DANIEL SITARU

CARTEA ROMÂNEASCĂ EDUCAŢIONAL

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MOTTO: “The best way to follow your dreams is to wake up!”

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Table of Contents Chapter O – Famous Theorems ................................................................... 6 Chapter I – Questions ................................................................................... 27 Chapter II – Solutions ................................................................................... 62 Bibliography ................................................................................................. 237

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Chapter 0 FAMOUS THEOREMS CAUCHY–SCHWARZ Inequality ( + ) ≤ ( + )( + ); , , , ∈ ℝ ( + + ) ≤ ( + + )( + + ); , , , , , ∈ ℝ ≤ ; , ∈ ℝ, ∈ 1,

+ ≥ ( + )+ ; , ∈ ℝ; , ∈ (0, ∞) + + ≥ ( + + )+ + ; , , ∈ ℝ; , , ∈ (0, ∞) + + ≥ ( + + )+ + ; , , , , , ∈ (0, ∞) + + ⋯ + ≥ ( + + ⋯ + )+ + ⋯ + ; ∈ ℝ; > 0; ∈ 1, + + + + + ≥ 32 ; (∀) , , ∈ (0, ∞)

+ + + + + ≥ 3+ 1 ; , , ∈ (0, ∞); ∈ ℕ∗ MINKOWSKI’s Inequality ( + ) + ( + ) ≤ + + + ( + + ) + ( + + ) ≤ + + + + + ( + ) + ( + ) + ( + ) ≤ + + + + + ( + ) + ( + ) + ⋯ + ( + ) ≤ + + ⋯ + + + + ⋯ + ; ∈ ℝ; ∈ 1, ; ∈ ℕ∗ | + | ≤ | | + | | > 1; , ∈ ℝ; ∈ 1, ; ∈ ℕ∗ HÖLDER’s Inequality + + ≥ ( + + )3( + + ) ; , , , , , ∈ (0, ∞)

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+ + ≥ ( + + )9( + + ) ; , , , , , ∈ (0, ∞) + + ≥ ( + + )3 ( + + ) ; , , , , , ∈ (0, ∞); ∈ ℕ; ≥ 2 + ≥ ( + )2 ( + ) ; , , , ∈ (0, ∞); ≥ 2; ∈ ℕ + + ≥ ( + + )( + + ) ; , , , , , ∈ (0, ∞) + + ≥ ( + + )( + + ) ; , , , , , ∈ (0, ∞) + + ≥ ( + + )( + + ) ; , , , , , ∈ (0, ∞); ∈ ℕ ≥ ; , , ∈ [0, ∞); ∈ ℕ∗

≥ ; , , , ∈ ℝ; ∈ ℕ∗ | | ≤ | | ⋅ | | ; , ∈ (1, ∞) 1 + 1 = 1; , ∈ ℝ; ∈ 1, ; ∈ ℕ∗ HUYGENS’s Inequality (1 + )(1 + ) ≥ 1 + ; , ∈ [0, ∞) (1 + )(1 + )(1 + ) ≥ 1 + ; , , ∈ [0, ∞) (1 + )(1 + )(1 + )(1 + ) ≥ 1 + ; , , , ∈ [0, ∞) (1 + ) ≥ 1 + ⋯ ; ∈ [0, ∞); ∈ ℕ; ≥ 2 ( + )( + ) ≥ + ( + )( + )( + ) ≥ + ( + )( + ) ⋅ … ⋅ ( + ) ≥ ⋅ … ⋅ + ⋅ … ⋅ ( + + )( + + ) ≥ + + ( + + )( + + ) ⋅ … ⋅ ( + + )≥ ⋅ … ⋅ + ⋅ … ⋅ + ⋅ … ⋅

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Generalization of HÖLDER’s Inequality ≥ + + ⋯ + = 1 CHEBYSHEV’s Inequality ( ≤ ) ∧ ( ≤ ) or ( ≥ ) ∧ ( ≥ )+ ≥ 12 ( + )( + ) ( ≤ ≤ ) ∧ ( ≤ ≤ ) or ( ≥ ≥ ) ∧ ( ≥ ≥ )+ + ≥ 13 ( + + )( + + ) ( ≤ ≤ ⋯ ≤ ) ∧ ( ≤ ≤ ⋯ ≤ ) or ( ≥ ≥ ⋯ ≥ ) ∧ ( ≥ ≥ ⋯ ≥ )+ + ⋯ + ≥ 1 ( + + ⋯ + )( + + ⋯ + ) ( ≤ ) ∧ ( ≥ ) or ( ≥ ) ∧ ( ≤ )+ ≤ 12 ( + )( + ) ( ≤ ≤ ) ∧ ( ≥ ≥ ) or ( ≥ ≥ ) ∧ ( + + )+ + ≤ 13 ( + + )( + + ) ( ≤ ≤ ⋯ ≤ ) ∧ ( ≥ ≥ ⋯ ≥ ) or ( ≥ ≥ ⋯ ≥ ) ∧∧ ( ≤ ≤ ⋯ ≤ )+ + ⋯ + ≤ 1 ( + + ⋯ + )( + + ⋯ + ), ∈ ℝ; ∈ ℕ∗; ∈ 1,

( ) ( ) ≥ ( ) ( ) ≥ ( ) ( )for ≤ ≤ ⋯ ≤ ; ≤ ≤ ⋯ ≤≥ 0; ∈ 1, ; ∈ ℕ∗; + + ⋯ + = 1 , non-decreasing

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SCHUR’s Inequality ( − )( − ) + ( − )( − ) + ( − )( − ) ≥ 0, , ∈ [0, ∞); > 0 + + + 3 ≥ ( + ) + ( + ) + ( + )≥ (− + + )( − + )( + − )( + + ) + 9 ≥ 4( + + )( + + )( − ) ( + − ) + ( − ) ( + − ) + ( − ) ( + − ) ≥ 0+ + + 9+ + ≥ 2( + + )+ + + + + + 4( + )( + )( + ) ≥ 2

, , ∈ (0, ∞) + + + ( + + ) ≥ ( + ) + ( + ) + ( + ), , ∈ [0, ∞) ≤, , ⋯ , eigenvalues of ; ∈ ℕ∗ ; ∈ (ℝ)

≤ ; 1 ≤ ≤ ; ∈ ℕ∗≥ ≥ ⋯ ≥ eigenvalues of =≥ ≥ ⋯ ≥

MILNE’s Inequality ( + + + ) + + + ≤ ( + )( + ) ( + + + + + ) + + + + + ≤ ( + + ) ⋅ ( + + ) ( + ) + ≤ , ∈ 1, ; ≥ 2; > 0; > 0

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REARRANGEMENT’s Inequality ≥ ≥≤ ≤ ⋯ ≤ ; ≤ ≤ ⋯ ≤= 1 2 ⋯ ⋯⋯ ⋯ ( ) ⋯ ⋯ ⋯ ∈ ; ∈ ℕ∗

( ) ≥ ( ) ≥ ( )with ( ) − ( ) non-decreasing; 1 ≤ ≤ STIRLING’s Inequality ≤ √2 ≤ ! ≤ √2 ≤ MAHLER’s Inequality ( + )( + ) ≥ + ( + )( + )( + ) ≥ + ( + )( + )( + )( + ) ≥ + ( + ) ≥ + , > 0; ∈ 2, ; ∈ ℕ; ≥ 2 WEIERSTRASS Inequality (1 − ) ≥ 1 − ; ≤ 1; ≥ 1 or ≤ 0; ∈ 1,

(1 − ) ≤ 1 − ; ∈ [0,1]; ≤ 1 ; ∈ (−∞, 1]; ∈ 1, ; ∈ ℕ∗ RADON’s Inequality + + ⋯ + ≥ ( + + ⋯ + )( + + ⋯ + ) , , … , , , , … , ∈ (0, ∞); ≥ 1; ≥ 1

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YOUNG’s Inequality + ≤ ≤ + , , ∈ (0, ∞); , ∈ (0, ∞); + = 1 JENSEN’s Inequality : → ℝ; convex on ;, , , , ⋯ , ∈ ; ∈ ℕ∗+2 ≤ ( ) + ( )2+ +3 ≤ ( ) + ( ) + ( )3+ + ⋯ + ≤ ( ) + ( ) + ⋯ + ( )

: → ℝ; concave on ;+2 ≥ ( ) + ( )2+ +3 ≥ ( ) + ( ) + ( )3+ + ⋯ + ≥ ( ) + ( ) + ⋯ + ( ) : → ℝ; convex on ;≥ 0; + + ⋯ + = 1( + + ⋯ + ) ≤ ( ) + ( ) + ⋯ + ( ) : → ℝ; concave on ≥ 0; + + ⋯ + = 1( + + ⋯ + ) ≥ ( ) + ( ) + ⋯ + ( ) WEIGHTED MEANS Inequality = +++ ≥ ⋅ ≥ + = + ++ ++ + ≥ ⋅ ⋅ ≥ + +

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∑∑ ≥ ∑

> 0; ∈ 1, ; ∈ ℕ∗. Convention: 0 = 1 ≥ ∑∑ MEANS Inequality

min ≤ ∑ 1 ≤∈ (0, ∞); ∈ 1, ; ∈ 1, ; ∈ ℕ∗

≤ ∑ ≤ 1 ≤ max POWER MEANS Inequality | | + | | ≤ | | + | |, , ∈ [0, ∞); + = 1; , ∈ ℝ

| | + | | ≤ | | + | |≥ > 0; , ∈ [0, ∞); + = 1; , ∈ ℝ | | + | | + | | ≤ | | + | | + | |, , ∈ [0, ∞); + + = 1; , , ∈ ℝ | | + | | + | | ≤ | | + | | + | |, , ∈ [0, ∞); + + = 1; , , ∈ ℝ

| | ≤ | |, ∈ [0, ∞); ≤ ; ∈ [0, ∞)∈ 1, ; ∈ ℕ∗; + + ⋯ + = 1

( ) = | |lim→ ( ) = | |lim→ ( ) = min , , ⋯ ,lim→ ( ) = max , , … ,

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LEHMER’s Inequality | | + | || | + | | ≤ | | + | || | + | | | | + | | + | || | + | | + | | ≤ | | + | | + | || | + | | + | | ∑ | |∑ | | ≤ ∑ | |∑ | | , ≤ ; ≥ 0; ∈ 1, ; ∈ ℕ CARLEMAN’s Inequality | | ≤ | | , ∈ ℕ∗; , , … , ∈ ℝ CALLEBAUT’s Inequality

≥ 1 ≥ ≥ ≥ 0; ∈ 1, ; ∈ ℕ∗ SUM & PRODUCT Inequality ≥ ( )≤ ( )0 ≤ ≤ ≤ ⋯ ≤ ; ∈ 1,= 1 2 3 ⋯ ⋯⋯ ⋯ ⋯ ( ) ⋯ ⋯ ⋯

− ≤ | − | ; | | ≤ 1; | | ≤ 1∈ 1, ; ∈ ℕ∗; , ∈ ℝ; ( + ) ≥ (1 + ) ; ≥ 1; > 0; > 0

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SQUARE & ROOT Inequality 2√ + 1 − 2√ < 1√ < √ + 1 − √ − 1 < 2√ − 2√ − 1; ≥ 1 LOGARITHM MEAN Inequality ≤ √ +2 ≤ −ln − ln ≤ √ +2 ≤ +2 ;, ∈ (0, ∞)

HEINZ’s Inequality ≤ +2 ≤ +2 ; , > 0, ∈ [0,1] MACLAURIN’s Inequality + +3 ≥ + +3 ≥ √+ + +4 ≥ + + + + +6 ≥ + + +4 ≥≥ √∑ ≥ ∑ 2 ≥ ∑ 3 ≥ ⋯ ≥ ∑ ⋅ … ⋅− 1 ≥ ⋅ … ⋅

≥ ; ≥ ; 1 ≤ <= 1 ⋅ ⋅ … ⋅⋯ ; ≥ 0 BERNOULLI’s Inequality (1 + ) ≥ 1 + ; ≥ −1; ∈ (−∞, 0] ∪ [1, ∞) (1 + ) ≤ 1 + ; ≥ −1; ∈ [0,1] (1 + ) ≤ 1 + (2 − 1) ; ∈ [0,1]; ∈ (−∞, 0] ∪ [1, ∞) (1 + ) ≤ 11 − ; ∈ [−1,0]; ∈ ℕ (1 + ) ≤ 1 + 1 − ( − 1) ; ∈ −1, 1− 1 ; > 1 (1 + ) ≥ (1 + ( + 1) ) ; ∈ ℝ; ∈ ℕ

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( + ) ≤ + ( + ) ; , ≥ 0; ∈ ℕ 1 + ≥ 1 + for ( ) > 0; > > 0( ) − < − < < 0( ) − > − > > 0 1 + ≤ 1 + for ( ) < 0 < , − > > 0( ) < 0 < , − < < 0 TRIGONOMETRIC Inequalities − 2 ≤ cos ≤ cos1 − ≤ √cos ≤ − 6 ≤ cos √3 ≤ sin HIPERBOLIC Inequalities cos ≤ sin ℎ ≤ cos 2 ≤ sin ≤ ( cos + 2 ) 3⁄ ≤ sin ℎ 2 ≤ sin ≤ cos 2 ≤ ≤ + 3 ≤ tan ; ∈ 0, 2 cos ℎ + sin ℎ ≤ ; ∈ ℝ; ∈ [−1,1] ACZEL’s Inequality

− ≥ − −if > or >

ABEL’s Inequality min ≤ ≤ max≥ ≥ ⋯ ≥ ≥ 0; ∈ ℕ∗

KY FAN’s Inequality ∏∏ (1 − ) ≤ ∑∑ (1 − ) ; ∈ 0, 12∈ [0,1]; + + ⋯ + = 1

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SHAPIRO’s Inequality + ≥ 2 ; > 0; = ; =≤ 12, uneven or ≤ 23, even; ∈ ℕ∗; ≥ 3

CHONG’s Inequality ( ) ≥ ; ≥ ; > 0

= 1 2 3 ⋯ ⋯⋯ ⋯ ⋯ ( ) ⋯ ⋯ ⋯ ∈ ; ∈ ℕ∗

SURANYI’s Inequality If , , … , > 0, ∈ ℕ∗ then: ( − 1)( + + ⋯ + ) + … ≥ ≥ ( + + ⋯ + )( + + ⋯ + ) HÖLDER’s Reversed Inequality + + ≥ ( + + )( + + ) ⋅ 3 , , , , , , , ∈ (0, ∞) VIÈTE INTERFERENCES – 1 + + = , + + = , = + + = − 2 + + = − 3 + 3 + + = ( − 2 ) − 2( − 2 ) + + = − 2 + + + + + = − 3 ( + )( + )( + ) = − (2 − − )(2 − − )(2 − − ) = 2 − 9 + 27

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( + )( + )( + ) = − 2 − 2 + 4 − + + + + + = − 3 VIÈTE INTERFERENCES – 2 + + = , + + = , = 1 + 1 + 1 = , 1 + 1 + 1 = − 2 + + + + + = − 2 − 2 + 4 − 3 + + + + + = − 2 − + + = − 2 VIÈTE INTERFERENCES – 3 + + = , + + = , = ( − )+ + ( − )+ + ( − )+ = 2( − 3 − 2 )− + + + + = + − 6 + 9 ( − ) + ( − ) + ( − ) = 2 − 4 ( − ) + ( − ) + ( − ) = 2 − 3 − 3 ( − ) ( − ) ( − ) = − 4 − 4 + 18 − 27 + + + + + = = − 2 − 3 + 6 + 3 − 7 VIÈTE INTERFERENCES – 4 + + = , + + = , = + + = − 3 + 3 + + = − 4 + 2 + 4

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≥ 3 , ≥ 27 , ≥ 3 , ≥ 9 2 + 9 ≥ 7 , + 3 ≥ 4 + 4 + 6 ≥ 5 ≥ 0, (4 − )4 , (4 − )( − )6 PQR METHOD – 1 + + = , + + = −3 , = + + = + 23 + + = + 3 ( + ) + ( + ) + ( + ) = ( − )3 − 3 ( + )( + )( + ) = ( − )3 − ( + ) + ( + ) + ( + ) = ( + 2 )( − )9 − + + = − + 8 + 29 + 4 PQR METHOD – 2 If + + = , + + = −3 , = , ≥ 0 then: ( + ) ( − 2 )27 ≤ ≤ ( − ) ( + 2 )27 If + + = 1 then: (1 + ) (1 − 2 )27 ≤ ≤ (1 − ) (1 + 2 )27 If = 1 then: ( + ) ( − 2 )27 ≤ 1 ≤ ( − ) ( + 2 )27

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EXPONENTIAL Inequalities ≥ 1 + ≥ 1 + ; 1 + ≥ 1 − ; > 1; | | ≤ ≥ ; (∀) ∈ ℝ ! + 1 ≤ ≤ 1 + ; , ∈ (0, ∞) ≥ 1 + + 2 ; ≥ 0; ≤ 1 + + 2 ; ≤ 0 ≤ 1 − 2 ; ∈ [0, 1,59], 2 ≤ 1 − 2 ; ∈ [0, 1] 12 − < < − + 1; ∈ (0, 1) ( − 1) ≤ − 1 ; , ∈ [1, ∞) + > 1; > 1 + > ; , ∈ (0, ∞) 2 − − ≤ 1 + ≤ + ; , ∈ ℝ ≤ + ; , ∈ ℝ LOGARITHMIC Inequalities − 1 ≤ ln ≤ − 12 ≤ − 1; ln ≤ − 1 ; , ∈ (0, ∞) 21 + ≤ ln(1 + ) ≤ √ + 1 ; ≥ 0 21 + ≥ ln(1 + ) ≥ √ + 1 ; ∈ (−1,0]

ln( + 1) < 1 + ln ≤ 1 ≤ 1 + ln ln(1 + ) ≥ 2 ; ∈ [0, 2,51], ln(1 + ) ≤ 2 ; ∈ (−1,0] ∪ (2,51, ∞) ln(1 + ) ≥ − 2 + 4 ; ∈ [0; 0,45] ln(1 + ) ≤ − 2 + 4 ; ∈ (−∞, 0) ∪ (0,45; ∞) ln(1 − ) ≥ − − 2 − 2 ; ∈ [0,0,43] ln(1 − ) ≤ − − 2 − 2 ; ∈ (−∞, 0) ∪ (0,43; 1)

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BINOMIAL Inequalities max ; ( − + 1)! ≤ ≤ ! ≤ ( ) ≤ ( − ) ≤ 2 , 4 ! ≤ for √ ≥ ≥ 0 4√ 1 − 18 ≤ 2 ≤ 4√ 1 − 19 ≤ ++ for ≥ ≥ 0; ≥ ≥ 0 √2 ≤ ≤ ; = 2 ( )2 (1 − ) ; ( ) = − log ( (1 − ) ) ≤ + 1; ≤ 2 ; ≥ ≥ 0 ≤ ; ≥ ≥ 1

≤ 1 + − 2 + 1 ; 2 ≥ ≥ 0 ≤ ≤ 1 −1 − 2 ; ∈ 0, 12 GIBBS’s Inequality ln ≥ ln ; > 0; > 0∈ 1, ; ∈ ℕ∗≤ ; concave;

= + + ⋯ + ; = + + ⋯ +

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KANTOROVICH’s Inequality ≤ ( + )4, ∈ (0, ∞); 0 < ≤ ≤ < ∞∈ 1, ; ∈ ℕ∗

KARAMATA’s Inequality ≥ ; + ≥ + ; + + ⋯ + ≥ + + ⋯ ++ + ⋯ + = + + ⋯ + ; convex( ) + ( ) + ⋯ + ( ) ≥ ( ) + ( ) + ⋯ + ( ) MURRAY–KLAMKIN Inequality ∏ (1 + )(1 + ) ≥ ∏ (1 − )( − 1) ; ≥ 2∈ (0, ∞); ∈ 1, ; + + ⋯ + = 1; ∈ ℕ∗ KURLIANCIK’s Inequality

1 + 1 + ⋯ + 1 < 2 ; > 0 TIBERIU POPOVICIU’s Inequality 13 ( ) + ( ) + ( ) + + +3 ≥ 23 +2 + +2 + +2 : [ , ] → ℝ; convex DIAZ–METCALF Inequality

+ ≤ ( + ) , ∈ ℝ∗; ≤ ≤ ; ∈ 1, ; ∈ ℕ∗ GREUB–RHEIBOLDT Inequality [ , ]; [ , ] ⊂ (0, ∞); ∈ [ , ]; ∈ [ , ]; ∈ ℝ; ∈ 1, ; ∈ ℕ∗

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≤ ( + )4 POLYA–SZEGO Inequality [ , ]; [ , ] ⊂ (0, ∞); ∈ [ , ]; ∈ [ , ]; ∈ ℝ; ∈ 1, ; ∈ ℕ∗ ≤ ( + )4 SCHWEITZER’s Inequality [ , ] ⊂ (0, ∞); ∈ [ , ]; ∈ 1, ; ∈ ℕ∗ 1 ≤ ( + )4 HLAWKA’s Inequality | + + | + | | + | | + | | ≥ | + | + | + | + | + |; , , ∈ ℂ ABEL’s Inequality = ( − )( + + ⋯ + )

BECKENBACH’s Inequality ∑ ( + )∑ + ∑ ≤ ∑∑ + ∑∑ , , … , , , , … , > 0; ∈ ℕ∗ USEFUL INEQUALITIES √ < 1 + 2 ; ≥ 2; √ ! > ; ≥ 1 2√ + 1 + √ − 1 > 1√ ; ≥ 2; ≥ 1 tan > sin ; ∈ 0, 2 ; sin + tan > 2 ; ∈ 0, 2

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2 − ≥ cos ; ∈ 0, 2 , cos ≥ 1 − 2 ; ∈ ℝ − sin ≤ ; ∈ 0, 2 , tan − ≤ ; ∈ [0,1) + 2√ + 1 ≥ 2; ∈ ℝ; 1 + ≤ 12 ; ∈ ℝ 2 + ≥ √ + ; + ≥ +2 + + ≤ 14 1 + 1 ; + √ ≥ √ + +4 + + − ≤ −2 ; ≥ > 0; √ − 1 ≤ 12 ; ≥ 1; + ≥ +2 + ; , ≥ 0 + + ≥ 3 ; , , ∈ [0, ∞) 1(1 + ) + 1(1 + ) ≥ 11 + ; , ∈ (0, ∞); ≥ 1 ++ + ≥ +3 ; , ∈ (0, ∞) + + ≥ 2 −3 ; , ∈ (0, ∞) + + ≥ √32 ( + ); , ∈ [0, ∞) + + ≤ 3( − + ); , ∈ ℝ + ≤ + ; , ∈ ℝ∗ √ + 1+ 2 ≤ 12 ; ∈ ℝ

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3( + )2( + + ) ≥ − + ≥ ; , ∈ ℝ √+ 1 < 1√ + 2 ; ∈ ℕ∗, − 1 < 2 − 1 ; > 2 1 < √ ≤ √3; ≥ 4, + + ≥ 3; , ∈ ℝ √ + ≤ 2 ; , ≥ 0, ≤ ( !) ; ∈ ℕ∗ ≥ 1; , , , ∈ ℝ∗ tan ≥ ≥ sin ≥ + 1 ; ∈ 0, 2 ( + )( + ) ≥ + √ ; , , ∈ [0, ∞), ≤ + ; ∈ ℝ ++ ≥ +2 ; , ∈ ℝ; + ≠ 0

SCHWEITZER’s Inequality – Integral Form , , , ∈ (0, ∞); < ; < , : [ , ] → [ , ]; continous function ( ) 1( ) ≤ ( + ) ( − ) KANTOROVICH’s Inequality – Integral Form , , , ∈ (0, ∞); < ; < , : [ , ] → [ , ]; ℎ: [ , ] → ℝ, , , ℎ continuous functions ( ⋅ ℎ) ( ) ℎ ( ) ≤ ( + )4 ℎ( ) GREUB–RHEINBOLDT Inequality – Integral Form , , , , , ∈ (0, ∞); < ; < ; < : [ , ] → [ , ]; : [ , ] → [ , ], ℎ: [ , ] → ℝ; , , ℎ continuous functions

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( ℎ) ( ) ( ℎ)( ) ≤ ( + )4 ( ℎ)( ) PÓLYA–SZEGŐ Inequality – Integral Form , , , , , ∈ (0, ∞); < ; < ; < : [ , ] → [ , ]; : [ , ] → [ , ] , continuous functions ( ) ( ) ≤ ( + )4 ( )( ) MEANS Inequality – Integral Form ≤ −( ) ≤ ( ) ≤ 1− ( ) ; : [ , ] → [ , ]

CAUCHY–SCHWARZ Inequality – Integral Form | ( ) ( )| ≤ ( ) ( ) , , : [ , ] → ℝ

HÖLDER’s Inequality – Integral Form | ( ) ( )| ≤ | ( )| | ( )| , > 1; 1 + 1 = 1 MINKOWSKI’s Inequality – Integral Form

[ ( ) + ( )] ≤ ( ) + ( ) | ( ) + ( )| ≤ | ( )| + | ( )| ≥ 1; , : [ , ] → ℝ

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CHEBYSHEV’s Inequality – Integral Form , : [ , ] → ℝ monotone to the contrary( ) ( ) ≤ 1− ( ) ( ) , : [ , ] → ℝ monotone in the same sense( ) ( ) ≥ 1− ( ) ( ) JENSEN’s Inequality – Integral Form : [ , ] → [ , ]; : [ , ] → ℝ − convex1− ( ) ≤ 1− ( ) YOUNG’s Inequality – Integral Form : (0, ∞) → (0, ∞) continuous and increasing∈ (0, ∞); ∈ (0, ∞) ; (0) = 0≤ ( ) + ( ) CAUCHY’s Inequality – Integral Form : [ , ] → ℝ; increasing, ( ) ≤ ( ) ≤ ( ) HERMITE–HADAMARD Inequality – Integral Form +2 ≤ 1− ( ) ≤ ( ) + ( )2 : [ , ] → ℝ; ∈ ([ , ]); convex