An extremal problem for polynomials with nonnegative coefficients

Abstract

Let W n {W_n} be the set of all algebraic polynomials of exact degree n n whose coefficients are all nonnegative. For the norm in L 2 [ 0 , ∞ ) {L^2}[0,\infty ) with generalized Laguerre weight function w ( x ) = x α e − x ( α > − 1 ) w(x) = {x^\alpha }{e^{ - x}}\quad (\alpha > - 1) , the extremal problem C n ( α ) = sup P ∈ W n ( ‖ P ′ ‖ / ‖ P ‖ ) 2 {C_n}(\alpha ) = {\sup _{P \in {W_n}}}{(\left \| {P’} \right \|/\left \| P \right \|)^2} is solved, which completes a result of A. K. Varma [7].