Processing math: 0%
Eufisky - The lost book

谢之题解:重积分的应用举例

7.设uiLpi(Ω),pi>0,i=1,2,,m,且mi=11pi=1.证明


解.利用P270的Holder不等式以及数学归纳法即可.假设m-1时不等式成立,则有

\begin{align*}&\iint\limits_\Omega {{u_1} \cdots {u_{m - 1}}{u_m}dxdy} \le \iint\limits_\Omega {\left| {{u_1} \cdots {u_{m - 1}}{u_m}} \right|dxdy} \\= &{\left\| {{u_1} \cdots {u_{m - 1}}{u_m}} \right\|_1} \le {\left\| {{u_1} \cdots {u_{m - 1}}} \right\|_{1/\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }}{\left\| {{u_m}} \right\|_{{p_m}}}\\\le& {\left( {{{\left\| {u_1^{1/\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }} \right\|}_{{p_1}\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }} \cdots {{\left\| {u_{m - 1}^{1/\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }} \right\|}_{{p_{m - 1}}\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }}} \right)^{\sum\limits_{i = 1}^{m - 1} {\frac{1}{{{p_i}}}} }}{\left\| {{u_m}} \right\|_{{p_m}}}\\= &{\left\| {{u_1}} \right\|_{{p_1}}} \cdots {\left\| {{u_{m - 1}}} \right\|_{{p_{m - 1}}}}{\left\| {{u_m}} \right\|_{{p_m}}}.\end{align*}

另解:利用n元的Holder不等式

\prod\limits_{i = 1}^m {x_i^{{\theta _i}}} \le \sum\limits_{i = 1}^m {{\theta _i}{x_i}} ,\quad \text{其中}\sum\limits_{i = 1}^m {{\theta _i}} = 1,{\theta _i} \ge 0.

{x_i} = \frac{{{{\left| {{u_i}} \right|}^{{p_i}}}}}{{\left\| {{u_i}} \right\|_{{p_i}}^{{p_i}}}},{\theta _i} = \frac{1}{{{p_i}}}再积分即可.


证明1 < \iiint\limits_{{{\left[ {0,1} \right]}^3}} {\left( {\cos \left( {xyz} \right) + \sin \left( {xyz} \right)} \right)dxdydz} < \sqrt 2 .


证.注意到1 < \cos \left( {xyz} \right) + \sin \left( {xyz} \right) = \sqrt 2 \sin \left( {xyz + \frac{\pi }{4}} \right) < \sqrt 2 即可.


证明

{\left\{ {{{\int_a^b {dx\left[ {\int_c^d {f\left( {x,y} \right)dy} } \right]} }^2}} \right\}^{1/2}} \le \int_c^d {dy{{\left[ {\int_a^b {{f^2}\left( {x,y} \right)dx} } \right]}^{1/2}}} ,

其中f是连续函数.


证.利用Cauchy-Schwarz不等式我们有

\begin{align*}&{\left\{ {\int_c^d {dy{{\left[ {\int_a^b {{f^2}\left( {x,y} \right)dx} } \right]}^{1/2}}} } \right\}^2}\\= &\int_c^d {dy{{\left[ {\int_a^b {{f^2}\left( {x,y} \right)dx} } \right]}^{1/2}}} \cdot \int_c^d {dz{{\left[ {\int_a^b {{f^2}\left( {x,z} \right)dx} } \right]}^{1/2}}} \\\ge& \int_c^d {dy} \int_c^d {dz} \int_a^b {f\left( {x,y} \right)f\left( {x,z} \right)dx} = {\int_a^b {dx\left[ {\int_c^d {f\left( {x,y} \right)dy} } \right]} ^2}.\end{align*}

 

2013武大数分压轴题

(13年武大数分)求\displaystyle I = \iint\limits_\Sigma  {{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{ - \frac{3}{2}}}{{\left( {\frac{{{x^2}}}{{{a^4}}} + \frac{{{y^2}}}{{{b^4}}} + \frac{{{z^2}}}{{{c^4}}}} \right)}^{ - \frac{1}{2}}}dS} ,其中\sum为椭球面: \displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1(a,b,c>0).


下面是自己的解答:

x = a\sin \varphi \cos \theta ,y = b\sin \varphi \sin \theta ,z = c\cos \varphi ,其中0\leq \theta\leq 2\pi,0\leq\varphi \leq\pi,经计算得到
\frac{{\partial \left( {y,z} \right)}}{{\partial \left( {\varphi ,\theta } \right)}} = bc{\sin ^2}\varphi \cos \theta ,\frac{{\partial \left( {z,x} \right)}}{{\partial \left( {\varphi ,\theta } \right)}} = ac{\sin ^2}\varphi \sin \theta ,\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {\varphi ,\theta } \right)}} = ab\sin \varphi \cos \varphi ,
所以
\begin{align*}EG - {F^2} &= {\left( {\frac{{\partial \left( {y,z} \right)}}{{\partial \left( {\varphi ,\theta } \right)}}} \right)^2} + {\left( {\frac{{\partial \left( {z,x} \right)}}{{\partial \left( {\varphi ,\theta } \right)}}} \right)^2} + {\left( {\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {\varphi ,\theta } \right)}}} \right)^2}\\& = {\left( {abc} \right)^2}{\sin ^2}\varphi \left( {\frac{{{{\sin }^2}\varphi {{\cos }^2}\theta }}{{{a^2}}} + \frac{{{{\sin }^2}\varphi {{\sin }^2}\theta }}{{{b^2}}} + \frac{{{{\cos }^2}\varphi }}{{{c^2}}}} \right).\end{align*}
而这时被积函数化为
\begin{align*}&{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - \frac{3}{2}}}{\left( {\frac{{{x^2}}}{{{a^4}}} + \frac{{{y^2}}}{{{b^4}}} + \frac{{{z^2}}}{{{c^4}}}} \right)^{ - \frac{1}{2}}}\\= &{\left( {{a^2}{{\sin }^2}\varphi {{\cos }^2}\theta  + {b^2}{{\sin }^2}\varphi {{\sin }^2}\theta  + {c^2}{{\cos }^2}\varphi } \right)^{ - \frac{3}{2}}}\\&{\left( {\frac{{{{\sin }^2}\varphi {{\cos }^2}\theta }}{{{a^2}}} + \frac{{{{\sin }^2}\varphi {{\sin }^2}\theta }}{{{b^2}}} + \frac{{{{\cos }^2}\varphi }}{{{c^2}}}} \right)^{ - \frac{1}{2}}}.\end{align*}
因此
I = abc\iint\limits_{\left[ {0,\pi } \right] \times \left[ {0,2\pi } \right]} {{{\left( {{a^2}{{\sin }^2}\varphi {{\cos }^2}\theta  + {b^2}{{\sin }^2}\varphi {{\sin }^2}\theta  + {c^2}{{\cos }^2}\varphi } \right)}^{ - \frac{3}{2}}}\sin \varphi d\varphi d\theta }
 
注意到这么一个事实,当M+Nx^2不取0M\neq 0时,我们有
\int {{{\left( {M + N{x^2}} \right)}^{ - 3/2}}dx}  = \frac{1}{M} \cdot \frac{x}{{\sqrt {M + N{x^2}} }} + C.
 
\begin{align*}I &= abc\int_0^{2\pi } {d\theta } \int_0^\pi  {{{\left( {{a^2}{{\sin }^2}\varphi {{\cos }^2}\theta  + {b^2}{{\sin }^2}\varphi {{\sin }^2}\theta  + {c^2}{{\cos }^2}\varphi } \right)}^{ - \frac{3}{2}}}\sin \varphi d\varphi } \\&=  - abc\int_0^{2\pi } {d\theta } \int_0^\pi  {{{\left( {{a^2}{{\sin }^2}\varphi {{\cos }^2}\theta  + {b^2}{{\sin }^2}\varphi {{\sin }^2}\theta  + {c^2}{{\cos }^2}\varphi } \right)}^{ - \frac{3}{2}}}d\left( {\cos \varphi } \right)} \\&=  - abc\int_0^{2\pi } {d\theta } \int_0^\pi  {{{\left[ {\left( {{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta } \right) + \left( {{c^2} - {a^2}{{\cos }^2}\theta  - {b^2}{{\sin }^2}\theta } \right){{\cos }^2}\varphi } \right]}^{ - \frac{3}{2}}}d\left( {\cos \varphi } \right)} \\&= abc\int_0^{2\pi } {d\theta } \int_{ - 1}^1 {{{\left[ {\left( {{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta } \right) + \left( {{c^2} - {a^2}{{\cos }^2}\theta  - {b^2}{{\sin }^2}\theta } \right){x^2}} \right]}^{ - \frac{3}{2}}}dx} \\&= abc\int_0^{2\pi } {\frac{2}{{\left( {{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta } \right)c}}d\theta }  = 4ab\int_0^\pi  {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta } .\end{align*}
\begin{align*}&\int_0^\pi  {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta }  = \int_0^{\frac{\pi }{2}} {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta }  + \int_{\frac{\pi }{2}}^\pi  {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta } \\= &\int_0^{\frac{\pi }{2}} {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta }  + \int_0^{\frac{\pi }{2}} {\frac{1}{{{a^2}{{\sin }^2}\theta  + {b^2}{{\cos }^2}\theta }}d\theta } \\= &\int_0^{ + \infty } {\frac{1}{{{a^2} + {b^2}{x^2}}}d}  + \int_0^{ + \infty } {\frac{1}{{{a^2}{x^2} + {b^2}}}dx}  = \frac{1}{{ab}}\left. {\arctan \left( {\frac{b}{a}x} \right)} \right|_0^{ + \infty } + \frac{1}{{ab}}\left. {\arctan \left( {\frac{a}{b}x} \right)} \right|_0^{ + \infty }\\= &\frac{\pi }{{ab}}.\end{align*}
进而得到
I = 4ab\int_0^\pi  {\frac{1}{{{a^2}{{\cos }^2}\theta  + {b^2}{{\sin }^2}\theta }}d\theta }  = 4\pi .

另外有更好的方法:(Hansschwarzkopf)

注意到\Sigma 在点(x,y,z)处的单位外法向量是
n=\frac{\left(\frac{x}{a^2},\frac{y}{b^2},\frac{z}{c^2}\right)}{\sqrt{\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}}},
1=x\cdot \frac{x}{a^2}+y\cdot\frac{y}{b^2}+z\cdot\frac{z}{c^2}.
从而原积分可写成第二型曲面积分
\iint\limits_\Sigma \frac{xdydz+yd zd x+zdxdy}{\sqrt{(x^2+y^2+z^2)^3}}.
作小球面S_\varepsilon: x^2+y^2+z^2=\varepsilon^2. 运用Gauss公式可知
\iint\limits_\Sigma \frac{xd yd z+yd zd x+zd xd y}{\sqrt{(x^2+y^2+z^2)^3}} =\iint\limits_{S_\varepsilon} \frac{xdyd z+ydzd x+zd xd y}{\sqrt{(x^2+y^2+z^2)^3}}=4\pi.
\iint\limits_\Sigma\frac{d S}{\sqrt{(x^2+y^2+z^2)^3}\sqrt{\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}}}=4\pi.

 

2011年南开大学高等代数试题

 

参考:http://www.math.org.cn/forum.php?mod=viewthread&tid=21850&extra=&page=2

谢惠民一道全微分题

前几天徐半仙问了我谢惠民下册P325页上一道难度稍大的全微分题目,利用今晚美好的独处时间(笑哭),做了下,解答如下:

对于以下一阶微分形式\omega ,求函数M(x,y)\neq0, 使得在适当的区域内M\omega 为全微分,并求其原函数:

(1) \displaystyle \omega = \left[ { - y\sqrt {{x^2} + {y^2} + 1} - x\left( {{x^2} + {y^2}} \right)} \right]dx + \left[ {x\sqrt {{x^2} + {y^2} + 1} - y\left( {{x^2} + {y^2}} \right)} \right]dy;

 

(2) \displaystyle \omega = x\left[ {{{\left( {ay + bx} \right)}^3} + a{y^3}} \right]dx + y\left[ {{{\left( {ay + bx} \right)}^3} + b{x^3}} \right]dy.

解:(1)取M = \frac{1}{{\left( {{x^2} + {y^2}} \right)\sqrt {{x^2} + {y^2} + 1} }},我们有

M\omega = \left( { - \frac{y}{{{x^2} + {y^2}}} - \frac{x}{{\sqrt {{x^2} + {y^2} + 1} }}} \right)dx + \left( {\frac{x}{{{x^2} + {y^2}}} - \frac{y}{{\sqrt {{x^2} + {y^2} + 1} }}} \right)dy.

则有P = - \frac{y}{{{x^2} + {y^2}}} - \frac{x}{{\sqrt {{x^2} + {y^2} + 1} }},Q = \frac{x}{{{x^2} + {y^2}}} - \frac{y}{{\sqrt {{x^2} + {y^2} + 1} }},且\frac{{\partial P}}{{\partial y}} = \frac{{{y^2} - {x^2}}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} + \frac{{xy}}{{{{\left( {{x^2} + {y^2} + 1} \right)}^{3/2}}}} = \frac{{\partial Q}}{{\partial x}}.

此时原函数为

\begin{align*}\varphi \left( {x,y} \right) = &\int_{{x_0}}^x {P\left( {x,{y_0}} \right)dx} + \int_{{y_0}}^y {Q\left( {x,y} \right)dy} + C'\\= &\int_{{x_0}}^x {\left( { - \frac{{{y_0}}}{{{x^2} + y_0^2}} - \frac{x}{{\sqrt {{x^2} + y_0^2 + 1} }}} \right)dx} + \int_{{y_0}}^y {\left( {\frac{x}{{{x^2} + {y^2}}} - \frac{y}{{\sqrt {{x^2} + {y^2} + 1} }}} \right)dy} + C'\\= &\left( { - \arctan \frac{x}{{{y_0}}} - \sqrt {{x^2} + y_0^2 + 1} + \arctan \frac{{{x_0}}}{{{y_0}}} + \sqrt {x_0^2 + y_0^2 + 1} } \right)\\&+ \left( {\arctan \frac{y}{x} - \sqrt {{x^2} + {y^2} + 1} - \arctan \frac{{{y_0}}}{x} + \sqrt {{x^2} + y_0^2 + 1} } \right) + C'\\=& \arctan \frac{y}{x} - \sqrt {{x^2} + {y^2} + 1} + C.\end{align*}

 

值得一提的是:本题的积分因子是通过Wolfram Alpha求解出ODE,然后分别对x,y求偏导得来的.

 

(2)丁同仁书上一定理:


齐次方程P(x,y)dx+Q(x,y)dy=0有积分因子M=\frac{1}{xP+yQ}.


 

定理的证明:作变换y=ux,则由P\left( {x,y} \right)dx + Q\left( {x,y} \right)dy = 0是齐次方程,我们有P\left( {x,ux} \right)dx + Q\left( {x,ux} \right)\left( {udx + xdu} \right) = \left[ {{x^m}P\left( {1,u} \right) + u{x^m}Q\left( {1,u} \right)} \right]dx + {x^{m + 1}}Q\left( {1,u} \right)du = 0.

 

方程两边同乘\frac{1}{{xP + yQ}} = \frac{1}{{{x^{m + 1}}\left[ {P\left( {1,u} \right) + uQ\left( {1,u} \right)} \right]}},则有

\frac{1}{x}dx + \frac{{Q\left( {1,u} \right)}}{{P\left( {1,u} \right) + uQ\left( {1,u} \right)}}du = 0.显然此方程为全微分方程.证毕.

 

M = \frac{1}{{xP + yQ}} = \frac{1}{{\left( {{x^2} + {y^2}} \right){{\left( {ay + bx} \right)}^3} + {x^2}{y^2}\left( {ay + bx} \right)}}.

则有

P' = \frac{{x{{\left( {ay + bx} \right)}^3} + ax{y^3}}}{{\left( {{x^2} + {y^2}} \right){{\left( {ay + bx} \right)}^3} + {x^2}{y^2}\left( {ay + bx} \right)}},Q' = \frac{{y{{\left( {ay + bx} \right)}^3} + b{x^3}y}}{{\left( {{x^2} + {y^2}} \right){{\left( {ay + bx} \right)}^3} + {x^2}{y^2}\left( {ay + bx} \right)}}.

 

我猜此时一定成立\frac{{\partial P'}}{{\partial y}} = \frac{{\partial Q'}}{{\partial x}}.

 

事实上

\begin{align*}\frac{{\partial P'}}{{\partial y}} = & - 2xy{\left( {ay + bx} \right)^6} - 2{x^3}y{\left( {ay + bx} \right)^4} + \left( {5a{x^3}{y^2} + ax{y^4}} \right){\left( {ay + bx} \right)^3}\\& - 3{a^2}x{y^3}\left( {{x^2} + {y^2}} \right){\left( {ay + bx} \right)^2} + a{x^3}{y^4}\left( {ay + bx} \right) - {a^2}{x^3}{y^5}\\\frac{{\partial Q'}}{{\partial x}} = &- 2xy{\left( {ay + bx} \right)^6} - 2x{y^3}{\left( {ay + bx} \right)^4} + \left( {5b{x^2}{y^3} + b{x^4}y} \right){\left( {ay + bx} \right)^3}\\& - 3{b^2}{x^3}y\left( {{x^2} + {y^2}} \right){\left( {ay + bx} \right)^2} + b{x^4}{y^3}\left( {ay + bx} \right) - {b^2}{x^5}{y^3}.\end{align*}

 

于是

\begin{align*}\varphi \left( {x,y} \right) &= \int_{{x_0}}^x {P'\left( {x,{y_0}} \right)dx} + \int_{{y_0}}^y {Q'\left( {x,y} \right)dy} + C'\\&= \frac{1}{2}\ln \left[ {\left( {{x^2} + {y^2}} \right){{\left( {ay + bx} \right)}^3} + {x^2}{y^2}\left( {ay + bx} \right)} \right] - \frac{3}{2}\ln \left( {ay + bx} \right) + C.\end{align*}

 

事实上,我们还可取M = \frac{1}{{{{\left( {ay + bx} \right)}^3}}},由此得到

\varphi \left( {x,y} \right) = \frac{{{x^2} + {y^2}}}{2} + \frac{{{x^2}{y^2}}}{{2{{\left( {ay + bx} \right)}^2}}} + C.


一道杂志征解题的解答

这道题来自MAA的杂志The American Mathematical Monthly, Vol. 122, No. 5 (May 2015), pp. 500-507,可以参考链接http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.500?seq=1#page_scan_tab_contents


\int_0^\infty  {\frac{1}{x}dx} \int_0^x {\frac{{\cos \left( {x - y} \right) - \cos x}}{y}dy} .


解.(翻译而来)令f\left( {x,y} \right) = \frac{{\cos \left( {x - y} \right) - \cos x}}{y}.对x>0,我们有\int_0^x {f\left( {x,y} \right)dy}  = \int_0^1 {\frac{{\cos \left( {1 - t} \right)x - \cos x}}{y}dt}  = x\int_0^1 {\frac{1}{t}} \int_{1 - t}^1 {\sin ux\, dudt} .

因而对R>0,

\int_0^R {\frac{1}{x}} \int_0^x {f\left( {x,y} \right)\,dydx}  = \int_0^R {\int_0^1 {\frac{1}{t}} \int_{1 - t}^1 {\sin ux\,dudtdx} } .

|\sin ux|\leq1,该三重积分是绝对收敛的.由Fubini定理可知积分能交换次序

\begin{align*}&\int_0^R {\frac{1}{x}} \int_0^x {f\left( {x,y} \right)\,dydx}  = \int_0^1 {\int_{1 - t}^1 {\frac{1}{t}} \int_0^R {\sin ux\,dxdudt} } \\=& \int_0^1 {\frac{{1 - \cos Ru}}{u}\int_{1 - u}^1 {\frac{1}{t}} \,dtdu} \\= & - \int_0^1 {\frac{{\ln \left( {1 - u} \right)}}{u}\,du}  + \int_0^1 {\frac{{\ln \left( {1 - u} \right)}}{u}\cos Ru\,du} .\end{align*}
我们知|\ln (1-u)/u|\in L^1([0,1]),由Riemann-Lebesgue引理可知
\mathop {\lim }\limits_{R \to \infty } \int_0^1 {\frac{{\ln \left( {1 - u} \right)}}{u}\cos Ru\,du}  = 0.
由于{\sum\limits_{n = 1}^\infty  {\frac{{{t^{n - 1}}}}{n}} }一致收敛,故可逐项积分.因此我们有
\begin{align*}&\mathop {\lim }\limits_{R \to \infty } \int_0^R {\frac{1}{x}} \int_0^x {f\left( {x,y} \right)dydx}  =  - \int_0^1 {\frac{{\ln \left( {1 - u} \right)}}{u}du} \\= &\int_0^1 {\sum\limits_{n = 1}^\infty  {\frac{{{t^{n - 1}}}}{n}} dt}  = \sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}}  = \frac{{{\pi ^2}}}{6}.\end{align*}

 

哆嗒数学网里代数龙发的一系列级数题

练习题1.证明:\sum\limits_{n=1}^{\infty}\frac{1}{(n+1)\sqrt[p]{n}}\leq p,\,\,(p\ge1).

证:由Lagrange中值定理,我们有

\sqrt[p]{{n + 1}} - \sqrt[p]{n} = \frac{1}{p}{\xi ^{1/p - 1}} \ge \frac{1}{p}{\left( {n + 1} \right)^{1/p - 1}},\quad \xi  \in \left( {n,n + 1} \right).

因此\frac{1}{{\left( {n + 1} \right)\sqrt[p]{n}}} = \frac{{\sqrt[p]{{n + 1}} - \sqrt[p]{n}}}{{\sqrt[p]{n} \cdot \sqrt[p]{{n + 1}}}} \cdot \frac{{{{\left( {n + 1} \right)}^{1/p - 1}}}}{{\sqrt[p]{{n + 1}} - \sqrt[p]{n}}} \le p\frac{{\sqrt[p]{{n + 1}} - \sqrt[p]{n}}}{{\sqrt[p]{n} \cdot \sqrt[p]{{n + 1}}}} = p\left( {\frac{1}{{\sqrt[p]{n}}} - \frac{1}{{\sqrt[p]{{n + 1}}}}} \right).

立即有

\sum\limits_{n = 1}^\infty  {\frac{1}{{\left( {n + 1} \right)\sqrt[p]{n}}}}  \le p\sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{\sqrt[p]{n}}} - \frac{1}{{\sqrt[p]{{n + 1}}}}} \right)}  = p.


练习题2.设\displaystyle S_n=\sum\limits_{k=1}^{n}a_k, p>1,c>1,证明:\sum\limits_{n=1}^{\infty}\frac{S_n^p}{n^c}\le K\sum\limits_{n=1}^{\infty}\frac{(na_n)^p}{n^c},并求出K的最优值.


练习题3.设a_n是有界的正数列,p>0,证明:

\frac{1}{a_1^p}+\sum\limits_{n=1}^{\infty}\frac{a_1a_2 \cdots a_n}{a_{n+1}^p} \ge \sum\limits_{n=0}^{\infty}(\frac{p}{p+1})^{n-p}.

练习题4.设(0,+\infty)上的函数列f_n由下式定义:f_1(x)=x,f_{n+1}(x)=(f_n(x)+\frac{1}{n})f_n(x).证明:存在唯一的正数a,使得对于所有n0<f_n(x)<f_{n+1}(a)<1.


练习题5.\displaystyle\sum\limits_{n=1}^{\infty}a_n为正项收敛级数,\displaystyle r_n=\sum\limits_{k=n}^{\infty}a_k,0<p<1,证明:\sum\limits_{n=1}^{\infty}\frac{a_n}{r_n^p}<\frac{1}{1-p}\left(\sum\limits_{n=1}^{\infty}a_n \right)^{1-p}.


练习题6.设a>0,a_n是一个数列,并且a_n>0,a_{n+1}\ge a_n,证明:\sum\limits_{n=1}^{\infty}\frac{a_n-a_{n-1}}{a_na_{n-1}^a}收敛.

证:首先可以确定给定的级数是正项级数.

(1)当0<a<1时,我们利用Lagrange中值定理,有\frac{{a_n^a - a_{n - 1}^a}}{{{a_n} - {a_{n - 1}}}} = a{\xi ^{a - 1}} \ge aa_n^{a - 1},\quad \xi  \in \left( {{a_{n - 1}},{a_n}} \right).

因此\frac{{{a_n} - {a_{n - 1}}}}{{{a_n}a_{n - 1}^a}} = \frac{{a_n^a - a_{n - 1}^a}}{{a_n^aa_{n - 1}^a}} \cdot \left( {\frac{{{a_n} - {a_{n - 1}}}}{{a_n^a - a_{n - 1}^a}} \cdot a_n^{a - 1}} \right) \le \frac{1}{a}\frac{{a_n^a - a_{n - 1}^a}}{{a_n^aa_{n - 1}^a}} = \frac{1}{a}\left( {\frac{1}{{a_{n - 1}^a}} - \frac{1}{{a_n^a}}} \right).

\sum\limits_{n = 1}^\infty  {\frac{{{a_n} - {a_{n - 1}}}}{{{a_n}a_{n - 1}^a}}}  \le \frac{1}{a}\sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{a_{n - 1}^a}} - \frac{1}{{a_n^a}}} \right)}  = \frac{1}{a}\left( {\frac{1}{{a_0^a}} - \mathop {\lim }\limits_{n \to \infty } \frac{1}{{a_n^a}}} \right).

由于\{a_n\}是单增的正数列,则{\mathop {\lim }\limits_{n \to \infty } \frac{1}{{a_n^a}}}必定存在,由此可知原正项级数收敛;

(2)当a\geq1时,由\sum\limits_{n = 1}^\infty  {\frac{{{a_n} - {a_{n - 1}}}}{{{a_n}a_{n - 1}^a}}}  = \sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{a_{n - 1}^a}} - \frac{{a_{n - 1}^{1 - a}}}{{{a_n}}}} \right)}  \le \sum\limits_{n = 1}^\infty  {\left( {\frac{1}{{a_{n - 1}^a}} - \frac{1}{{a_n^a}}} \right)}  = \frac{1}{a}\left( {\frac{1}{{a_0^a}} - \mathop {\lim }\limits_{n \to \infty } \frac{1}{{a_n^a}}} \right)同样可知原正项级数收敛.

综上,级数\sum\limits_{n=1}^{\infty}\frac{a_n-a_{n-1}}{a_na_{n-1}^a}收敛.


练习题7.设\displaystyle S(x)=\sum\limits_{n=1}^{\infty}\frac{2n}{(n^2 +x^2)^2},证明:\frac{1}{x^2 +\frac{1}{2\zeta(3)}}<S(x)<\frac{1}{x^2 +\frac{1}{6}},其中\displaystyle \zeta(3)=\sum\limits_{n=1}^{\infty}\frac{1}{n^3}.


练习题8.给定序列\{a_n\},且a_n满足a_1=2,a_2=8,a_n=4a_{n-1}-a_{n-2}(n=3,4,\ldots),证明:\sum\limits_{n=1}^{\infty}\text{arccot}\,\,a_n^2=\frac{\pi}{12}.


证.由{a_n} + {a_{n - 2}} = 4{a_{n - 1}}可知{a_n}\left( {{a_n} + {a_{n - 2}}} \right) = 4{a_{n - 1}}{a_n} = {a_{n - 1}}\left( {{a_{n + 1}} + {a_{n - 1}}} \right),递推得a_n^2 - {a_{n + 1}}{a_{n - 1}} = a_{n - 1}^2 - {a_n}{a_{n - 2}} = \cdots = a_2^2 - {a_3}{a_1} = 4.

 

注意到\mathrm{arccot\,} x的一个公式

\mathrm{arccot\,} x-\mathrm{arccot\,} y=\mathrm{arccot\,}\left( \frac{1+xy}{y-x}\right).

因此有

\begin{align*}\mathrm{arccot\,} a_n^2 &= \mathrm{arccot\,} \frac{{{a_n} \cdot 4{a_n}}}{4} = \mathrm{arccot\,} \frac{{{a_n}\left( {{a_{n + 1}} + {a_{n - 1}}} \right)}}{4} =\mathrm{arccot\,} \frac{{{a_n}\left( {{a_{n + 1}} + {a_{n - 1}}} \right)}}{{a_n^2 - {a_{n + 1}}{a_{n - 1}}}}\\& = \mathrm{arccot\,} \frac{{1 + \frac{{{a_{n + 1}}}}{{{a_{n - 1}}}}}}{{\frac{{{a_n}}}{{{a_{n - 1}}}} - \frac{{{a_{n + 1}}}}{{{a_n}}}}} = \mathrm{arccot\,} \frac{{{a_{n + 1}}}}{{{a_n}}} -\mathrm{arccot\,} \frac{{{a_n}}}{{{a_{n - 1}}}}.\end{align*}

易得\mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}}}}{{{a_n}}} = 2 + \sqrt 3 .

\sum\limits_{n = 1}^\infty {\mathrm{arccot\,} a_n^2} = \mathrm{arccot\,} a_1^2 + \sum\limits_{n = 2}^\infty {\mathrm{arccot\,} a_n^2} = \mathop {\lim }\limits_{n \to \infty } \mathrm{arccot\,} \frac{{{a_{n + 1}}}}{{{a_n}}} - \mathrm{arccot\,} \frac{{{a_2}}}{{{a_1}}} + \mathrm{arccot\,} a_1^2 = \frac{\pi }{{12}}.


练习题9.设\displaystyle a_n=\arctan \frac{1}{n^2 +n +1},证明: \sum\limits_{k=1}^{\infty}\frac{a_k^{1/2}}{k^2} \le \sqrt{\frac{\pi}{3}}.


证.注意到

\begin{align*}\sum\limits_{k = 1}^\infty  {{a_k}}  &= \sum\limits_{k = 1}^\infty  {\arctan \frac{1}{{{k^2} + k + 1}}}  = \sum\limits_{k = 1}^\infty  {\left( {\arctan \frac{1}{k} - \arctan \frac{1}{{k + 1}}} \right)}  = \frac{\pi }{4}\\\sum\limits_{k = 1}^\infty  {\frac{1}{{{k^4}}}}  &= \zeta \left( 4 \right) = \frac{{{\pi ^4}}}{{90}}.\end{align*}

由Cauchy-Schwarz不等式可知

\sum\limits_{k = 1}^N {\frac{1}{{{k^4}}}}  \cdot \sum\limits_{k = 1}^N {{a_k}}  \ge {\left( {\sum\limits_{k = 1}^N {\frac{{a_k^{1/2}}}{{{k^2}}}} } \right)^2}.

N\to\infty,我们有\sum\limits_{k = 1}^\infty  {\frac{{a_k^{1/2}}}{{{k^2}}}}  \le \sqrt {\frac{{{\pi ^4}}}{{90}} \cdot \frac{\pi }{4}}  = \sqrt {\frac{{{\pi ^5}}}{{360}}}  < \sqrt {\frac{\pi }{3}} .

也可通过放缩实现\sum\limits_{k = 1}^\infty  {\frac{1}{{{k^4}}}}  = 1 + \sum\limits_{k = 2}^\infty  {\frac{1}{{{k^4}}}}  < 1 + \sum\limits_{k = 2}^\infty  {\frac{1}{{\left( {k + 1} \right)\left( {k + 2} \right)}}}  = \frac{4}{3}.


练习题10.设\displaystyle a_n > 0, S_n=\sum\limits_{k=1}^na_k,证明:

(1)\displaystyle\sum\limits_{n=1}^{\infty}\frac{n}{S_n} \le 2 \sum\limits_{n=1}^{\infty}\frac{1}{a_n};
(2)\displaystyle\sum\limits_{n=1}^{\infty}\frac{2n+1}{S_n} \le 4 \sum\limits_{n=1}^{\infty}\frac{1}{a_n}.

证.(1)由柯西不等式我们得

\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} \sum\limits_{m = 1}^n {{a_m}}  \ge {\left( {1 + 2 +  \cdots  + n} \right)^2} = \frac{1}{4}{n^2}{\left( {n + 1} \right)^2},

\frac{n}{{{a_1} + {a_2} +  \cdots  + {a_n}}} \le \frac{4}{{n{{\left( {n + 1} \right)}^2}}}\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} .

因此

\begin{align*}\sum\limits_{n = 1}^\infty  {\frac{n}{{{a_1} + {a_2} +  \cdots  + {a_n}}}}  &\le 4\sum\limits_{n = 1}^\infty  {\frac{1}{{n{{\left( {n + 1} \right)}^2}}}\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} }  = 4\sum\limits_{m = 1}^\infty  {\frac{{{m^2}}}{{{a_m}}}\sum\limits_{n = m}^\infty  {\frac{1}{{n{{\left( {n + 1} \right)}^2}}}} } \\&\le 4\sum\limits_{m = 1}^\infty  {\frac{{{m^2}}}{{{a_m}}}\sum\limits_{n = m}^\infty  {\frac{1}{2}\left[ {\frac{1}{{{n^2}}} - \frac{1}{{{{\left( {n + 1} \right)}^2}}}} \right]} }  = 2\sum\limits_{m = 1}^\infty  {\frac{1}{{{a_m}}}} .\end{align*}

这里用到了\frac{1}{{n{{\left( {n + 1} \right)}^2}}} \le \frac{1}{2}\frac{{2n + 1}}{{{n^2}{{\left( {n + 1} \right)}^2}}} = \frac{1}{2}\left[ {\frac{1}{{{n^2}}} - \frac{1}{{{{\left( {n + 1} \right)}^2}}}} \right].

 

注意到a_n=n^\alpha,\alpha>1时有

\mathop {\lim }\limits_{\alpha \to 1} \frac{{\sum\limits_{n = 1}^\infty {\frac{n}{{{a_1} + {a_2} + \cdots + {a_n}}}} }}{{\sum\limits_{j = 1}^\infty {\frac{1}{{{a_j}}}} }} = \mathop {\lim }\limits_{\alpha \to 1} \mathop {\lim }\limits_{N \to \infty } \frac{{\sum\limits_{n = 1}^N {\frac{n}{{{1^\alpha } + {2^\alpha } + \cdots + {n^\alpha }}}} }}{{\sum\limits_{n = 1}^N {\frac{1}{{{n^\alpha }}}} }} = \mathop {\lim }\limits_{\alpha \to 1} \mathop {\lim }\limits_{N \to \infty } \frac{{\frac{N}{{\frac{1}{{\alpha + 1}}{N^{\alpha + 1}} + O\left( {{N^\alpha }} \right)}}}}{{\frac{1}{{{N^\alpha }}}}} = 2.

(2)如法炮制.由柯西不等式我们得

\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} \sum\limits_{m = 1}^n {{a_m}} \ge {\left( {1 + 2 + \cdots + n} \right)^2} = \frac{1}{4}{n^2}{\left( {n + 1} \right)^2},

\frac{{2n + 1}}{{{a_1} + {a_2} + \cdots + {a_n}}} \le \frac{{4\left( {2n + 1} \right)}}{{{n^2}{{\left( {n + 1} \right)}^2}}}\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} .

因此

\begin{align*}\sum\limits_{n = 1}^\infty {\frac{{2n + 1}}{{{a_1} + {a_2} + \cdots + {a_n}}}} &\le 4\sum\limits_{n = 1}^\infty {\frac{{2n + 1}}{{{n^2}{{\left( {n + 1} \right)}^2}}}\sum\limits_{m = 1}^n {\frac{{{m^2}}}{{{a_m}}}} } = 4\sum\limits_{m = 1}^\infty {\frac{{{m^2}}}{{{a_m}}}\sum\limits_{n = m}^\infty {\frac{{2n + 1}}{{{n^2}{{\left( {n + 1} \right)}^2}}}} } \\&= 4\sum\limits_{m = 1}^\infty {\frac{{{m^2}}}{{{a_m}}}\sum\limits_{n = m}^\infty {\left[ {\frac{1}{{{n^2}}} - \frac{1}{{{{\left( {n + 1} \right)}^2}}}} \right]} } = 4\sum\limits_{m = 1}^\infty {\frac{1}{{{a_m}}}}.\end{align*}


练习题11.设\displaystyle a_n \ge 0, n=1,2,\ldots,\sum\limits_{n=1}^{\infty}a_n < \infty,证明:

\sum\limits_{n=1}^{\infty}(a_1a_2\cdots a_n)^{\frac{1}{n}} \le e \sum\limits_{n=1}^{\infty}a_n,且证明e是最优值.

此题再拓展下求证:\sum\limits_{n=1}^{\infty}(a_1a_2\cdots a_n)^{\frac{1}{n}} \le e \sum\limits_{n=1}^{\infty}[1-\frac{1}{2(n+1)}]a_n.

 


练习题12.如果正项级数\displaystyle \sum\limits_{n=1}^{\infty}\frac{1}{p_n}收敛,证明:级数\displaystyle \sum\limits_{n=1}^{\infty}\frac{n^2}{(p_1+p_2+\cdots+p_n)^2}p_n也收敛.


练习题13.设\displaystyle \sum\limits_{n=1}^{\infty}a_n为正项级数,且\displaystyle \sum\limits_{k=1}^{n}(a_k-a_n)n有界,a_n单调递减趋于0,证明:级数\displaystyle \sum\limits_{n=1}^{\infty}a_n收敛.


练习题14.设级数\displaystyle \sum\limits_{n=1}^{\infty}a_n收敛, \displaystyle \sum\limits_{n=1}^{\infty}(b_{n+1}-b_n)绝对收敛,证明:级数\displaystyle \sum\limits_{n=1}^{\infty}a_nb_n收敛.


练习题15.设a_n>0,\left\{ a_n-a_{n+1}\right\}为一个严格递减的数列.如果\sum_{n=1}^{\infty}a_n收敛。试证:\lim\limits_{n \to \infty}\left( \cfrac{1}{a_{n+1}}-\cfrac{1}{a_n}\right)=+\infty.


练习题16.能否构造一个收敛数列\sum\limits_{n=1}^{\infty}a_n,使得级数\sum\limits_{n=1}^{\infty}a_n^3发散.


练习题17.设\lim \limits_{n\rightarrow +\infty}x_n=+\infty,正项级数\sum\limits_{n=1}^{\infty}y_n收敛,设n_0是某一自然数,

若当n>n_0时有x_n <x_{n+1},x_n< \frac{1}{2}(x_{n-1}+x_{n+1}),y_{n+1}< y_n,
求证:\lim \limits_{n\rightarrow +\infty}\frac{x_ny_n}{x_{n+1}-x_n}=0.

练习题18.设\sum\limits_{n=1}^{\infty}a_n是一正项收敛级数,且有a_{n+1}< \frac{1}{2}(a_n+a_{n+2}),\,\frac{1}{a_{n+2}}-\frac{1}{a_{n+1}}\le \frac{1}{3}(\frac{1}{a_{n+3}}-\frac{1}{a_{n}}),

求极限\lim\limits_{n \rightarrow +\infty}\frac{\displaystyle a_na_{n+2}(a_n-a_{n+1})}{\displaystyle a_na_{n+1}-2a_na_{n+2}+a_{n+1}a_{n+2}}.

裴礼文上的一道积分不等式

证明:对n\geq 3\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin{(2n+1)t}}{\sin{t}}\right|dt<\pi\left(1+\frac{\ln{n}}{2}\right).


Poof.For all x,

\left|\frac{\sin((2n+1)x)}{\sin(x)}\right|=\left|\sum_{k=-n}^{n}e^{i2kx}\right|\le2n+1
Note that \displaystyle\sum_{k=-n}^{n}e^{i2kx}=1+2\sum_{k=1}^n\cos(2kx).
 
For 0\le x\le\pi/2, we have \sin(x)\ge2x/\pi. Therefore,
\left|\frac{\sin((2n+1)x)}{\sin(x)}\right|\le\frac\pi{2x}
Thus,
\begin{align*}\int_0^{\pi/2}\left|\frac{\sin((2n+1)x)}{\sin(x)}\right|\,\mathrm{d}x&\le\int_0^{\pi/(4n+2)}(2n+1)\,\mathrm{d}x+\int_{\pi/(4n+2)}^{\pi/2}\frac\pi{2x}\,\mathrm{d}x\\&=\frac\pi2+\frac\pi2\log\left(2n+1\right)\end{align*}
For n\ge3, 2n+1\le\frac73n. Therefore,
\begin{align*}\frac\pi2+\frac\pi2\log\left(2n+1\right)&\le\frac\pi2\left(1+\log\left(\frac73\right)+\log(n)\right)\\[6pt]&\le\pi\left(1+\frac{\log(n)}{2}\right).\end{align*}

Show that for p>1 and x \ge 0,\dfrac{2}{\pi}\int_{x}^{px}\left(\dfrac{\sin{t}}{t}\right)^2\,\mathrm dt\le 1-\dfrac{1}{p}

Proof.This is quite a difficult problem, and I found it very enjoyable.  Here is the solution I found:  

First, we give some simple bounds when x is large, or px is small.  If x\geq\frac{2}{\pi}, then by using the bound |\sin(t)|\leq1,
we have that
\frac{2}{\pi}\int_{x}^{px}\left(\frac{\sin(t)}{t}\right)^{2}dt\leq\frac{2}{\pi}\int_{x}^{px}\frac{1}{t^{2}}dt=\frac{2}{\pi x}\left(1-\frac{1}{p}\right)\leq1-\frac{1}{p}.
Similarly, if px\leq\frac{\pi}{2}, then since \frac{\text{sin}(t)}{t}\leq1,
it follows that 
\frac{2}{\pi}\int_{x}^{px}\left(\frac{\sin(t)}{t}\right)^{2}dt\leq\frac{2}{\pi}\left(px-x\right)=\frac{2xp}{\pi}\left(1-\frac{1}{p}\right)\leq\left(1-\frac{1}{p}\right).
Now, assume that 0\leq x\leq\frac{2}{\pi}, and that px\geq\frac{\pi}{2}.
Then notice that 
\frac{2}{\pi}\int_{x}^{px}\left(\frac{\sin(t)}{t}\right)^{2}dt=1-\frac{2}{\pi}\int_{px}^{\infty}\left(\frac{\sin(t)}{t}\right)^{2}dt-\frac{2}{\pi}\int_{0}^{x}\left(\frac{\sin(t)}{t}\right)^{2}dt
since \int_{0}^{\infty}\left(\frac{\sin(t)}{t}\right)^{2}dt=\frac{\pi}{2}.
We will now find a bound on the other two terms. Working over an interval
of length \pi, by pulling out a lower bound for \frac{1}{t^{2}},
we have that for any y 
 
\int_{y}^{y+\pi}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\frac{1}{\left(y+\pi\right)^{2}}\int_{0}^{\pi}\sin^{2}(t)dt\geq\frac{\pi}{2}\int_{y+\pi}^{y+2\pi}\frac{1}{t^{2}}dt,
and so 
\frac{2}{\pi}\int_{px}^{\infty}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\int_{px+\pi}^{\infty}\frac{1}{t^{2}}dt=\frac{1}{px+\pi}.
Since the function \frac{\sin(t)}{t} is monotonically decreasing
on the interval \left[0,\frac{2}{\pi}\right], it follows that for
x\leq\frac{2}{\pi} we have 
 
 
\frac{1}{x}\int_{0}^{x}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\frac{\pi}{2}\int_{0}^{\frac{2}{\pi}}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\frac{\pi}{2}\cdot\frac{5}{3\pi},
 
and hence 
 
\frac{2}{\pi}\int_{0}^{x}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\frac{5x}{3\pi}. 
 
Now, notice that since px\geq\frac{\pi}{2}, and p>1, by plugging them in directly, we have that 
 
\frac{5\left(xp\right)^{2}}{3\pi}+\frac{2}{3}px+p-\pi>\frac{5\pi}{12}+\frac{\pi}{3}+1-\pi=1-\frac{\pi}{4}>0. 
 
Rearranging the above by dividing through by both (px+\pi)  and p, we obtain the inequality
 
\frac{5}{3\pi}x+\frac{1}{px+\pi}>\frac{1}{p}, 
for px\geq\frac{\pi}{2}, and p>1. It then follows that
\frac{2}{\pi}\int_{px}^{\infty}\left(\frac{\sin(t)}{t}\right)^{2}dt+\frac{2}{\pi}\int_{0}^{x}\left(\frac{\sin(t)}{t}\right)^{2}dt\geq\frac{1}{p},
 for x\leq\frac{2}{\pi}, and px\geq\frac{\pi}{2}, and hence
we have shown that for all x\geq0, and all p>1
\int_{x}^{px}\left(\frac{\sin(t)}{t}\right)^{2}dt\leq1-\frac{1}{p},
as desired. 

Show that if f\in \mathcal C^{n+1}([a,b]) and f(a)=f^{'}(a)=\cdots=f^\left(n\right)(a)=0, then the following statements are ture:
 
\mathbf a)
 
\forall r\in[1,\infty),the inequality \left(\int_{a}^{b}|f(x)|^rdx\right)^{\frac{1}{r}} \leq \frac{(b-a)^{n+\frac{1}{r}}}{n!(nr+1)^{\frac{1}{r}}}\int_{a}^{b}|f^{(n+1)}(x)|dxholds.
 
\mathbf b)
 
\forall r\in[1,\infty),the inequality \left(\int_{a}^{b}|f(x)|^rdx\right)^{\frac{1}{r}} \leq \frac{2^{\frac{1}{r}}(b-a)^{n+\frac{1}{r}+\frac{1}{2}}}{n!\sqrt{2n+1}(2nr+r+1)^{\frac{1}{r}}}\left(\int_{a}^{b}|f^{(n+1)}(x)|^{2}dx\right)^{\frac{1}{2}}holds.

Proof.We have by Taylor's Theorem with Integral form of the Remainder

\begin{align*}f(x) = \int_a^x\dfrac{f^{(n+1)}(t)}{n!}(x-t)^ndt\end{align*}
 
Then we have
\begin{align*}\int_a^b |f(x)|^rdx &= \int_a^b \left|\int_a^x\dfrac{f^{(n+1)}(t)}{n!}(x-t)^ndt\right|^rdx \\&\leq \int_a^b \left(\int_a^x \left|\dfrac{f^{(n+1)}(t)}{n!}\right| \left|(x-t)^n \right|dt\right)^rdx \\&\leq \int_a^b \left(\int_a^x \left|\dfrac{f^{(n+1)}(t)}{n!}\right|dt (x-a)^n\right)^rdx \\&\leq \left(\int_a^b \left|\dfrac{f^{(n+1)}(t)}{n!}\right|dt\right)^r\int_a^b \left( (x-a)^n\right)^rdx \\& = \left(\int_a^b \left|\dfrac{f^{(n+1)}(t)}{n!}\right|dt\right)^r\frac{(b-a)^{nr+1}}{nr+1} .\end{align*}
 
So we get
\begin{align*}\left(\int_a^b |f(x)|^rdx\right)^{1/r} \leq \left(\frac{(b-a)^{nr+1}}{nr+1}\right)^{1/r} \int_a^b \left|\dfrac{f^{(n+1)}(x)}{n!}\right|dx\end{align*}
which is \mathbf a)
 
To get \mathbf b) we can proceed similarly using Holder's inequality
 
\begin{align*}\int_a^b |f(x)|^rdx &= \int_a^b \left|\int_a^x\dfrac{f^{(n+1)}(t)}{n!}(x-t)^ndt\right|^rdx \\&\leq \int_a^b \left(\int_a^x \left|\dfrac{f^{(n+1)}(t)}{n!}\right|^2dt \int_a^x \left|(x-t)^{2n} \right|dt\right)^{r/2} dx \\&\leq \left(\int_a^b \left|\dfrac{f^{(n+1)}(t)}{n!}\right|^2dt\right)^{r/2} \int_a^b  \left(\int_a^x \left|(x-t)^{2n} \right|dt\right)^{r/2} dx \\&= \left(\int_a^b \left|\dfrac{f^{(n+1)}(t)}{n!}\right|^2dt\right)^{r/2} \int_a^b  \left(\frac{(x-a)^{2n+1}}{2n+1}\right)^{r/2} dx\\& = \left(\int_a^b \left|\dfrac{f^{(n+1)}(t)}{n!}\right|^2dt\right)^{r/2} \left(\frac{1}{2n+1}\right)^{r/2} \left(\frac{(b-a)^{nr+\frac{r}{2} +1}}{nr+\frac{r}{2} +1}\right)\end{align*}

Let f be a twice continuously differentiable function from [0,1] into R,Give that
f(0)+2f(\frac{1}{2})+f(1)=0
show that
\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2

Proof.1) Let g_1(x)=x(x-1/2), g_2(x)=(x-1)(x-1/2). By two integration by parts, we have

 
\int_0^{1/2}f^{\prime\prime}(x)g_1(x)dx=-\frac{f(1/2)+f(0)}{2}+2\int_0^{1/2}f(x)dx
and
\int_{1/2}^{1}f^{\prime\prime}(x)g_2(x)dx=-\frac{f(1/2)+f(1)}{2}+2\int_0^{1/2}f(x)dx
Hence
\int_0^{1/2}f^{\prime\prime}(x)g_1(x)dx+\int_{1/2}^{1}f^{\prime\prime}(x)g_2(x)dx=2\int_0^1f(x)dx
 
2) By Cauchy-Schwarz:
 
(\int_{0}^{1/2}f^{\prime\prime}(x)g_1(x)dx)^2\leq (\int_{0}^{1/2}f^{\prime\prime}(x)^2dx)\frac{1}{15.2^6}
 
(\int_{1/2}^{1}f^{\prime\prime}(x)g_2(x)dx)^2\leq (\int_{1/2}^{1}f^{\prime\prime}(x)^2dx)\frac{1}{15.2^6}
 
3) Use now \sqrt{U}+\sqrt{V}\leq \sqrt{2}\sqrt{U+V} with \displaystyle U=\int_{0}^{1/2}f^{\prime\prime}(x)^2dx and \displaystyle V=\int_{1/2}^{1}f^{\prime\prime}(x)^2dx to finish the proof. 

Let f be a positive-valued,concave function on [0,1],Prove that
6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx.

Proof.Let c=\int_0^1 f(x)\,dx and g=f/c, so \int_0^1 g(x)\,dx=1. Then by Holder's inequality,

1\le \left(\int_0^1 g(x)^3\,dx\right)^{1/3}\left(\int_0^1 1^{3/2}\,dx\right)^{2/3} .
Therefore \int_0^1 f(x)^3\,dx=c^3\int_0^1g(x)^3\,dx\ge c^3, and 
8\int_0^1 f(x)^3\,dx + 1-6\left(\int_0^1 f(x)\,dx\right)^2\ge 8c^3+1-6c^2 =: h(c).
For c>0 the right-hand side is minimized when 0=h'(c)=24c^2-12c, meaning c=1/2 (noting h'(c)<0 for c<1/2 and h'(c)>0 for c>1/2). Thus h(1/2)=8(1/2)^3+1-6(1/2)^2=1/2\le h(c) for all c>0.
Actually, then it follows  
6\left(\int_0^1 f(x)\,dx\right)^2 \le \frac12 + 8\int_0^1f(x)^3\,dx.
Concavity of f is not needed.

链接:http://math.stackexchange.com/questions/763253/how-prove-this-integral-inequality-6-left-int-01fxdx-right2-le-1-8-i?rq=1

三角多项式不等式

逻辑丁的提问:证明\sum\limits_{k = 1}^{+\infty} {\frac{{\sin kx}}{{{k^a}}}}  > 0,x \in \left( {0,\pi } \right),a \in \left( {0,\frac{1}{2}} \right]证明在(0,\pi)上勒贝格可积.

一个很好的函数

一个白衣书生出游,于湖光山色之中寻得一寺,寺中有一老僧。二人相谈甚欢,便携手入院。老僧见书生谈吐不凡,遂生考较之意。见院内瓜果藤蔓,老僧出上联曰:”一阶石桌两个闲人三尺小院,四顾春色静看五月石榴。”书生羽扇轻摇,蛋定一笑:”这有何难:五维空间四次齐次三角函数,二重积分必然一致连续。

哆塔微博上告知了一个很好的实函数y = x\left( {\sqrt {\cos \left( {2\pi x} \right) - 1}  + 1} \right) + 0 \cdot \ln x.

图象是(1,1),(2,2),\cdots,(n,n),\cdots这些离散的点.
 

谢惠民上册的一道不等式题

往事如烟!


 

谢上P9的一道不等式题,以前写过,但文件丢失,先前的解答难以回忆起,现在重新给出解答。


用向前-向后数学归纳法证明:设\displaystyle 0<x_i\leq \frac12,i=1,2,\cdots,n,则

\frac{{\prod\limits_{i = 1}^n {{x_i}} }}{{{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^n}}} \le \frac{{\prod\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} }}{{{{\left[ {\sum\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} } \right]}^n}}}.

(这个不等式是由在美国数学界有重大影响的华裔数学家樊畿(Fan Ky)得到的,关于它的许多研究和推广见[30].)


首先,

\frac{{{{\left[ {\sum\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} } \right]}^n}}}{{{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^n}}} \le \frac{{\prod\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} }}{{\prod\limits_{i = 1}^n {{x_i}} }} \Leftrightarrow {\left[ {\frac{n}{{\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right]^n} \le \prod\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}} - 1} \right)} .

 

n=2时,即证

\left( {\frac{1}{{{x_1}}} - 1} \right)\left( {\frac{1}{{{x_2}}} - 1} \right) \ge {\left[ {\frac{2}{{{x_1} + {x_2}}} - 1} \right]^2}.

展开得

\frac{1}{{{x_1}{x_2}}} - \frac{1}{{{x_1}}} - \frac{1}{{{x_2}}} + 1 \ge \frac{4}{{{{\left( {{x_1} + {x_2}} \right)}^2}}} - \frac{4}{{{x_1} + {x_2}}} + 1.

等价于证明

\frac{1}{{{x_1}{x_2}}} - \frac{4}{{{{\left( {{x_1} + {x_2}} \right)}^2}}} \ge \frac{1}{{{x_1}}} + \frac{1}{{{x_2}}} - \frac{4}{{{x_1} + {x_2}}} \Leftrightarrow \frac{{{{\left( {{x_1} - {x_2}} \right)}^2}}}{{{{\left( {{x_1} + {x_2}} \right)}^2}{x_1}{x_2}}} \ge \frac{{{{\left( {{x_1} - {x_2}} \right)}^2}}}{{\left( {{x_1} + {x_2}} \right){x_1}{x_2}}}.

注意到x_1+x_2\leq 1,上式显然成立.

 

n=2的已知情况出发,可以得到如下n=4时的情形:

\begin{align*}&\prod\limits_{i = 1}^4 {\left( {\frac{1}{{{x_i}}} - 1} \right)} = \prod\limits_{i = 1}^2 {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \prod\limits_{i = 3}^4 {\left( {\frac{1}{{{x_i}}} - 1} \right)} \ge {\left[ {\frac{2}{{\sum\limits_{i = 1}^2 {{x_i}} }} - 1} \right]^2}{\left[ {\frac{2}{{\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right]^2}\\= &{\left[ {\left( {\frac{1}{{\frac{1}{2}\sum\limits_{i = 1}^2 {{x_i}} }} - 1} \right)\left( {\frac{1}{{\frac{1}{2}\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right)} \right]^2} \le {\left[ {{{\left( {\frac{2}{{\frac{1}{2}\sum\limits_{i = 1}^2 {{x_i}} + \frac{1}{2}\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right)}^2}} \right]^2} = {\left[ {\frac{4}{{\sum\limits_{i = 1}^4 {{x_i}} }} - 1} \right]^4}.\end{align*}

同样可知,若n=2^k时不等式已成立,则可得到n=2^{k+1}时的不等式

\begin{align*}&\prod\limits_{i = 1}^{{2^{k + 1}}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} = \prod\limits_{i = 1}^{{2^k}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \prod\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \ge {\left[ {\frac{{{2^k}}}{{\sum\limits_{i = 1}^{{2^k}} {{x_i}} }} - 1} \right]^{{2^k}}}{\left[ {\frac{{{2^k}}}{{\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right]^{{2^k}}}\\= & {\left[ {\left( {\frac{1}{{\frac{1}{{{2^k}}}\sum\limits_{i = 1}^{{2^k}} {{x_i}} }} - 1} \right)\left( {\frac{1}{{\frac{1}{{{2^k}}}\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right)} \right]^{{2^k}}} \le {\left[ {{{\left( {\frac{2}{{\frac{1}{{{2^k}}}\sum\limits_{i = 1}^{{2^k}} {{x_i}} + \frac{1}{{{2^k}}}\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right)}^2}} \right]^{{2^k}}} = {\left[ {\frac{{{2^{k + 1}}}}{{\sum\limits_{i = 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right]^{{2^{k + 1}}}}.\end{align*}

这样就证明了当n2的所有方幂时平均值不等式已成立.这是“向前”部分.

 

第二步要证明,当平均值不等式对某个n>2成立时,则它对n-1也一定成立.这是证明中的“向后”部分.写出

\begin{align*}&{\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{n - 1}} = {\left[ {\frac{n}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} + \frac{1}{{n - 1}}\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^n} \cdot {\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{ - 1}}\\\le &\prod\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \left( {\frac{1}{{\frac{1}{{n - 1}}\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right) \cdot {\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{ - 1}} = \prod\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{{x_i}}} - 1} \right)} .\end{align*}

于是n-1时不等式也成立.合并以上向前和向后两部分,可见不等式对每个自然数n成立.


事实上,我们有

{\left[ {\frac{n}{{\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right]^n} \le \prod\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}} - 1} \right)}  \Leftrightarrow \frac{1}{n}\sum\limits_{i = 1}^n {\ln \left( {\frac{1}{{{x_i}}} - 1} \right)}  \ge \ln \left( {\frac{1}{{\frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right).结合函数y = \ln \left( {\frac{1}{x} - 1} \right)的凹凸性便可得证.