Abstract/Details

EXTREMAL PROBLEMS FOR CONSTRAINED POLYNOMIALS AND UNIVALENT FUNCTIONS (APPROXIMATION THEORY, COMPLEX VARIABLES, JACOBI POLYNOMIALS, CHEBYSHEV)

LACHANCE, MICHAEL ANTHONY. 
 University of South Florida ProQuest Dissertations Publishing,  1979. 8428366.

Abstract (summary)

We generalize Lorentz's notion of incomplete polynomials and study the following extremal problems: (I) For (alpha), (beta) (GREATERTHEQ) 0 and any integer m (GREATERTHEQ) 0, determine

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

(II) For integers s, m (GREATERTHEQ) 0, determine

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

Problem (I) may be viewed as an L(,(INFIN)) version of the L(,2) problem solved by the classical Jacobi polynomials and also gives rise to constrained analogs of the classical Chebyshev polynomials. Problem (II) is a generalization of problems posed independently by Halasz and by Blatt. These authors wished to determine e(,l,m), m (GREATERTHEQ) 0; Rahman and Stenger, and Rahman and Schmeisser obtained estimates from above and below on

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

For Problem (I) and for the more general cases of Problem (II), we examine in detail the corresponding extremal functions for fixed choices for ((alpha),(beta),m) and (s,m). Furthermore, we study the limiting behavior of these functions as (alpha) + (beta) + m and s + m tend, respectively, to infinity. Upper and lower bounds for (epsilon)(,m)('((alpha),(beta))) and e(,s,m) are also given.

In addition, we determine upper bounds on the growth in modulus of constrained polynomials of the form

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

with respect to the real interval {-1,1}, and of polynomials with a prescribed zero on the unit circle, of the form

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

These bounds appear to improve some known estimates and are further shown to be best possible in a limiting sense.

Finally, we discount an attractive conjecture for the class of univalent functions all of whose derivatives are univalent. That is, if E denotes the set of functions

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

where f('(n))(z) is univalent in (VBAR)z(VBAR) < 1, for each n (GREATERTHEQ) 0, and if (alpha) := sup{(VBAR)a(,2)(VBAR): f (epsilon) E}, then Shah and Trimble proved that (alpha) (GREATERTHEQ) (pi)/2. We show (alpha) > (pi)/2 + .02, and thus that (e('(pi)z)-1)/(pi) is not extremal in E.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
EXTREMAL PROBLEMS FOR CONSTRAINED POLYNOMIALS AND UNIVALENT FUNCTIONS (APPROXIMATION THEORY, COMPLEX VARIABLES, JACOBI POLYNOMIALS, CHEBYSHEV)
Author
LACHANCE, MICHAEL ANTHONY
Number of pages
125
Degree date
1979
School code
0206
Source
DAI-B 45/09, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-204-17600-3
University/institution
University of South Florida
University location
United States -- Florida
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
8428366
ProQuest document ID
302954509
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
https://www.proquest.com/docview/302954509