Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]