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Logarithmic coefficients problems in families related to starlike and convex functions | | S. Ponnusamy
; N. L. Sharma
; K.-J. Wirths
; | | Date: |
3 Nov 2018 | | Abstract: | Let $es$ be the family of analytic and univalent functions $f$ in the unit
disk $D$ with the normalization $f(0)=f’(0)-1=0$, and let
$gamma_n(f)=gamma_n$ denote the logarithmic coefficients of $fin {es}$. In
this paper, we study bounds for the logarithmic coefficients for certain
subfamilies of univalent functions. Also, we consider the families $F(c)$ and
$G(delta)$ of functions $fin {es}$ defined by $$ {
m Re} left (
1+frac{zf’’(z)}{f’(z)}
ight )>1-frac{c}{2}, mbox{ and } , {
m Re} left
( 1+frac{zf’’(z)}{f’(z)}
ight )<1+frac{delta}{2},quad zin D $$ for some
$cin(0,3]$ and $deltain (0,1]$, respectively. We obtain the sharp upper
bound for $|gamma_n|$ when $n=1,2,3$ and $f$ belongs to the classes $F(c)$
and $G(delta)$, respectively. The paper concludes with the following two
conjectures: egin{itemize}
item If $finF (-1/2)$, then $ displaystyle |gamma_n|le
frac{1}{n}left(1-frac{1}{2^{n+1}}
ight)$ for $nge 1$,
and $$ sum_{n=1}^{infty}|gamma_{n}|^{2} leq frac{pi^2}{6}+frac{1}{4}
~{
m Li,}_{2}left(frac{1}{4}
ight)
-{
m Li,}_{2}left(frac{1}{2}
ight), $$ where ${
m Li}_2(x)$ denotes the
dilogarithm function.
item If $fin G(delta)$, then $ displaystyle |gamma_n|,leq
,frac{delta}{2n(n+1)}$ for $nge 1$. end{itemize} | | Source: | arXiv, 1811.1203 | | Services: | Forum | Review | PDF | Favorites | |
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