Let $P\_n$ be the [Legendre polynomial](https://en.wikipedia.org/wiki/Legendre_polynomials) 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.



Bounds on Legendre polynomials on the complex plane
Setting $Q\_{n}(z)=(n+1/2)^{1/2}P\_{n}(z)$ for the orthonormalized Legendre polynomial, and 
$$\kappa\_{n}(z)=\sum\_{j=0}^{n}|Q\_{j}(z)|^{2},$$
for the inverse of the Christoffel function, it is known that 
$$\kappa\_{n}^{\frac{1}{2n}}(z)\to\left|z+\sqrt{z^{2}-1}\right|,$$
uniformly on $\mathbb{C}$ as $n\to\infty$, so that $\kappa\_{n}(z)$ grows geometrically fast outside of $[-1,1]$. For $x\in[-1,1]$, a deeper result is needed, which says that
$$\lim\_{n\to\infty}\frac{\kappa\_{n}(x)}{n}=\frac{1}{\pi\sqrt{1-x^{2}}},$$
see, for instance, Chapter V.6. of the book



> 
> G. Freud, Orthogonal Polynomials, Pergamon Press, New York, 1971.
> 
> 
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Hence, for $x\in[-1,1]$,
$$\lim\_{n\to\infty}\frac1n\sum\_{j=0}^{n}\left(j+\frac12\right)|P\_{j}(x)|^{2}=\frac{1}{\pi\sqrt{1-x^{2}}},$$
from which one may derive, together with
[Abel's summation formula](https://en.wikipedia.org/wiki/Abel%27s_summation_formula), that $\sum\_{j=0}^{n}\sqrt{j}|P\_{j}(x)|^{2}$ behaves, for $x\in(-1,1)$ and $n$ large, like $C\_{x}\sqrt{n}$ where $C\_{x}$ is some positive, bounded below, constant depending on $x$.


Hence your estimate in a neighborhood of $[-1,1]$ holds true with the exponent $a=1/2$ which is optimal.


