  We sharpen the bound $n^{2k}$ on the maximum modulus of the $k^{{\rm th}}$
radial derivative of the Zernike circle polynomials (disk polynomials) of
degree $n$ to $n^2(n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$. This bound
is obtained from a result of Koornwinder on the non-negativity of connection
coefficients of the radial parts of the circle polynomials when expanded into a
series of Chebyshev polynomials of the first kind. The new bound is shown to be
sharp for, for instance, Zernike circle polynomials of degree $n$ and azimuthal
order $m$ when $m=O(\sqrt{n})$ by using an explicit expression for the
connection coefficients in terms of squares of Jacobi polynomials evaluated at
0.
  Keywords: Zernike circle polynomial, disk polynomial, radial derivative,
Chebyshev polynomial, connection coefficient, Gegenbauer polynomial.
