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| On the Springer's Conjecture of the Coefficients of Univalent Meromorphic Functions |
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Citation： |
Yao Biyun.On the Springer's Conjecture of the Coefficients of Univalent Meromorphic Functions[J].Chinese Annals of Mathematics B,1982,3(1):85~88 |
| Page view： 816
Net amount： 802 |
Authors： |
Yao Biyun; |
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| Abstract： |
Let $\sigma$ denote the family of univalent functions
$\[F(z) = z + \sum\limits_{n = 1}^\infty {\frac{{{b_n}}}{{{z^n}}}} \]$
in l< |z| <\infty if G(w) is the inverse of a function $F(z) \in \sigma ^'$, the expansion of G(w) in some neighborhood of w=\infty is
$\[G(w) = w - \sum\limits_{n = 1}^\infty {\frac{{{B_n}}}{{{w^n}}}} \]$
It is well known that |B_1|\leq 1 for any F(z) \in \sigma ^'. Springer^[1] proved that | B_3| \leq 1 and conjectured that
$\[|{B_{2n - 1}}| \le \frac{{(2n - 2)!}}{{n!(n - 1)!}}{\rm{ }}(n = 3,4, \cdots )\]$ (1)
Kubota^[2] proved (1) for n=3, 4, 5. Schober^[3] proved (1) for n = 6, 7. Ren Fuyao[4,5] has verified (1) for n=6, 7, 8. In this article we are going to verify (1) for n=9. |
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