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

### 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

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

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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

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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

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

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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

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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.