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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.

##### MSC:
 42A05 Trigonometric polynomials, inequalities, extremal problems 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 26D05 Inequalities for trigonometric functions and polynomials 26D15 Inequalities for sums, series and integrals of real functions 30C45 Special classes of univalent and multivalent functions 33C45 Orthogonal polynomials and functions of hypergeometric type
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##### References:
 [1] Andrews, G. E.; Askey, R.; Roy, R.: Special functions. (1999) · Zbl 0920.33001 [2] Askey, R.; Steinig, J.: Some positive trigonometric sums. Trans. amer. Math. soc. 187, No. 1, 295-307 (1974) · Zbl 0244.42002 [3] Brown, G.; Hewitt, E.: A class of positive trigonometric sums. Math. ann. 268, 91-122 (1984) · Zbl 0522.42001 [4] Koumandos, S.: An extension of vietoris’s inequalities. The Ramanujan J. 14, No. 1, 1-38 (2007) · Zbl 1138.42002 [5] S. Koumandos, Monotonicity of some functions involving the gamma and psi function, submitted for publication. · Zbl 1210.33002 [6] Koumandos, S.; Ruscheweyh, S.: Positive Gegenbauer polynomial sums and applications to starlike functions. Constr. approx. 23, No. 2, 197-210 (2006) · Zbl 1099.30006 [7] Pommerenke, C.: Univalent functions. (1975) · Zbl 0298.30014 [8] Ruscheweyh, S.: Linear operators between classes of prestarlike functions. Comment. math. Helv. 52, 497-509 (1977) · Zbl 0372.30007 [9] S. Ruscheweyh, Convolutions in geometric function theory, Seminaire de Mathematiques Superievres, vol. 83, Les Presses de l’Université de Montréal, 1982. · Zbl 0499.30001 [10] Ruscheweyh, S.; Salinas, L.: On starlike functions of order $\lambda \in$[12,1). Ann. univ. Mariae Curie-sklodowska 54, 117-123 (2000) · Zbl 0989.30010 [11] Ruscheweyh, S.; Salinas, L.: Stable functions and vietoris’ theorem. J. math. Anal. appl. 291, 596-604 (2004) · Zbl 1052.30011 [12] Vietoris, L.: Über das vorzeichen gewisser trigonometrischer summen. S.-B. Österreich. akad. Wiss. 167, 125-135 (1958) · Zbl 0088.27402 [13] Zygmund, A.: Trigonometric series. (2002) · Zbl 1084.42003