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Zbl 1200.42002
Koumandos, Stamatis; Lamprecht, Martin
On a conjecture for trigonometric sums and starlike functions. II.
(English)
[J] J. Approx. Theory 162, No. 5, 1068-1084 (2010). ISSN 0021-9045

For $\mu>0$ let $s_n^\mu(z)=\sum_{k=0}^n (\mu)_k z^k/k!$, where $(\mu)_k=\mu(\mu+1)\dots (\mu+k-1)$. Denote by $\mathcal{A}$ the class of analytic functions in the unit disk $\Bbb{D}=\{z\in\Bbb{C}:|z|<1\}$. For $f,g\in\mathcal{A}$ we write $f\prec g$ ($f$ is subordinate to $g$ in $\Bbb{D}$) if there exists $w\in\mathcal{A}$ with $|w(z)|\leq z$ for $z\in\Bbb{D}$, such that $f=g\circ w$. Also, for $\rho\in (0,1]$ we let $\mu^*(\rho)$ be the unique solution of the integral equation $\int_0^{(\rho+1)\pi}\sin(t-\rho\pi)t^{\mu-1}dt=0$.\par The paper under review is a continuation of the work of {\it S. Koumandos} and {\it S. Ruscheweyh} [J. Approximation Theory 149, No. 1, 42--58 (2007; Zbl 1135.42001)], where its authors guessed that:\par Conjecture. For $\rho\in(0,1]$ the number $\mu^*(\rho)$ is indeed maximal number $\mu(\rho)$ such that for all $n\in\Bbb{N}$ and $\mu\in (0,\mu(\rho)]$ we have $$(1-z)^\rho s_n^\mu(z)\prec \Big(\frac{1+z}{1-z}\Big)^\rho.$$\par In the present paper, the authors give a proof of this conjecture in case $\rho=1/4$. To do this, they prove (based on the properties of completely monotonic functions) a sharp trigonometric inequality. Main result of paper has some applications concerning starlike functions, as more as a corollary, it gives an inequality concerning Gegenbauer polynomials.
[Mehdi Hassani (Zanjan)]
MSC 2000:
*42A05 Trigonometric polynomials
30A10 Inequalities in the complex domain
26D05 Inequalities for trigonometric functions and polynomials
42A32 Trigonometric series of special types
26D20 Analytical inequalities involving real functions

Keywords: positive trigonometric inequality; completely monotonic function; starlike function; Gegenbauer polynomial

Citations: Zbl 1135.42001

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