On a conjecture of S.P. Robinson
In his thesis, S.P. Robinson made a conjecture concerning the polynomials namely that zPβ n is prestarlike of order (3− β)/2. These polynomials are closely related to the de la Vallée Poussin means (the case β= 1). We prove this conjecture in a more general form and show that these functions constitute a sort of two-dimensional subordination chain. These results are then compared with similar ones for Cesáro means of various orders.
On cyclic variation-diminishing transforms
We give a new and more manageable characterization for Cyclic Pólya Frequency functions of order 3 (CPF3). Our result also improves present knowledge concerning smoothness properties in CPF. In particular, a conjecture of Mairhuber, Schoenberg, and Williamson, On variation-diminishing transformations on the circle, Rend. Circ. Mat. Palermo (2) 8 (1959), 1-30, about discontinuous CPF functions is established.
