Questions on proving and manipulating inequalities.

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prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
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Can the following inequality be directly infered?

If we have a condition as follows $$\log(1+\mathbf{h}_2^* \mathbf{\Sigma} \mathbf{h}_2) \leq \log(1+\mathbf{h}_1^* \mathbf{\Sigma} \mathbf{h}_1)$$ where $\Sigma$ is positive semi definite matrix ...
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Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...