[math analysis] sup f< sup g=>∫f^n<∫g^n

[mathdunce]

Homework Statement

Homework Equations

The Attempt at a Solution

 

[micromass]

We know that

$$\int_0^1{f^n}\leq (\sup f)^n$$.

So we need to show that

$$(\sup f)^n\leq \int_0^1{g^n}$$

Since sup(f)<sup(g), there exists a neighbourhood ]a,b[ such that

$$\forall x\in ]a,b[:~\sup(f)<g(x)$$

Now we can use

$$\int_0^1{g_n}\geq \int_a^b{g_n}\geq (b-a)\inf_{x\in ]a,b[}{g^n(x)}$$.

so you must prove now that there exists an n such that

$$(\sup f)^n<(b-a)\inf_{x\in ]a,b[}{g^n(x)}$$

 

[mathdunce]

We know that

$$\int_0^1{f^n}\leq (\sup f)^n$$.

So we need to show that

$$(\sup f)^n\leq \int_0^1{g^n}$$

Since sup(f)<sup(g), there exists a neighbourhood ]a,b[ such that

$$\forall x\in ]a,b[:~\sup(f)<g(x)$$

Now we can use

$$\int_0^1{g_n}\geq \int_a^b{g_n}\geq (b-a)\inf_{x\in ]a,b[}{g^n(x)}$$.

so you must prove now that there exists an n such that

$$(\sup f)^n<(b-a)\inf_{x\in ]a,b[}{g^n(x)}$$

Hi Micromass. Thank you for your help with https://www.physicsforums.com/showthread.php?t=451462
I now know how to solve the first question, but I still do not know know to link them with e$$^{mn+c}$$. I tried the mean value theorem of integral without success. Could you please give me another hint? Thank you!

 

[mathdunce]

Hi Micromass. Thank you for your help with https://www.physicsforums.com/showthread.php?t=451462
I now know how to solve the first question, but I still do not know know to link them with e$$^{mn+c}$$. I tried the mean value theorem of integral without success. Could you please give me another hint? Thank you!

Oh, I think I know how to do the second one, too. Thanks. I have not written it down formally yet.