Vol. 44, No. 2, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Extensions of inequalities of the Laguerre and Turán type

Merrell Lee Patrick

Vol. 44 (1973), No. 2, 675–682
Abstract

It is shown that

2∑k (− 1)jtk ( 2k )
-(2k)!-  j    F(n+j)(z)F(n+2k−j)(z) ≧ 0,
j=0

for −∞ < z < ,n 1 and k 0, where F(z) is an entire function of a special type. For k = 1 this simply is the well known Laguerre inequality

   (n+1)   2    (n)    (n+2)
(F     (z)) − F   (z)F     (z) ≧ 0

−∞ < z < ,n 0. From these inequalities we obtain the inequalities

∑2k (− 1)j+k (    )
-------   2k   un+j(x )un+2k−j(x) ≧ 0
j=0  (2k )!    j

which hold for such values of x, for which the functions un = un(x) have a generating function of the type

∑∞     n
un z--= F(z).
n=0   n!

Mathematical Subject Classification
Primary: 33A70
Secondary: 30A66
Milestones
Received: 4 October 1971
Published: 1 February 1973
Authors
Merrell Lee Patrick