On a conjecture of S.P. Robinson

Abstract

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