Given vectors $a,b, \epsilon \in \mathbb{R}^n$, I am trying to get a good upper bound for
$$ \sqrt{\frac{1}{n}\sum_{i=0}^n \big((a_i + \epsilon_i b_i)^p - a_i^p\big)^2} $$
which depends on $\epsilon_G := max_{i=1,\dots,n} \epsilon_i$ and $p \in \mathbb{N}$
So far I have used the binomial formula and the identity for $a^p - b^p$ but the bounds obtained are not tight enough.
This is related to the equation I am working with
$$ E_{TG}(\phi, p) = || \big(\lambda\phi(x) + \epsilon(x)^T \nabla\phi(x^{t})\big)^p - (\lambda \phi(.))^p ||_G $$
where the $||.||_G$ norm is the norm of a function on a discrete grid G of coordinates defined as
$$ ||\phi||_G = \sqrt{\frac{1}{|G|} \sum_{x \in G} (\phi(x))^2} $$
The function $\phi := v^T \psi$ where vector $v \in \mathbb{R}^D$ has $||v|| = 1$, and $\psi: \mathbb{R}^n \xrightarrow{} \mathbb{R}^D$ is a set of basis functions(example, polynomial basis)
$x^t$ is a flow function which is known explicitly
$\epsilon_G = max_{x \in G} ||\epsilon(x)|| $
The upper bound I have obtained is
$$ E_{TG}(\phi, p) \leq C_{TG}(p, \lambda) \epsilon_G $$
where $C_{TG} = \left\lVert ||J_{\psi}(.^t)||_2 \bigg(\sum_{i=0}^{p-1} ||\psi(T^t (.)||^{p-1-i} ||\psi(.)||^i \lambda^i \bigg) \right\rVert _G$
where $J_\psi$ is the Jacobian of the basis functions and $||.||_2$ is the spectral norm.
The problem is when such a system is simulated with data and the Euler method for flow $T^t$, I get a constant which increases exponentially with increasing $p$ but growth $E_{TG}$ with respect to p is small. I would like to get an upper bound which tracks the growth of the error with $p$