Bounds for higher order divided differences of analytic functions on a disk (Q662644)

For a function \(f\) analytic on the disk \(E_R= \{|z|< R\}\), the order \(n\) divided difference is defined by the recurrence \[ [f(z); z_0]= f(z_0), \] \[ [f(z); z_0, z_1]= (f(z_0)- f(z_1))/(z_0- z_1) \] and \[ [f(z); z_0,z_1,\dots, z_n]= ([f(z); z_0,\dots, z_{n-1}]- [f(z); z_1,\dots, z_n])/(z_0- z_n), \] with appropriate modifications if any of the \(z_i\) coincide. The authors show how bounds for the absolute value and for the real part of the order \(n\) divided difference of \(f\) may be obtained from bounds for the absolute value of the order \(n\) derivative \(f^{(n)}(z)\) on \(E_R\). They also use these bounds to obtain bounds for the remainder term in Newton's approximation formula. Interestingly, in many of their results equality occurs when the interpolation points lie on a ray (rather than a circle).