✣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