How to prove that $\int_{\Omega}(a+b|u|^{(p-1)})^{q}\le C\int_{\Omega}(1+|u|^{(p-1)q})$ where $ u\in L^{p}(\Omega)$, and a,b are constants. q is a conjugate exponent of p
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Any bounds on $a, b, p$, or $q$? – marty cohen Jul 31 '13 at 05:25
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@marty-cohen here q is conjugate exponent of p – nanthini Jul 31 '13 at 05:48
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Note: This answer may show how little I know about Lebesgue integration. I am treating this just as an inequality between the integrands and assuming some bounds on the parameters.
This is true if $(1+x^u)^v < C(1+x^{uv}) $, where $x > 0$ and $C$ depends on $u$ and $v$ and $u = [-1$ and $v = q$.
If we let $x^u = y$, this becomes $(1+y)^v < C(1+y^v)$, or $1+y < C(1+y^v)^{1/v}$, where the $C$'s are different and depend on $u$ and $v$.
By the arithmetic means inequality, if $v > 1$, $\dfrac{1+y}{2} < \left(\dfrac{1+y^v}{2}\right)^{1/v} $ or $1+y < 2^{1-1/v}(1+y^v)^{1/v} $
So, if $p>1$ and $q > 1$, this seems to be true.
marty cohen
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I screwed up here. Could you (1) fix $u=[-1$ to $u=p-1$ and (2) change $C$ to some other letter since it's not the same $C$ as in the question? And I think "generalized means inequality" is more common than "arithmetic means inequality". I don't recall ever voting on any of your posts before, by the way. – epimorphic Mar 01 '15 at 17:15
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About the parameters: The fact that $p$ and $q$ are Hölder conjugates ($1/p+1/q = 1$ and $p,q \in [1,\infty]$) and that we presumably don't want to deal with infinite exponents would imply that $p,q > 1$. And I think we would also want $a,b \geq 0$ so that their exponentiation by non-integer powers is nice. – epimorphic Mar 01 '15 at 17:30