Vol. 44, No. 2, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
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