Root and sign of a complicated bivariate function

Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let $$\Phi(p,i) := \frac{1}{2^p+1} + \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),$$ where $\lg x$ is the binary logarithm. With the help of a Computer Algebra System, it seems that

• If $0 \leqslant p \leqslant 3$, then $\Phi(p,i) < 0$.

• If $4 \leqslant p$, there exists $i_p$ such that $\Phi(p,i_p) = 0$ and $\Phi(p,i) > 0$ for $1 \leqslant i < i_p$, and $\Phi(p,i) < 0$ for $i_p < i \leqslant 2^p$.

How can I prove this?

Just in case, the partial derivative with respect to $i$ is: $$\frac{\partial\Phi}{\partial i}(p,i) = \frac{1}{i(2^p+i)\ln 2} - \frac{2}{(i+1)^3},$$ where $\ln x$ is the natural logarithm.

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