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Consider $1\le p,q < \infty , t\in \mathbb R , f,g>0$ then is $g(x)^q[f^pg^{-q} (x)]^t$ convex in the following context?

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    There's a mistake with $a(x)$, it should be $a(x)=\ln[f^p(x)g^{-q}(x)]$ instead. I just used the identity $\alpha^\beta=\exp(\beta\ln\alpha)$ provided \alpha>0.2012-07-13

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Take $f^p(x)=g^q(x)=2+\cos x$, then $g^q[f^pg^{-q}]^t$ is not convex.