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On a conjecture for trigonometric sums and starlike functions. (English) Zbl 1135.42001
Summary: We pose and discuss the following conjecture: let $s_n^\mu(z):= \sum_{k=0}^n \frac{(\mu)_k}{k!} z^k$, and for $\rho\in(0,1]$ let $\mu^*(\rho)$ be the unique solution $\mu\in(0,1]$ of $$\int_0^{(\rho+1)\pi} \sin(t-\rho\pi) t^{\mu-1}\,dt=0.$$ Then for $0<\mu\le \mu^*(\rho)$ and $n\in\Bbb N$ we have $|\arg[(1-z)^\mu s_n^\mu(z)]|\le \rho\pi/2$, $|z|<1$. We prove this for $\rho=\frac12$, and in a somewhat weaker form, for $\rho=\frac34$. Far reaching extensions of our conjectures and results to starlike functions of order $1-\mu/2$ are also discussed. Our work is closely related to recent investigations concerning the understanding and generalization of the celebrated Vietoris’ inequalities.

42A05Trigonometric polynomials, inequalities, extremal problems
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
26D05Inequalities for trigonometric functions and polynomials
26D15Inequalities for sums, series and integrals of real functions
30C45Special classes of univalent and multivalent functions
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
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