# How to get this inequality given another inequality (norms and logarithms)

Suppose I have proved for all $f$ positive with $\lVert f \rVert_{L^2} = 1$ that $$\int f^2\log(f) \leq -C_1\log(\epsilon) + \epsilon\lVert \nabla f \rVert_{L^2}^2.$$ I want to prove that for all $f$ with norm not necessarily $1$ that $$\int f^2\log\left(\frac{|f|^2}{\lVert f \rVert_{L^2}^2}\right) \leq -C_1\log(\epsilon)\lVert f \rVert_{L^2}^2 + \epsilon\lVert \nabla f \rVert_{L^2}^2$$

How do I do this? I Tried taking $f=\frac{f}{\lVert f \rVert_{L^2}}$ but I just can't get the log in left hand side right.

• Use $\log(x^2)=2\log(x)$ Jul 13 '14 at 20:43

We can let $f \mapsto \frac{f}{\lVert f \rVert_{L^2}}$ and as @mathematician suggested, we can multiply by $2 \times \lVert f \rVert_{L^2}$.