Doubt on Lebesgue integration

Question

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

Answer 1

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.