Show $int_0^a f(x),dx + int_0^b f^{-1}(x),dx ge ab$ for strictly increasing function $f(x)$
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Let $f: [0, infty) to [0, infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$int_0^a f(x),dx + int_0^b f^{-1}(x),dx ge ab$$ for any $a, b > 0$, and give a condition for equality to hold.
calculus real-analysis integration inverse
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Let $f: [0, infty) to [0, infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$int_0^a f(x),dx + int_0^b f^{-1}(x),dx ge ab$$ for any $a, b > 0$, and give a condition for equality to hold.
calculus real-analysis integration inverse
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2
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Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
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– user217285
Sep 5 '15 at 20:59
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math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14
add a comment |
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Let $f: [0, infty) to [0, infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$int_0^a f(x),dx + int_0^b f^{-1}(x),dx ge ab$$ for any $a, b > 0$, and give a condition for equality to hold.
calculus real-analysis integration inverse
$endgroup$
Let $f: [0, infty) to [0, infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$int_0^a f(x),dx + int_0^b f^{-1}(x),dx ge ab$$ for any $a, b > 0$, and give a condition for equality to hold.
calculus real-analysis integration inverse
calculus real-analysis integration inverse
edited Sep 5 '15 at 21:21
Peter Woolfitt
18.6k54480
18.6k54480
asked Sep 5 '15 at 20:43
user268284
2
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Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
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– user217285
Sep 5 '15 at 20:59
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math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14
add a comment |
2
$begingroup$
Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
$endgroup$
– user217285
Sep 5 '15 at 20:59
$begingroup$
math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14
2
2
$begingroup$
Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
$endgroup$
– user217285
Sep 5 '15 at 20:59
$begingroup$
Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
$endgroup$
– user217285
Sep 5 '15 at 20:59
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math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14
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math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14
add a comment |
3 Answers
3
active
oldest
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Note that
$$int_0^cf(x),dx+int_0^{f(c)}f^{-1}(x),dx=cf(c)$$
You can think of this as the area of the rectangle with $(0,0)$ for one corner and $(c,f(c))$ for the opposite corner.
Without loss of generality, let $f(a)le b$. Then
$$int_0^af(x),dx+int_0^bf^{-1}(x),dx=af(a)+int_{f(a)}^bf^{-1}(x),dx$$
Now since $f(x)$ is strictly increasing, we have $f^{-1}(x)>a$ for $x>f(a)$. Hence
$$int_{f(a)}^bf^{-1}(x),dxge aleft(b-f(a)right)$$
with equality if and only if $b=f(a)$.
Putting this together, we get
$$int_0^af(x),dx+int_0^bf^{-1}(x),dxge af(a)+a(b-f(a))=ab$$
with equality if and only if $b=f(a)$.
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Well done! +1....
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– Mark Viola
Sep 5 '15 at 22:12
add a comment |
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As @Peter Woolfitt provide the solution, Here is my view
Set $f(x)=y,x=varphi(y)$
$varphi(f(x))=x$
Set $displaystyle F(t)=int_0^af(x)dx+int_0^tvarphi(y)dy-at$
take the derivative of this with respect to $F(t)$
we have $F'(t)=varphi(t)-a$
and by the conditions $varphi(0)=0$, $varphi(t)$ monotonic increasing.
Set $F'(t)=0$, we got $varphi(t_0)=a$, and $t_0=varphi^{-1}(a)=f(a)$
When $t in (0,t_0)$, $F(t)$ monotonic decreasing, $t in (t_0,+infty)$, $F(t)$ monotonic increasing
$$
F(t_0)=int_0^af(x)dx+int_0^{t_0}varphi(y)dy-at_0\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{f(a)}varphi(y)dy-af(a)\
$$
change of variable
$$
F(t_0)=int_0^af(x)dx+int_0^{a}varphi(f(x))d(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{a}xd(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+xf(x)Big|_0^a-int_0^{a}f(x)d(x)-af(a)=0\
$$
Finally, we got $F(t)geqslant 0$ when $t_0=f(a)=b$, $a=varphi(b)$, the equation is satisfied
$$
int_0^af(x)dx+int_0^bvarphi(y)dy=ab
$$
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add a comment |
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We remark that the integrals are well-defined by Theorem 6.9 of Rudin's Principles of Mathematical Analysis.
Theorem 6.9 (Rudin). If $f$ is monotonic on $[a, b]$, and if $alpha$ is continuous on $[a, b]$, then $f in mathscr{R}(alpha)$. (We still assume, of course, that $alpha$ is monotonic.)
By definition, there exists a sequence of partitions $(P_n)$ of $[0, a]$ such that$$U(P_n, f) to int_0^a f(x),dx$$and sequence of partitions $(Q_n)$ of $[0, b]$ such that$$L(Q_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Let $P_n' = P_n cap f^{-1}(Q_n)$, $tilde{P}_n = P_n' cap [0, a]$, $Q_n' = Q_n cap f(P_n)$, and $tilde{Q}_n = Q_n' cap [0, b]$. Then $U(tilde{P}_n, f) le U(P_n, f)$, $L(tilde{Q}_n, f^{-1}) ge L(Q_n, f^{-1})$ and so we have$$U(tilde{P}_n, f) to int_0^a f(x),dx,text{ }L(tilde{Q}_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Replace $P_n$ by $tilde{P}_n$ and $Q_n$ with $tilde{Q}_n$. Because $f$ is strictly increasing,$$U(P_n, f) = sum_{i=1}^p f(x_i)(x_i - x_{i-1}),$$where $P_n = {x_0, dots, x_p}$, and similarly,$$L(Q_n, f^{-1}) = sum_{j=1}^q f^{-1}(y_{j-1})(y_j - y_{j-1}),$$where $Q_n = {y_0, dots, y_q}$. Also,$$ab = left(sum_{i=1}^p (x_i - x_{i-1})right)left( sum_{j=1}^q(y_j - y_{j-1})right).$$Let $P_n' = {x_0, dots, x_p, dots, x_{p'}}$ and $Q_n' = {y_0, dots, y_q, dots, y_{q'}}$. Realizing that, we can write$$U(P_n, f) = sum_{i=1}^p sum_{y_j le f(x_i)} (y_j - y_{j-1})(x_i - x_{i-1}) = sum_{i=1}^p sum_{y_j le f(x_i)} (x_i - x_{i-1})(y_j - y_{j-1})$$and$$L(Q_n, f^{-1}) = sum_{j=1}^q sum_{x_i le f^{-1}(y_{j-1})}(x_i - x_{i-1})(y_j - y_{j-1}) = sum_{j=1}^q sum_{f(x_i) < y_j} (x_i - x_{i-1})(y_j - y_{j-1}),$$we obtain $ab le U(P_n, f) + L(Q_n, f^{-1})$ since the left-hand side involves summing over fewer terms. Taking limits gives the required identity.
When $f(a) = b$, we obtain equality because the sums are equal. To see the converse, we use an argument like the one in Exercise 6.2 of Rudin's Principles of Mathematical Analysis.
Exercise 6.2 (Rudin). Suppose $f ge 0$, $f$ is continuous on $[a, b]$, and $int_a^b f(x),dx = 0$. Then $f(x) = 0$ for all $x in [a, b]$. (See here for a solution.)
If $f(a) > b$, find a neighborhood $N$ of which this is true; one of the sums in the above argument will consist of more terms and the extra contribution will not converge to zero as this contribution is giving $int_N (f(x) - b),dx > 0$. Similarly, if $f(a) < b$.
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$begingroup$
Note that
$$int_0^cf(x),dx+int_0^{f(c)}f^{-1}(x),dx=cf(c)$$
You can think of this as the area of the rectangle with $(0,0)$ for one corner and $(c,f(c))$ for the opposite corner.
Without loss of generality, let $f(a)le b$. Then
$$int_0^af(x),dx+int_0^bf^{-1}(x),dx=af(a)+int_{f(a)}^bf^{-1}(x),dx$$
Now since $f(x)$ is strictly increasing, we have $f^{-1}(x)>a$ for $x>f(a)$. Hence
$$int_{f(a)}^bf^{-1}(x),dxge aleft(b-f(a)right)$$
with equality if and only if $b=f(a)$.
Putting this together, we get
$$int_0^af(x),dx+int_0^bf^{-1}(x),dxge af(a)+a(b-f(a))=ab$$
with equality if and only if $b=f(a)$.
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Well done! +1....
$endgroup$
– Mark Viola
Sep 5 '15 at 22:12
add a comment |
$begingroup$
Note that
$$int_0^cf(x),dx+int_0^{f(c)}f^{-1}(x),dx=cf(c)$$
You can think of this as the area of the rectangle with $(0,0)$ for one corner and $(c,f(c))$ for the opposite corner.
Without loss of generality, let $f(a)le b$. Then
$$int_0^af(x),dx+int_0^bf^{-1}(x),dx=af(a)+int_{f(a)}^bf^{-1}(x),dx$$
Now since $f(x)$ is strictly increasing, we have $f^{-1}(x)>a$ for $x>f(a)$. Hence
$$int_{f(a)}^bf^{-1}(x),dxge aleft(b-f(a)right)$$
with equality if and only if $b=f(a)$.
Putting this together, we get
$$int_0^af(x),dx+int_0^bf^{-1}(x),dxge af(a)+a(b-f(a))=ab$$
with equality if and only if $b=f(a)$.
$endgroup$
$begingroup$
Well done! +1....
$endgroup$
– Mark Viola
Sep 5 '15 at 22:12
add a comment |
$begingroup$
Note that
$$int_0^cf(x),dx+int_0^{f(c)}f^{-1}(x),dx=cf(c)$$
You can think of this as the area of the rectangle with $(0,0)$ for one corner and $(c,f(c))$ for the opposite corner.
Without loss of generality, let $f(a)le b$. Then
$$int_0^af(x),dx+int_0^bf^{-1}(x),dx=af(a)+int_{f(a)}^bf^{-1}(x),dx$$
Now since $f(x)$ is strictly increasing, we have $f^{-1}(x)>a$ for $x>f(a)$. Hence
$$int_{f(a)}^bf^{-1}(x),dxge aleft(b-f(a)right)$$
with equality if and only if $b=f(a)$.
Putting this together, we get
$$int_0^af(x),dx+int_0^bf^{-1}(x),dxge af(a)+a(b-f(a))=ab$$
with equality if and only if $b=f(a)$.
$endgroup$
Note that
$$int_0^cf(x),dx+int_0^{f(c)}f^{-1}(x),dx=cf(c)$$
You can think of this as the area of the rectangle with $(0,0)$ for one corner and $(c,f(c))$ for the opposite corner.
Without loss of generality, let $f(a)le b$. Then
$$int_0^af(x),dx+int_0^bf^{-1}(x),dx=af(a)+int_{f(a)}^bf^{-1}(x),dx$$
Now since $f(x)$ is strictly increasing, we have $f^{-1}(x)>a$ for $x>f(a)$. Hence
$$int_{f(a)}^bf^{-1}(x),dxge aleft(b-f(a)right)$$
with equality if and only if $b=f(a)$.
Putting this together, we get
$$int_0^af(x),dx+int_0^bf^{-1}(x),dxge af(a)+a(b-f(a))=ab$$
with equality if and only if $b=f(a)$.
edited Sep 5 '15 at 22:12
answered Sep 5 '15 at 21:20
Peter WoolfittPeter Woolfitt
18.6k54480
18.6k54480
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Well done! +1....
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– Mark Viola
Sep 5 '15 at 22:12
add a comment |
$begingroup$
Well done! +1....
$endgroup$
– Mark Viola
Sep 5 '15 at 22:12
$begingroup$
Well done! +1....
$endgroup$
– Mark Viola
Sep 5 '15 at 22:12
$begingroup$
Well done! +1....
$endgroup$
– Mark Viola
Sep 5 '15 at 22:12
add a comment |
$begingroup$
As @Peter Woolfitt provide the solution, Here is my view
Set $f(x)=y,x=varphi(y)$
$varphi(f(x))=x$
Set $displaystyle F(t)=int_0^af(x)dx+int_0^tvarphi(y)dy-at$
take the derivative of this with respect to $F(t)$
we have $F'(t)=varphi(t)-a$
and by the conditions $varphi(0)=0$, $varphi(t)$ monotonic increasing.
Set $F'(t)=0$, we got $varphi(t_0)=a$, and $t_0=varphi^{-1}(a)=f(a)$
When $t in (0,t_0)$, $F(t)$ monotonic decreasing, $t in (t_0,+infty)$, $F(t)$ monotonic increasing
$$
F(t_0)=int_0^af(x)dx+int_0^{t_0}varphi(y)dy-at_0\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{f(a)}varphi(y)dy-af(a)\
$$
change of variable
$$
F(t_0)=int_0^af(x)dx+int_0^{a}varphi(f(x))d(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{a}xd(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+xf(x)Big|_0^a-int_0^{a}f(x)d(x)-af(a)=0\
$$
Finally, we got $F(t)geqslant 0$ when $t_0=f(a)=b$, $a=varphi(b)$, the equation is satisfied
$$
int_0^af(x)dx+int_0^bvarphi(y)dy=ab
$$
$endgroup$
add a comment |
$begingroup$
As @Peter Woolfitt provide the solution, Here is my view
Set $f(x)=y,x=varphi(y)$
$varphi(f(x))=x$
Set $displaystyle F(t)=int_0^af(x)dx+int_0^tvarphi(y)dy-at$
take the derivative of this with respect to $F(t)$
we have $F'(t)=varphi(t)-a$
and by the conditions $varphi(0)=0$, $varphi(t)$ monotonic increasing.
Set $F'(t)=0$, we got $varphi(t_0)=a$, and $t_0=varphi^{-1}(a)=f(a)$
When $t in (0,t_0)$, $F(t)$ monotonic decreasing, $t in (t_0,+infty)$, $F(t)$ monotonic increasing
$$
F(t_0)=int_0^af(x)dx+int_0^{t_0}varphi(y)dy-at_0\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{f(a)}varphi(y)dy-af(a)\
$$
change of variable
$$
F(t_0)=int_0^af(x)dx+int_0^{a}varphi(f(x))d(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{a}xd(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+xf(x)Big|_0^a-int_0^{a}f(x)d(x)-af(a)=0\
$$
Finally, we got $F(t)geqslant 0$ when $t_0=f(a)=b$, $a=varphi(b)$, the equation is satisfied
$$
int_0^af(x)dx+int_0^bvarphi(y)dy=ab
$$
$endgroup$
add a comment |
$begingroup$
As @Peter Woolfitt provide the solution, Here is my view
Set $f(x)=y,x=varphi(y)$
$varphi(f(x))=x$
Set $displaystyle F(t)=int_0^af(x)dx+int_0^tvarphi(y)dy-at$
take the derivative of this with respect to $F(t)$
we have $F'(t)=varphi(t)-a$
and by the conditions $varphi(0)=0$, $varphi(t)$ monotonic increasing.
Set $F'(t)=0$, we got $varphi(t_0)=a$, and $t_0=varphi^{-1}(a)=f(a)$
When $t in (0,t_0)$, $F(t)$ monotonic decreasing, $t in (t_0,+infty)$, $F(t)$ monotonic increasing
$$
F(t_0)=int_0^af(x)dx+int_0^{t_0}varphi(y)dy-at_0\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{f(a)}varphi(y)dy-af(a)\
$$
change of variable
$$
F(t_0)=int_0^af(x)dx+int_0^{a}varphi(f(x))d(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{a}xd(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+xf(x)Big|_0^a-int_0^{a}f(x)d(x)-af(a)=0\
$$
Finally, we got $F(t)geqslant 0$ when $t_0=f(a)=b$, $a=varphi(b)$, the equation is satisfied
$$
int_0^af(x)dx+int_0^bvarphi(y)dy=ab
$$
$endgroup$
As @Peter Woolfitt provide the solution, Here is my view
Set $f(x)=y,x=varphi(y)$
$varphi(f(x))=x$
Set $displaystyle F(t)=int_0^af(x)dx+int_0^tvarphi(y)dy-at$
take the derivative of this with respect to $F(t)$
we have $F'(t)=varphi(t)-a$
and by the conditions $varphi(0)=0$, $varphi(t)$ monotonic increasing.
Set $F'(t)=0$, we got $varphi(t_0)=a$, and $t_0=varphi^{-1}(a)=f(a)$
When $t in (0,t_0)$, $F(t)$ monotonic decreasing, $t in (t_0,+infty)$, $F(t)$ monotonic increasing
$$
F(t_0)=int_0^af(x)dx+int_0^{t_0}varphi(y)dy-at_0\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{f(a)}varphi(y)dy-af(a)\
$$
change of variable
$$
F(t_0)=int_0^af(x)dx+int_0^{a}varphi(f(x))d(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+int_0^{a}xd(f(x))-af(a)\
$$
$$
F(t_0)=int_0^af(x)dx+xf(x)Big|_0^a-int_0^{a}f(x)d(x)-af(a)=0\
$$
Finally, we got $F(t)geqslant 0$ when $t_0=f(a)=b$, $a=varphi(b)$, the equation is satisfied
$$
int_0^af(x)dx+int_0^bvarphi(y)dy=ab
$$
answered Dec 14 '18 at 0:45
Jack throneJack throne
184
184
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add a comment |
$begingroup$
We remark that the integrals are well-defined by Theorem 6.9 of Rudin's Principles of Mathematical Analysis.
Theorem 6.9 (Rudin). If $f$ is monotonic on $[a, b]$, and if $alpha$ is continuous on $[a, b]$, then $f in mathscr{R}(alpha)$. (We still assume, of course, that $alpha$ is monotonic.)
By definition, there exists a sequence of partitions $(P_n)$ of $[0, a]$ such that$$U(P_n, f) to int_0^a f(x),dx$$and sequence of partitions $(Q_n)$ of $[0, b]$ such that$$L(Q_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Let $P_n' = P_n cap f^{-1}(Q_n)$, $tilde{P}_n = P_n' cap [0, a]$, $Q_n' = Q_n cap f(P_n)$, and $tilde{Q}_n = Q_n' cap [0, b]$. Then $U(tilde{P}_n, f) le U(P_n, f)$, $L(tilde{Q}_n, f^{-1}) ge L(Q_n, f^{-1})$ and so we have$$U(tilde{P}_n, f) to int_0^a f(x),dx,text{ }L(tilde{Q}_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Replace $P_n$ by $tilde{P}_n$ and $Q_n$ with $tilde{Q}_n$. Because $f$ is strictly increasing,$$U(P_n, f) = sum_{i=1}^p f(x_i)(x_i - x_{i-1}),$$where $P_n = {x_0, dots, x_p}$, and similarly,$$L(Q_n, f^{-1}) = sum_{j=1}^q f^{-1}(y_{j-1})(y_j - y_{j-1}),$$where $Q_n = {y_0, dots, y_q}$. Also,$$ab = left(sum_{i=1}^p (x_i - x_{i-1})right)left( sum_{j=1}^q(y_j - y_{j-1})right).$$Let $P_n' = {x_0, dots, x_p, dots, x_{p'}}$ and $Q_n' = {y_0, dots, y_q, dots, y_{q'}}$. Realizing that, we can write$$U(P_n, f) = sum_{i=1}^p sum_{y_j le f(x_i)} (y_j - y_{j-1})(x_i - x_{i-1}) = sum_{i=1}^p sum_{y_j le f(x_i)} (x_i - x_{i-1})(y_j - y_{j-1})$$and$$L(Q_n, f^{-1}) = sum_{j=1}^q sum_{x_i le f^{-1}(y_{j-1})}(x_i - x_{i-1})(y_j - y_{j-1}) = sum_{j=1}^q sum_{f(x_i) < y_j} (x_i - x_{i-1})(y_j - y_{j-1}),$$we obtain $ab le U(P_n, f) + L(Q_n, f^{-1})$ since the left-hand side involves summing over fewer terms. Taking limits gives the required identity.
When $f(a) = b$, we obtain equality because the sums are equal. To see the converse, we use an argument like the one in Exercise 6.2 of Rudin's Principles of Mathematical Analysis.
Exercise 6.2 (Rudin). Suppose $f ge 0$, $f$ is continuous on $[a, b]$, and $int_a^b f(x),dx = 0$. Then $f(x) = 0$ for all $x in [a, b]$. (See here for a solution.)
If $f(a) > b$, find a neighborhood $N$ of which this is true; one of the sums in the above argument will consist of more terms and the extra contribution will not converge to zero as this contribution is giving $int_N (f(x) - b),dx > 0$. Similarly, if $f(a) < b$.
$endgroup$
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$begingroup$
We remark that the integrals are well-defined by Theorem 6.9 of Rudin's Principles of Mathematical Analysis.
Theorem 6.9 (Rudin). If $f$ is monotonic on $[a, b]$, and if $alpha$ is continuous on $[a, b]$, then $f in mathscr{R}(alpha)$. (We still assume, of course, that $alpha$ is monotonic.)
By definition, there exists a sequence of partitions $(P_n)$ of $[0, a]$ such that$$U(P_n, f) to int_0^a f(x),dx$$and sequence of partitions $(Q_n)$ of $[0, b]$ such that$$L(Q_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Let $P_n' = P_n cap f^{-1}(Q_n)$, $tilde{P}_n = P_n' cap [0, a]$, $Q_n' = Q_n cap f(P_n)$, and $tilde{Q}_n = Q_n' cap [0, b]$. Then $U(tilde{P}_n, f) le U(P_n, f)$, $L(tilde{Q}_n, f^{-1}) ge L(Q_n, f^{-1})$ and so we have$$U(tilde{P}_n, f) to int_0^a f(x),dx,text{ }L(tilde{Q}_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Replace $P_n$ by $tilde{P}_n$ and $Q_n$ with $tilde{Q}_n$. Because $f$ is strictly increasing,$$U(P_n, f) = sum_{i=1}^p f(x_i)(x_i - x_{i-1}),$$where $P_n = {x_0, dots, x_p}$, and similarly,$$L(Q_n, f^{-1}) = sum_{j=1}^q f^{-1}(y_{j-1})(y_j - y_{j-1}),$$where $Q_n = {y_0, dots, y_q}$. Also,$$ab = left(sum_{i=1}^p (x_i - x_{i-1})right)left( sum_{j=1}^q(y_j - y_{j-1})right).$$Let $P_n' = {x_0, dots, x_p, dots, x_{p'}}$ and $Q_n' = {y_0, dots, y_q, dots, y_{q'}}$. Realizing that, we can write$$U(P_n, f) = sum_{i=1}^p sum_{y_j le f(x_i)} (y_j - y_{j-1})(x_i - x_{i-1}) = sum_{i=1}^p sum_{y_j le f(x_i)} (x_i - x_{i-1})(y_j - y_{j-1})$$and$$L(Q_n, f^{-1}) = sum_{j=1}^q sum_{x_i le f^{-1}(y_{j-1})}(x_i - x_{i-1})(y_j - y_{j-1}) = sum_{j=1}^q sum_{f(x_i) < y_j} (x_i - x_{i-1})(y_j - y_{j-1}),$$we obtain $ab le U(P_n, f) + L(Q_n, f^{-1})$ since the left-hand side involves summing over fewer terms. Taking limits gives the required identity.
When $f(a) = b$, we obtain equality because the sums are equal. To see the converse, we use an argument like the one in Exercise 6.2 of Rudin's Principles of Mathematical Analysis.
Exercise 6.2 (Rudin). Suppose $f ge 0$, $f$ is continuous on $[a, b]$, and $int_a^b f(x),dx = 0$. Then $f(x) = 0$ for all $x in [a, b]$. (See here for a solution.)
If $f(a) > b$, find a neighborhood $N$ of which this is true; one of the sums in the above argument will consist of more terms and the extra contribution will not converge to zero as this contribution is giving $int_N (f(x) - b),dx > 0$. Similarly, if $f(a) < b$.
$endgroup$
add a comment |
$begingroup$
We remark that the integrals are well-defined by Theorem 6.9 of Rudin's Principles of Mathematical Analysis.
Theorem 6.9 (Rudin). If $f$ is monotonic on $[a, b]$, and if $alpha$ is continuous on $[a, b]$, then $f in mathscr{R}(alpha)$. (We still assume, of course, that $alpha$ is monotonic.)
By definition, there exists a sequence of partitions $(P_n)$ of $[0, a]$ such that$$U(P_n, f) to int_0^a f(x),dx$$and sequence of partitions $(Q_n)$ of $[0, b]$ such that$$L(Q_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Let $P_n' = P_n cap f^{-1}(Q_n)$, $tilde{P}_n = P_n' cap [0, a]$, $Q_n' = Q_n cap f(P_n)$, and $tilde{Q}_n = Q_n' cap [0, b]$. Then $U(tilde{P}_n, f) le U(P_n, f)$, $L(tilde{Q}_n, f^{-1}) ge L(Q_n, f^{-1})$ and so we have$$U(tilde{P}_n, f) to int_0^a f(x),dx,text{ }L(tilde{Q}_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Replace $P_n$ by $tilde{P}_n$ and $Q_n$ with $tilde{Q}_n$. Because $f$ is strictly increasing,$$U(P_n, f) = sum_{i=1}^p f(x_i)(x_i - x_{i-1}),$$where $P_n = {x_0, dots, x_p}$, and similarly,$$L(Q_n, f^{-1}) = sum_{j=1}^q f^{-1}(y_{j-1})(y_j - y_{j-1}),$$where $Q_n = {y_0, dots, y_q}$. Also,$$ab = left(sum_{i=1}^p (x_i - x_{i-1})right)left( sum_{j=1}^q(y_j - y_{j-1})right).$$Let $P_n' = {x_0, dots, x_p, dots, x_{p'}}$ and $Q_n' = {y_0, dots, y_q, dots, y_{q'}}$. Realizing that, we can write$$U(P_n, f) = sum_{i=1}^p sum_{y_j le f(x_i)} (y_j - y_{j-1})(x_i - x_{i-1}) = sum_{i=1}^p sum_{y_j le f(x_i)} (x_i - x_{i-1})(y_j - y_{j-1})$$and$$L(Q_n, f^{-1}) = sum_{j=1}^q sum_{x_i le f^{-1}(y_{j-1})}(x_i - x_{i-1})(y_j - y_{j-1}) = sum_{j=1}^q sum_{f(x_i) < y_j} (x_i - x_{i-1})(y_j - y_{j-1}),$$we obtain $ab le U(P_n, f) + L(Q_n, f^{-1})$ since the left-hand side involves summing over fewer terms. Taking limits gives the required identity.
When $f(a) = b$, we obtain equality because the sums are equal. To see the converse, we use an argument like the one in Exercise 6.2 of Rudin's Principles of Mathematical Analysis.
Exercise 6.2 (Rudin). Suppose $f ge 0$, $f$ is continuous on $[a, b]$, and $int_a^b f(x),dx = 0$. Then $f(x) = 0$ for all $x in [a, b]$. (See here for a solution.)
If $f(a) > b$, find a neighborhood $N$ of which this is true; one of the sums in the above argument will consist of more terms and the extra contribution will not converge to zero as this contribution is giving $int_N (f(x) - b),dx > 0$. Similarly, if $f(a) < b$.
$endgroup$
We remark that the integrals are well-defined by Theorem 6.9 of Rudin's Principles of Mathematical Analysis.
Theorem 6.9 (Rudin). If $f$ is monotonic on $[a, b]$, and if $alpha$ is continuous on $[a, b]$, then $f in mathscr{R}(alpha)$. (We still assume, of course, that $alpha$ is monotonic.)
By definition, there exists a sequence of partitions $(P_n)$ of $[0, a]$ such that$$U(P_n, f) to int_0^a f(x),dx$$and sequence of partitions $(Q_n)$ of $[0, b]$ such that$$L(Q_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Let $P_n' = P_n cap f^{-1}(Q_n)$, $tilde{P}_n = P_n' cap [0, a]$, $Q_n' = Q_n cap f(P_n)$, and $tilde{Q}_n = Q_n' cap [0, b]$. Then $U(tilde{P}_n, f) le U(P_n, f)$, $L(tilde{Q}_n, f^{-1}) ge L(Q_n, f^{-1})$ and so we have$$U(tilde{P}_n, f) to int_0^a f(x),dx,text{ }L(tilde{Q}_n, f^{-1}) to int_0^b f^{-1}(x),dx.$$Replace $P_n$ by $tilde{P}_n$ and $Q_n$ with $tilde{Q}_n$. Because $f$ is strictly increasing,$$U(P_n, f) = sum_{i=1}^p f(x_i)(x_i - x_{i-1}),$$where $P_n = {x_0, dots, x_p}$, and similarly,$$L(Q_n, f^{-1}) = sum_{j=1}^q f^{-1}(y_{j-1})(y_j - y_{j-1}),$$where $Q_n = {y_0, dots, y_q}$. Also,$$ab = left(sum_{i=1}^p (x_i - x_{i-1})right)left( sum_{j=1}^q(y_j - y_{j-1})right).$$Let $P_n' = {x_0, dots, x_p, dots, x_{p'}}$ and $Q_n' = {y_0, dots, y_q, dots, y_{q'}}$. Realizing that, we can write$$U(P_n, f) = sum_{i=1}^p sum_{y_j le f(x_i)} (y_j - y_{j-1})(x_i - x_{i-1}) = sum_{i=1}^p sum_{y_j le f(x_i)} (x_i - x_{i-1})(y_j - y_{j-1})$$and$$L(Q_n, f^{-1}) = sum_{j=1}^q sum_{x_i le f^{-1}(y_{j-1})}(x_i - x_{i-1})(y_j - y_{j-1}) = sum_{j=1}^q sum_{f(x_i) < y_j} (x_i - x_{i-1})(y_j - y_{j-1}),$$we obtain $ab le U(P_n, f) + L(Q_n, f^{-1})$ since the left-hand side involves summing over fewer terms. Taking limits gives the required identity.
When $f(a) = b$, we obtain equality because the sums are equal. To see the converse, we use an argument like the one in Exercise 6.2 of Rudin's Principles of Mathematical Analysis.
Exercise 6.2 (Rudin). Suppose $f ge 0$, $f$ is continuous on $[a, b]$, and $int_a^b f(x),dx = 0$. Then $f(x) = 0$ for all $x in [a, b]$. (See here for a solution.)
If $f(a) > b$, find a neighborhood $N$ of which this is true; one of the sums in the above argument will consist of more terms and the extra contribution will not converge to zero as this contribution is giving $int_N (f(x) - b),dx > 0$. Similarly, if $f(a) < b$.
edited Sep 6 '15 at 18:10
answered Sep 6 '15 at 4:27
user268197
add a comment |
add a comment |
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Wiki has a nice proof without words of this: en.wikipedia.org/wiki/Young's_inequality#/media/…
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– user217285
Sep 5 '15 at 20:59
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math.stackexchange.com/questions/1132153/…
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– StubbornAtom
Aug 2 '18 at 16:14