Common holomorphic neighborhood for a parameter family

A family $f_\lambda$ has a common bounded holomorphic neighborhood of a compact real interval $I$ if there exist an open $\Omega\subset\mathbb C$ , a radius $r>0$ , and $M<\infty$ such that every closed radius- $r$ disc centered on $I$ lies in $\Omega$ , every $f_\lambda$ is holomorphic there, and $\sup_{\lambda,z\in\Omega}|f_\lambda(z)|\le M$ .

Consequence and failure of uniformity

Cauchy estimates give $\sup_{\lambda,x\in I}|f_\lambda^{(m)}(x)|\le M m!r^{-m}$ at every order. Individual real analyticity gives no such common radius: $f_n(z)=1/(1+n^2z^2)$ has poles at $\pm i/n$ , approaching the real interval containing zero. A bound only on the real interval also cannot replace a bound on the complex domain.