Let $P_n$ be the Legendre polynomial of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to prove that $$\sum_{i=1}^{n}\sqrt i |P_i(z)|^2 >> n^{a}$$ for some $a>0$ and for all $z$ with $\Re(z)\in [-1+b, 1-b], |\Im(z)|\le 1/n$, for a given $b>0$. I also need upper bounds on $|P_i(z)|$, $|P'_i(z)|$, and $|P''_i(z)|$ for such $z$ as above. Any suggestion is greatly appreciated.

I know some bounds on the real line, such as the Bernstein and Markov inequalities, which say that $P_i(x)$ is of order $i^{-1/2}$ for $x \in [-1+b, 1-b]$. But I have no clue how to work with $x$ in the complex plane.