On a conjecture of S.P. Robinson

In his thesis, S.P. Robinson made a conjecture concerning the polynomials Pnβ(z)=1+∑k=1n∏j=0k−1n−jβ+n+jzk,n∈N,β⩾1, namely that zPnβ 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.