Smale’s Mean Value Conjecture and Complex Dynamics

The domains where Smale's mean value problem does not hold

“…Ng [8] showed that for a large class of polynomials, including all odd polynomials with a zero constant term, one can replace 4 with 2. Miles-Leighton and Pilgrim [9] presented the results of a version of Smale's mean value problem. Aimo et al [10] presented the results of the dual Smale's mean value problem.…”

Smale’s mean value conjecture for finite Blaschke products

“…The conjecture was repeated in [31,28] and it is also listed as one of the three minor problems in Smale's famous problem list [32]. The conjecture is now known as Smale's mean value conjecture which has remained open since 1981 even though it was proven to be true for many classes of polynomials (see [5], [26], [14], [15], [17], [25], [27], [33], [34] and [37]).…”