A sequence of functions \(F_0(z),F_1(z),\dots,F_n(z)\) analytic in an open disc centered at \(c\) is called a \(\Delta(n,c)\) basis if there are \(n+1\) polynomials \(R_0(x),R_1(x),\dots,R_n(x)\) such that (i) the derivatives satisfy \(F_i^{(j)}(c)=R_j(i)\), \(i,j=0,1,\ldots,n\), and (ii) the polynomials \(R_i(x)\) form a basis of \(\mathcal P^n\). One of the main results of this paper states the following: if \(p(x)=\sum_{i=0}^n a_iF_i(x)\) where \(\{F_i(x)\}\) is a \(\Delta(n,c)\) basis, and if the multiplicity \(\mu\) of the zero \(c\) of \(p\) is at least one, then \[ \sum_{i=0}^n | a_i| ^2 \geq \frac{(n-\mu)!(n+\mu)!}{n!^2}| a_0| ^2 \geq e^{2\mu^2/(2n+1)}| a_0| ^2. \] Other inequalities of similar flavor are proved: they all more or less follow from a general theorem involving orthogonal families of polynomials, and specializing this theorem to some given family.