کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6417706 | 1339305 | 2015 | 12 صفحه PDF | دانلود رایگان |
The squared modulus of every real-valued on R function f from the Laguerre-Pólya class L-P obeys a MacLaurin-type series representation|f(x+iy)|2=âk=0âLk(f;x)y2k,x,yâR. If f is a polynomial with only real roots, then the sum becomes finite. The coefficients {Lk} are representable as non-linear differential operators acting on f, and by a classical result of Jensen Lk(f;x)â¥0 for fâL-P and xâR. A conjecture of M. Patrick from 1971 states that for f=Pn(α,β), the n-th Jacobi polynomial, with αâ¥Î²>â1, the functions Lk(f;x), 1â¤kâ¤nâ1, attain their maxima in [0,1] at x=1. The aim of this paper is to validate Patrick's conjecture. Moreover, we prove a refined version of this conjecture, showing that {Lk(f;x)}k=1nâ1 are strictly monotonically increasing functions on the positive semi-axis. Towards our proof of Patrick's conjecture we extend the Sonin-Pólya majorization approach to all coefficient functions {Lk(f;x)}k=0n, f=Pn(α,β).
Journal: Journal of Mathematical Analysis and Applications - Volume 428, Issue 2, 15 August 2015, Pages 750-761