Abstract:
Let $\mathscr A$ denote the set of normalized analytic functions $f(z)=z+\sum^\infty_{k=2}a_kz^k$ in the unit disk $|z|<1$, $s_n(z)$ represent the $n$nth partial sum of $f(z)$. Our first objective of this note is to obtain a bound for $|\frac{s_n(z)}{f(z)}-1|$ when $f\in\mathscr A$ is univalent in $\mathbb D$. Let $\mathscr U$ denote the set of all $f\in\mathscr A$ in $\mathbb D$ satisfying the condition
$$
\Big|f'(z)\Bigl(\frac z{f(z)}\Bigr)^2-1\Big|<1
$$
for $|z|<1$. In case $f''(0)=0$, we find that all corresponding sections $s_n$ of $f\in\mathscr U$ are in $\mathscr U$ in the disk $|z|<1-\frac{3\log n-\log(\log n)}n$ for $n\ge5$. We also show that $\operatorname{Re}(f(z)/s_n(z))>1/2$ in the disk $|z|<\sqrt{\sqrt5-2}$. Finally, we establish a necessary coefficient condition for functions in $\mathscr U$ and the related radius problem for an associated subclass of $\mathscr U$. In result, we see that if $f\in\mathscr U$ thenfor $n\ge3$ we have
$$
\Big|\frac{f(z)}{s_n(z)}-\frac43\Big|<\frac23\quad\text{for}\quad|z|<r_n:=1-\frac{2\log n}n.
$$
Keywords:coefficient inequality, partial sums, radius of univalence, analytic, univalent, and starlike functions.
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