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.