx −

m b3 − a3
2 3(b − a)

g(x)dx −

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

≤
b−a

g(x)dx − g
a

a+b
.
2






m
2

b

a2 + ab + b2 a2 + 2ab + b2

3
4

1

ba

g(x)dx g

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

t tự tự t tr t tự õ t rở
ữ s

sỷ f : R R ỗ tr
b õ ợ ồ x [a, b] ồ [f (t), f + (t)]

ỵ ỵ
[a, b]

ợ

t [a, b]

a <

t õ

b

a+b
t
f (t)
2

1

ba

f (x)dx.

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

a

t
a

a+b
f (t)dt.
2




✽

✣à♥❤ ❧þ ✶✳✶✳✼ ✭❬✹✱ ✣à♥❤ ❧þ ✷✹❪✮

❤➔♠

◆➳✉ f ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ [a, b] ⊂ R ✈➔
x−

ϕ(x) :=

❧ç✐ tr➯♥ [a, b]✱ t❤➻

a+b
f (x)
2
b

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

ϕ(x)dx ≥ ϕ
a

a+b
.
2

❙û ❞ö♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ ϕ t❛ t❤✉ ✤÷ñ❝✿

1
2

b−a
2 (f

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

b

f (x)dx ≥ 0.
a

●✐↔ sû f : [a, b] ⊂ R → R ❧➔ ❤➔♠ ❦❤↔ ✈✐

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

✭✶✳✽✮

|f (t)| dt .
a

❈❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r ✈î✐ p > 1 ✈➔ q > 1 t❤ä❛
♠➣♥

1 1
+ = 1✱ t❛ ❝â
p q
b

1
b−a
≤

x−
a

1
b−a

a+b
f (x)dx
2
p

b

x−

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

b
a+b
2

a+b
x−
2

(b − a)p+1
=
.
(a + 1)2p

p

dx

,


✾

❙✉② r❛✱

a+b
x−
dx
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

1
q

b

| f (x) |q dx
a
1
q

b

1 (b − a)
=
2 (p + 1) p1

1
q

b

1
b−a

| f (x) |q dx
a

✈➔ ❦❤✐ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✽✮ ✤÷ñ❝ s✉② r❛ tø ✭✶✳✻✮✳

❈❤♦ f : C ⊂ R → R ❧➔ ♠ët ❤➔♠ ❦❤↔ ✈✐ tr➯♥

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




 x − a,


 x − b,

a+b
,
2
a+b
x∈
,b .
2
x ∈ a,

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

✣à♥❤ ❧þ ✶✳✶✳✶✵ ✭❬✹✱ ✣à♥❤ ❧þ ✷✽❪✮

b

f

a+b
2

−

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

≤

b

p(x)f (x)dx
a

1
b−a

1
p

b

| p(x) |p dx
a

×

1
b−a

1
q

b

| f (x) |q dx
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)

❉♦ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✵✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ❧þ ✶✳✶✳✶✶ ✭❬✹✱ ❇ê ✤➲ ✹❪✮

●✐↔ sû f

✤➳♥ ❝➜♣ ❤❛✐ tr➯♥ [a, b]✳
(i) ◆➳✉ |f | ❦❤↔ t➼❝❤ tr➯♥ [a, b] t❤➻
b

1
2

(t − a) (b − t) f (t)dt =

: [a, b] ⊂ R → R

❧➔ ❤➔♠ ❦❤↔ ✈✐

b

b−a
(f (a) + f (b)) −
2

a

f (t)dt.
a

✭✶✳✶✶✮

(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

❈❤ù♥❣ ♠✐♥❤✳ (i) ❚❛ ❝â
1
2

b

(x − a)(b − x)f (x)dx
a
b
1
b
[−2x + (a + b)]f (x)dx
= (x − a)(b − x)f (x)|a −
2
a
1 b
=
[2x − (a + b)]f (x)dx
2 a
b
b
1
=
(2x − (a + b))f (x) − 2
f (x)dx
a
2
a
b
b−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

(x a)(b x)f (x)dx
a
b

(x a)(b x)dx.
a

t t

1
2

b
a

ba
(x a)(b x)f (x)dx =
(f (a) + f (b))
2



b
a

b

f (x)dx
a

(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 )

n!

1

n

(t 1)n f (n) (tx + (1 t)a)dt
0
1

(1 t)n f (n) (tx + (1 t)b)dt
0

x [a, b] n số tỹ n 1

a < b f (n) L1[a, b] |f (n)| n 1 ỗ

ỵ ỵ

n a, b C
tr [a, b] t

n1

b

f (t)dt
a

sỷ f

: (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

b

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 f (n) (tx + (1 − t)a) dt
n!
0
1
n+1
(b − x)
(1 − t)n f (n) (tx + (1 − t)b) dt.
+
n!
0

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

b

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
+

|f (n) (x)| + (n + 1)|f (n) (b)| .
(n + 2)!

❈❤ó þ ✶✳✶✳✶✹ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✹✮ ♥➳✉ ❝❤å♥ n = 1 t❛ ❝â ❜➜t ✤➥♥❣
t❤ù❝ ❞÷î✐ ✤➙②✿
b

f (t)dt − [(x − a)f (a) + (b − x)f (b)]
a

≤

(x − a)2
2|f (a)| + |f (x)|
6
(b − x)2
+
|f (x)| + 2|f (b)| .
6

❍➺ q✉↔ ✶✳✶✳✶✺ ✭①❡♠ ❬✼✱ ❇ê ✤➲ ✷✳✶❪✮

❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝

✭✶✳✶✺✮
✭✶✳✶✹✮✱

♥➳✉ t❛

✶✸

❝❤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

f (t)dt −
a

k=0
1

(x − a)k+1 f (k) (a) + (−1k )(b − x)k+1 f (k) (b)
(k − 1)!
(x − a)n+1 |f n (a)|q + |f n (x)|q
n!
2

1 p
≤
np + 1

(b − x)n+1 |f n (x)|q + |f n (b)|q
+
n!
2

1
q

1
q

,

✭✶✳✶✻✮

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
+

n!

1
q

1

|f (n) (tx + (1 − t)a)|q dt
0
1
p

1

(1 − t)np dt
0

1
q

1

|f (n) (tx + (1 − t)b)|q dt
0

.


✶✹

❱➻ |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

1
p

1
np + 1

1
q

1

t|f (n) (x)|q + (1 − t)|f (n) (a)|q dt
0
1
p

1
q

1

t|f (n) (x)|q + (1 − t)|f (n) (b)|q dt
0

(x − a)n+1 |f n (a)|q + |f n (x)|q
n!
2

(b − x)n+1 |f n (x)|q + |f n (b)|q

+
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

1

p
(b − a)
1
≤
16
2P + 1




 1q 
q  1q
q



q
 |f (a)|q + f ( a+b

f ( a+b
2 ) )
2 ) + |f (b)|





×
+
.


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

a, b ≥ 0.

✭✶✳✶✼✮


✶✺

✭❜✮ ❚r✉♥❣ ❜➻♥❤ ♥❤➙♥✿

G = G(a, b) :=

√

a, b ≥ 0.

ab,

✭❝✮ ❚r✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛✿

H = H(a, b) :=

2
1 1
+
a b

,

a, b > 0.

✭❞✮ ❚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,

1
p

,

✈î✐ p ∈ R\ {−1, 0} ✈➔ a, b > 0✳

1
t

◆❤➟♥ ①➨t ✶✳✷✳✶ ✭❛✮ ❱î✐ ❤➔♠ ❧ç✐ f (x) = ✱ t > 0✱ ♥➳✉ a = b t❛ ❝â
b

1
b−a

f (t)dt = L−1 (a, b).
a

✭❜✮ ❱î✐ ❤➔♠ ❧ç✐ ✭❧ã♠✮ f (x) = xp ✱ p ∈ (−∞, 0) ∪ [1, ∞) \ {−1} ✭❤♦➦❝ p ∈

(a, b)✮✱ t❛ ❝â
b

1
b−a

f (t)dt = Lpp (a, b)
a

♥➳✉ 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

b

a+b
−t ,
1

xp dx ≥ tp + ptp−1
a

✈î✐ ♠å✐ t ∈ [a, b]✳ ❉♦ ✤â✱
b

1
a−b

xp dx = Lpp (a, b) = Lpp .
a

❙✉② r❛✱ t❛ ♥❤➟♥ ✤÷ñ❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✷✵✮✳

❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✵✮✱ t❛ ❝â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ❝❤♦ ❝→❝
❣✐→ trà tr✉♥❣ ❜➻♥❤ ✭①❡♠ ❬✸❪✮✳

Lp − L
Lp − A A − G
L−G
,
≥
≥
≥ 0,

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✱

0 ≤ A(ap , bp ) − Lpp (a, b) ≤

p(b − a)
2(p + 1)

1
p

p

✭✶✳✷✹✮

[Lp (a, b)] q

✈î✐ q := p −p 1 ✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✣à♥❤ ❧þ ✶✳✶✳✽ →♣ ❞ö♥❣ ❝❤♦ ❤➔♠ ❧ç✐ f (x) = xp ✱ t❛ ❝â✿
p

b

p

1
a +b
−
2
b−a

xp dx ≤
a

(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

dx =
= Lpp (a, b)(b − a)
p+1

.


✶✼

✈➔ ❞♦ ✤â t❛ ❝â✿
1

p

p

A(a , b ) −

Lpp (a, b)

≤

p(b − a) p

p

1

q
Lpp (a, b) q

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

2(p + 1) p

p−1
p
2p
1−p (a,b)

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✶✳✶✳✽ ❝❤♦ ❤➔♠ ❧ç✐ f (x) :=
0≤

1
a

+
2

1
b

1
p

b

ln b − ln a
1 (b − a)
≤
b−a
2 (p + 1) p1

−

▼➦t ❦❤→❝✱

a

dx
x2 q

✭✶✳✷✺✮

.
t❛ ❝â✿

1
x
1
q

.

b

x−2q dx = (b − a)Lpp (a, b),
a

✈î✐

−2q =

2p
.
p−1

❚❛ t❤✉ ✤÷ñ❝

0 ≤ H−1 (a, b) − L−1 (a, b)
1

1

1 (b − a) p (b − a) q
L−2q
≤
1
−2q (a, b)
2
(p + 1) p
=

(b − a)
1

2(p + 1) p

✶✳✷✳✷

L

2p

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

2

≤

x ln(1 + x)dx
a

≤

b−a
[a ln(1 + a) + b ln(1 + b)] .
2

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè f (x) = x ln(1 + x) tr➯♥ (0, +∞). ❚❛ ❝â
f (x) =

x+2
> 0 ✈î✐ ♠å✐ x ∈ (0, +∞)
(x + 1)2

♥➯♥ f (x) ❧➔ ❤➔♠ ❧ç✐ ✈î✐ ♠å✐ x ∈ (0, +∞)✳ ▼➦t ❦❤→❝✱

f

a+b
2

=

a+b+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)
.

2

a

❍❛②
b

(b − a)f

a+b
2

≤

f (t)dt ≤ (b − a)

f (a) + f (b)
.
2

a

❉♦ ✤â✱

b2 − a2
a+b+2
ln
2
2

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

≤ eb − ea ≤ aea + beb .

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè f (x) = xex + 4x3 ex > 0 ✈î✐ ♠å✐ x ∈ (0, +∞).
2

2

❑❤✐ ✤â✱
2

f (x) = ex (32x4 + 12x2 + 1),
2

f (x) = ex (64x5 + 156x3 + 30x).
2

❱➻ f (x) = ex (64x5 + 156x3 + 30x) > 0 ✈î✐ ♠å✐ x ∈ (0, +∞) ♥➯♥ f (x) ❧➔
❤➔♠ ❧ç✐ tr➯♥ (0, +∞)✳ ▼➦t ❦❤→❝✱
2



f

a+b


a+b
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



a+b


2
a+b
1 b2
aeb
2
a2
e
≤
e −e
≤ a2
2
2
be
❍❛②

(a + b)e

(a+b)2
4

2

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

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


✷✵

❈❤ù♥❣ ♠✐♥❤✳ ❚r÷î❝ ❤➳t✱ ✈➻ 0 ≤ y ≤

b−a
min(p, q) ♥➯♥ t❛ ❝â ❤❛✐ tr÷í♥❣
p+q

❤ñ♣ 0 < p ≤ q ✈➔ 0 < q < p✳
pa + qb
❚❤❡♦ ❣✐↔ t❤✐➳t ✈➻ v =
♥➯♥ a ≤ v − y < v + y ≤ b.
p+q
❱➻ ❤➔♠ f ❧ç✐ tr➯♥ [a, b] ♥➯♥ ❤➔♠ f ❝ô♥❣ ❧ç✐ tr➯♥ [v − y, v + y] ⊂ [a, b]. ⑩♣
❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ ❧ç✐ f tr➯♥ [v − y, v + y]✱
v+y

1
f (v) ≤
2y

f (t)dt ≤

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 <
p+q

x3 − x2
x2 − x1
f (x1 ) +
f (x3 ).
x3 − x1
x3 − x1

❈❤å♥ x1 = a, x3 = b t❛ ♥❤➟♥ ✤÷ñ❝

v−y−a
b − (v − y)
f (a) +

f (b),
b−a
b−a
b − (v + y)
v+y−a
f (v + y) ≤
f (a) +
f (b).
b−a
b−a
f (v − y) ≤

❈ë♥❣ ✈➳ ✈î✐ ✈➳ ❝õ❛ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✼✮ ✈➔ ✭✶✳✷✽✮✱

[f (u − v) + f (u + v)] ≤

v−a
b−v
f (a) +
f (b) .
b−a
b−a

❑➳t ❤ñ♣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈î✐ ✭✶✳✷✻✮✱ s✉② r❛
v+y

1
f (v) ≤
2y

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

✭✶✳✷✼✮
✭✶✳✷✽✮




ữỡ

ỗ s rở ự ử
ữỡ tr J ỗ ởt số t t ừ

J ỗ tr sỗ t tự rtr
sỗ ự ử ởt số tr tr t
ở ừ ữỡ ữủ tờ ủ tứ t



J ỗ

ử tr ỗ t
ỗ ợ t J ỗ J ỗ s rở t J ỗ
s rở ởt ự t ỹ tr ở ừ ử ữủ
t tr



ỗ tr Rn

a, b Rn tt x = (1 )a + b ợ

0 1 ồ t õ ố a b ữủ ỵ [a, b]

C Rn ữủ ồ ởt t ỗ õ ự trồ
t ố t ý tở õ õ (1)a+b

C ợ ồ a, b C ồ 0 1

ử t s ỗ
ỷ ổ õ ỷ ổ