On the Kakeya–Eneström Theorem and Gegenbauer Polynomial Sums
Copyright policy )On the Kakeya–Eneström Theorem and Gegenbauer Polynomial Sums
An extension of the classical Kakeya–Enestrom theorem is given. As an application we show that for $\lambda \geqq \frac{1}{2}$, $ - 1 0$, $k = 0,1, \cdots ,n$, we have \[\sum_{k = 0}^n {a_k \frac{{C_k^{(\lambda )} (x)}}{{C_k^{(\lambda )} (1)}}z^k \ne 0,\quad | z | \leqq 1,} \] where $C_k^{(\lambda )} $, are the Gegenbauer or ultraspherical polynomials. This extends an old result due to G. Szego and settles two recent conjectures of R. Askey and J. Bustoz. Other related results are obtained as well.
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), Zeros of Polynomials, Ultraspherical Polynomials, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Polynomials and rational functions of one complex variable, Extreme Points, Kakeya-Enestroem Theorem, Spherical harmonics, Gegenbauer Polynomial, Starlike Functions
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), Zeros of Polynomials, Ultraspherical Polynomials, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Polynomials and rational functions of one complex variable, Extreme Points, Kakeya-Enestroem Theorem, Spherical harmonics, Gegenbauer Polynomial, Starlike Functions
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