Application of Chebyshev polynomials to classes of analytic functions. (Application des polynômes de Chebyshev à des classes de fonctions analytiques.) (English. French summary) Zbl 1317.30030
Summary: Our objective in this paper is to consider some basic properties of the familiar Chebyshev polynomials in the theory of analytic functions. We investigate some basic useful characteristics for a class \(\mathcal H(t)\), \( t\in(1/2,1]\), of functions \(f\), with \(f(0)=0\), \(f'(0)=1\), analytic in the open unit disc \(\mathbb U=\{z:|z|<1\}\) satisfying the condition that
\[ 1+\frac{zf''(z)}{f'(z)}\prec H(z,t)=\frac{1}{1-2tz+z^2}\qquad (z\in\mathbb U), \]
where \(H(z,t)\) is the generating function of the second kind of Chebyshev polynomials. The Fekete-Szegő problem in the class is also solved.
\[ 1+\frac{zf''(z)}{f'(z)}\prec H(z,t)=\frac{1}{1-2tz+z^2}\qquad (z\in\mathbb U), \]
where \(H(z,t)\) is the generating function of the second kind of Chebyshev polynomials. The Fekete-Szegő problem in the class is also solved.
MSC:
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
| 30C10 | Polynomials and rational functions of one complex variable |
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