关于不等式的专题讨论III(均值不等式，Jensen不等式，Schwarz不等式，Holder不等式，Young不等式，Minkowski不等式，其他常用不等式)

$f命题：$设$f,g in Rleft[ {a,b} ight]$，且${m_1} le fleft( x ight) le {M_1},{m_2} le gleft( x ight) le {M_2}$，则

[frac{1}{{b - a}}int_a^b {fleft( x ight)gleft( x ight)dx}  - frac{1}{{{{left( {b - a} ight)}^2}}}int_a^b {fleft( x ight)dx} int_a^b {gleft( x ight)dx}  le frac{{left( {{M_1} - {m_1}} ight)left( {{M_2} - {m_2}} ight)}}{4}]

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$f命题：$设连续函数$fleft( x ight)$$:left[ {0,infty } ight) o left[ {0,infty } ight)$满足$fleft( {fleft( x ight)} ight) = {x^m},forall x in left[ {0,infty } ight),m in {Z^ + }$，则[int_0^1 {{f^2}left( x ight)dx}  ge frac{{2m - 1}}{{{m^2} + 6m - 3}}]

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$f命题：$设$p > 1,fleft( x ight) in Cleft( {0, + infty } ight),int_0^{ + infty } {{{left| {fleft( t ight)} ight|}^p}dt} $收敛，证明：[{left{ {int_0^{ + infty } {{{[frac{1}{x}int_0^x {left| {fleft( t ight)dt} ight|} ]}^p}dx} } ight}^{frac{1}{p}}} leqslant frac{p}{{p - 1}}{left( {int_0^{ + infty } {{{left| {fleft( t ight)} ight|}^p}dt} } ight)^{frac{1}{p}}}]

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$f命题：$

附录

$f命题：$设$fleft( x ight) in {C^1}left[ {a,b} ight]$，且$fleft( a ight) = 0$，则[int_a^b {left| {fleft( x ight)f'left( x ight)} ight|dx}  le frac{{b - a}}{2}int_a^b {{{left[ {f'left( x ight)} ight]}^2}dx} ]

当且仅当$fleft( x ight){ m{ = }}kleft( {x - a} ight)$时等号成立

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$f命题：$设$fleft( x ight) in {C^1}left[ {a,b} ight]$，且$fleft( a ight) = fleft( b ight) = 0$，则[int_a^b {left| {fleft( x ight)f'left( x ight)} ight|dx}  le frac{{b - a}}{4}int_a^b {{{left[ {f'left( x ight)} ight]}^2}dx} ]

并且$frac{{b - a}}{4}$为最佳系数

$f(04中科院七)$设$0<x,y<pi$，求证：$sin xsin ysin left( {x + y} ight) leqslant frac{{3sqrt 3 }}{8}$

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$f(对数不等式)$当$x>-1$时，成立：$frac{x}{{1 + x}} leqslant ln left( {1 + x} ight) leqslant x$

$f(Jordan不等式)$当$0<x<frac{pi }{2}$时，成立：$frac{2}{pi }x leqslant sin x leqslant x$