Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions |
| |
Authors: | Stamatis Koumandos Stephan Ruscheweyh |
| |
Institution: | (1) Department of Mathematics and Statistics, The University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus;(2) Mathematisches Institut, Universitat Wurzburg, 97074 Wurzburg, Germany |
| |
Abstract: | Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of
$f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order
$\lambda$ iff $\lambda_0\leq\lambda<1$ where
$\lambda_0=0.654222...$ is the unique solution
$\lambda\in(\frac{1}{2},1)$ of the equation
$\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes
the unit disk in the complex plane $\C$. This result is the best
possible with respect to $\lambda_0$. In particular, it
shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we
have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all
$n\in\N,\ x\in-1,1]$, and
$0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements
an inequality of Brown, Wang, and Wilson (1993) and extends a
result of Ruscheweyh and Salinas (2000). |
| |
Keywords: | Positive cosine sums Trigonometric inequalities Gegenbauer polynomials Starlike functions |
本文献已被 SpringerLink 等数据库收录! |
|