Bất đẳng thức feng qi và các dạng mở rộng

✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ✖✖✖✖✖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❘❺◆ ❚❍➚ ❙■➊◆ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❋❊◆● ◗■ ❱⑨ ❈⑩❈ ❉❸◆● ▼Ð ❘❐◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ✣⑨ ◆➂◆● ✲ ◆❿▼ ✷✵✶✼ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ✖✖✖✖✖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❘❺◆ ❚❍➚ ❙■➊◆ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❋❊◆● ◗■ ❱⑨ ❈⑩❈ ❉❸◆● ▼Ð ❘❐◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ữớ ữợ r ũ ✣⑨ ◆➂◆● ✲ ◆❿▼ ✷✵✶✼ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✳ ❈→❝ sè ❧✐➺✉✱ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❛✐ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦➻ ❝æ♥❣ tr➻♥❤ ♥➔♦ ❦❤→❝✳ ❚→❝ ❣✐↔ ❚r➛♥ ❚❤à ❙✐➯♥ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐ ✏ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✈➔ ❝→❝ ❞↕♥❣ ♠ð rë♥❣✑ ❧➔ ❦➳t q✉↔ ❝õ❛ q✉→ tr➻♥❤ ❝è ❣➢♥❣ ❦❤æ♥❣ ♥❣ø♥❣ ❝õ❛ ❜↔♥ t❤➙♥ ✈➔ ✤÷đ❝ sü ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺ t ỗ ữớ t ◗✉❛ tr❛♥❣ ✈✐➳t ♥➔② ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✲ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈ø❛ q✉❛✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ tr÷í♥❣ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ❦❤♦❛ ❚♦→♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ♣❤➙♥ ❤✐➺✉ ✣↕✐ ❤å❝ ✣➔ ♥➤♥❣ t↕✐ ❑♦♥ t✉♠ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ✈✐➳t ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❚æ✐ ①✐♥ tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ t❤➛② ❣✐→♦ ❚❙ ❚r➛♥ ũ trỹ t t t ữợ ụ ♥❤÷ ❝✉♥❣ ❝➜♣ t➔✐ ❧✐➺✉ t❤ỉ♥❣ t✐♥ ❦❤♦❛ ❤å❝ ❝➛♥ t❤✐➳t ❝❤♦ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ỡ ỗ rữớ ❚❍P❚ ❍✉ý♥❤ ❚❤ó❝ ❑❤→♥❣✱ ▲ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✸✶✱ ✈➔ ❣✐❛ ✤➻♥❤ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ▲✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❚r➛♥ ❚❤à ❙✐➯♥ ❇❷◆● ❑➑ ❍■➏❯ N✿ R✿ ❚➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥✳ ❚➟♣ ❤ñ♣ ❝→❝ sè t❤ü❝✳ R+ = (0; +∞)✿ ❚➟♣ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ R0 = [0; +∞)✿ ❚➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ Rn ✿ ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n ✲ ❝❤✐➲✉✳ Z+ ✿ ❚➟♣ ❤ñ♣ [a; b]✿ ✣♦↕♥✳ ➙♠✳ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ (a; b)✿ ❑❤♦↔♥❣✳ (a; b] ; [a; b)✿ ◆û❛ ❦❤♦↔♥❣✳ ✿ ✣➦t✳ f (n) (x)✿ ✣↕♦ ❤➔♠ ❝➜♣ n ❝õ❛ ❤➔♠ f t↕✐ x✳ C [a; b]✿ ❚➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b]✳ C [a; b]✿ ❚➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ (a; b)✱ ❧✐➯♥ tö❝ tr➯♥ [a; b]✳ X ∼ U [a; b] ✿ X ❝â ♣❤➙♥ ❜è ✤➲✉ tr➯♥ ✤♦↕♥ [a; b]✳ E (X)✿ ❑ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✳ Eµ ✿ ❑➻ ữợ st LP (X, ,à) ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦ ▲❡❜❡s❣✉❡ p ✲ ❝❤✐➲✉✳ L1 (a; b)✿ ▲ỵ♣ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ▲❡❜❡sq✉❡ tr➯♥ ❦❤♦↔♥❣ (a; b) ✤♦↕♥ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ✶ ✷ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t tự s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ trà tr✉♥❣ ❜➻♥❤ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ ✣ë ✤♦ ✈➔ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹✳✶ ✣ë ✤♦ ✹ ✶✳✹✳✷ ❚➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ✶✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✻ q ✲ ✤↕♦ ❤➔♠ ✈➔ q ✲ t➼❝❤ ♣❤➙♥✳ h ✲ ✤↕♦ ❤➔♠ ✈➔ h ✲ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✼ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❦ý ✈å♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ỗ t tự s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✈➔ ❝→❝ ❞↕♥❣ ♠ð rë♥❣ ✶✶ ✷✳✶ ✶✶ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✈ỵ✐ ♠ð rë♥❣ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✶✸ ✷✳✸ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✈ỵ✐ ♠ð rë♥❣ tr➯♥ t➟♣ sè t❤ü❝ ✳ ✳ ✶✺ ✷✳✹ ❉↕♥❣ rí✐ r↕❝ ❝õ❛ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✹✳✶ ✷✳✺ ❚r÷í♥❣ ❤ñ♣ α > max {1, β} α > 0, β > ✳ ✷✳✹✳✷ ❚r÷í♥❣ ❤đ♣ ✷✳✹✳✸ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ rí✐ r↕❝ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ✤ë ✤♦ ①→❝ s✉➜t t❤➼❝❤ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✸✼ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ❝❤♦ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✻ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ❝❤♦ q ✲ t➼❝❤ ♣❤➙♥ ✈➔ h ✲ t➼❝❤ ♣❤➙♥ ✷✳✼ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ❝❤♦ t➼❝❤ ♣❤➙♥ ❤❛✐ ❧ỵ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✽ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ t÷ì♥❣ tü ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✺ ✹✼ ✹✽ ✺✶ ✶ ▼Ð ✣❺❯ ▲à❝❤ sû ❜➜t ✤➥♥❣ t❤ù❝ ❜➢t ỗ tứ rt sốt t q✉❛ t❤í✐ ❣✐❛♥ ❝❤♦ tỵ✐ t➟♥ ♥❣➔② ♥❛②✳ ❈â ➼t ♥❤➜t ❜❛ ❧➼ ❞♦ ❣✐↔✐ t❤➼❝❤ t↕✐ s❛♦ ❝❤ó♥❣ t❛ ❧✉ỉ♥ q✉❛♥ t➙♠ tỵ✐ ❜➜t ✤➥♥❣ t❤ù❝✳ ✣â ❝❤➼♥❤ ❧➔ tỹ ỵ tt q trồ t t tỗ t tr t ỳ ữớ q t tợ t tự r ✤➭♣ ①✉②➯♥ q✉❛ ❧à❝❤ sû ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ t❤➻ ❦❤ỉ♥❣ t❤➸ ❦❤ỉ♥❣ ♥❤➢❝ tỵ✐ ♠ët ❜ë ♣❤➟♥ ❧➔♠ ♥➯♥ ✈➫ ✤➭♣ ✤â✱ ❝❤➼♥❤ ❧➔ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥✳ ❇➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët ♣❤➛♥ q✉❛♥ trå♥❣ tr♦♥❣ t➼❝❤ ♣❤➙♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❦❤ỉ♥❣ ❝❤➾ tr♦♥❣ t♦→♥ ❤å❝ ♠➔ ❝á♥ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝✳ ❇➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❧➔ ❜➔✐ t♦→♥ ❦❤â t❤÷í♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❖❧②♠♣✐❝ t ữủ sỹ ữợ t t t❤➛② ❚❙✳ ❚r➛♥ ❚❤❛♥❤ ❚ị♥❣✱ ❝❤ó♥❣ tỉ✐ ✤➣ t✐➳♣ ❝➟♥ ữợ ự t t t❤ù❝ ❋❡♥❣ ◗✐ ✈➔ ❝→❝ ❞↕♥❣ ♠ð rë♥❣✑ ✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ t❤æ♥❣ q✉❛ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❝❤ó♥❣ tỉ✐ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❝â ❤➺ t❤è♥❣ ❝→❝ ❦➳t q✉↔ ✈➲ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐✳ ◆❣❤✐➯♥ ❝ù✉ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐✱ ❇➜t tự ợ rở tr trữớ ủ ❝→❝ sè tü ♥❤✐➯♥✱ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ✈ỵ✐ ♠ð rë♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝→❝ sè t❤ü❝✱ ❞↕♥❣ rí✐ r↕❝ ❝õ❛ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐✱ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ❝❤♦ q ✲ t➼❝❤ ♣❤➙♥ ✈➔ h ✲ t➼❝❤ ♣❤➙♥✱ ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐ ❝❤♦ t➼❝❤ ♣❤➙♥ ợ ởt số t tự t tữỡ tü ❇➜t ✤➥♥❣ t❤ù❝ ❋❡♥❣ ◗✐✳ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự ỡ ỗ t tự s ❇➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r✱ ✣à♥❤ ❧➼ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❈❛✉❝❤②✱ ✤ë ✤♦ ✈➔ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡✱ ✈➔ h q ✲ ✤↕♦ ❤➔♠ ✈➔ q ✲ t➼❝❤ ♣❤➙♥✱ h ✲ ✤↕♦ ❤➔♠ ✲ t➼❝❤ ♣❤➙♥✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❦➻ ỗ t tự s ỗ ❍➔♠ sè f ♥➳✉ ∀x1 , x2 ∈ [a; b] (x) [a; b] ⊂ R α + β = 1✱ ữủ ỗ tr ợ sè ❦❤æ♥❣ ➙♠ α, β ❝â tê♥❣ t❛ ✤➲✉ ❝â f (αx1 + βx2 ) ≤ αf (x1 ) + βf (x2 ) ❉➜✉ ✏ =✑ tr♦♥❣ ✭✶✳✶✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➔♠ sè f (x) ❧➔ ỗ tỹ sỹ t tr x1 = x2 ❑❤✐ ✤â✱ t❛ ♥â✐ [a; b]✳ f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ tr➯♥ [a; b] ⊂ R ♥➳✉ ∀x1 , x2 ∈ [a; b] ✈➔ ✈ỵ✐ ♠å✐ ❝➦♣ sè ❦❤æ♥❣ ➙♠ α, β ❝â tê♥❣ α+β = 1✱ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ✭❬✽❪✮ ❍➔♠ sè t❛ ✤➲✉ ❝â f (αx1 + βx2 ) ≥ αf (x1 ) + βf (x2 ) ✭✶✳✷✮ ✸ ❉➜✉ ✏ =✑ tr♦♥❣ ✭✶✳✷✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➔♠ sè f (x) ❧➔ ❧ã♠ t❤ü❝ sü ✭♥❣➦t✮ tr♦♥❣ ❇ê ✤➲ ✶✳✶✳✶✳ ✭❬✶✻❪✮ ●✐↔ sû (a; b) ♥➳✉ ϕ (u) ≤ ϕ (v) ỗ tr x1 = x2 õ t ♥â✐ [a; b]✳ (a; b)✳ ❑❤✐ ✤â ϕ ❧➔ ❤➔♠ ❝❤♦ a < u < v < b t❛ ❝â ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ ✈➔ ❝❤➾ ♥➳✉ ❝❤♦ u v s t tự s ỵ ✶✳✶✳✶✳ ✭❬✽❪✮ ❈❤♦ x1 , x2 , , xn ∈ [a; b] f (x) ỗ tr [a; b]✳ ●✐↔ sû ≥ 0, i = 1, n✱ s❛♦ ❝❤♦ a1 + a2 + + an = 1✱ t❛ ❧✉æ♥ ❝â n f n ≤ xi i=1 f (xi ) ✭✶✳✸✮ i=1 ❉↕♥❣ t➼❝❤ ♣❤➙♥ ❝õ❛ ❇➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s ỵ số M tr ỗ tr (a; b) (a; b) t  b ❧➔ ✤ë ✤♦ ❞÷ì♥❣ ❤ú✉ ❤↕♥ tr➯♥ ♠ët f ❧➔ ♠ët ❤➔♠ t❤ü❝ tr➯♥  L1 (a; b) σ ✲ ✤↕✐ ✈➔ ϕ ❧➔ b  f (x) dµ (x)    ϕa b ≤   dµ (x) ϕ (f (x)) dµ (x) a b ✭✶✳✹✮ dµ (x) a a ✶✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r ỵ số ổ a1 , a2 , , an ✈➔ b1 , b2 , , bn ❀ p, q ❧➔ ❝→❝ sè t❤ü❝ ❧ỵ♥ ❤ì♥ n s❛♦ ❝❤♦ p n k=1 t❛ ❧✉æ♥ ❝â q n apk ak b k ≤ k=1 1 + = 1✱ p q bqk ✭✶✳✺✮ k=1 ✣➥♥❣ t❤ù❝ tr♦♥❣ ✭✶✳✺✮ ①↔② r❛ ❦❤✐ tỗ t số A, B ổ p q p q ỗ tớ ổ s ❆❛k = Bbk ✭❛k ✈➔ bk t➾ ❧➺ ✈ỵ✐ ♥❤❛✉✮❀ k = 1, 2, , n ✹✽ t∈ / [0; 1) ✸✳ ◆➳✉ [c; d] f (x; y) ≥ [(b − a) (d − c)]t−2 ✈ỵ✐ (x; y) ∈ [a; b] × ✈➔ f (x; y) ≤ [(b − a) (d − c)]t−2 ✈ỵ✐ (x; y) ∈ [a; b] × t❤➻ ✭✷✳✻✻✮ ✤ó♥❣✳ ✹✳ ◆➳✉ [c; d] ✈➔ 01 ❧➔ ❝→❝ ❤➔♠ sè ❦❤æ♥❣ ➙♠ t❤ä❛ f (x) ≤M