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Hi Micromass. Thank you for your help with https://www.physicsforums.com/showthread.php?t=451462micromass said:We know that
[tex]\int_0^1{f^n}\leq (\sup f)^n[/tex].
So we need to show that
[tex](\sup f)^n\leq \int_0^1{g^n}[/tex]
Since sup(f)<sup(g), there exists a neighbourhood ]a,b[ such that
[tex]\forall x\in ]a,b[:~\sup(f)<g(x)[/tex]
Now we can use
[tex]\int_0^1{g_n}\geq \int_a^b{g_n}\geq (b-a)\inf_{x\in ]a,b[}{g^n(x)}[/tex].
so you must prove now that there exists an n such that
[tex](\sup f)^n<(b-a)\inf_{x\in ]a,b[}{g^n(x)}[/tex]
mathdunce said: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[tex]^{mn+c}[/tex]. I tried the mean value theorem of integral without success. Could you please give me another hint? Thank you!
In math analysis, "sup" stands for supremum, which is the least upper bound of a set. In this equation, it refers to the supremum of the function f.
The symbol "∫" is the integral sign, which represents the integration of a function over a given interval. In this equation, it is used to find the integral of both f^n and g^n.
The notation "f^g" is read as "f raised to the power of g". This means that the function f is being raised to the power of the function g.
Yes, for example, if f(x) = x^2 and g(x) = 2x, then the equation would be written as: sup(x^2) < sup(2x) => ∫(x^2)^n < ∫(2x)^n.
This equation is known as the Holder's inequality and it tells us that if f and g are two non-negative functions on a given interval, with f^n and g^n being integrable, then the supremum of f must be less than or equal to the supremum of g in order for the integral of f^n to be less than or equal to the integral of g^n. In other words, this inequality helps us compare the behavior of two functions on a given interval.