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K. -J. Wirths

Publications and source records attributed to K. -J. Wirths.

5 recordsLinked to original sources
arXiv2018

Logarithmic coefficients problems in families related to starlike and convex functions

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 $γ_n(f)=γ_n$ denote the logarithmic coefficients of $f\in {\es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $\F(c)$ and $\G(δ)$ of functions $f\in {\es}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )<1+\fracδ{2},\quad z\in \D $$ for some $c\in(0,3]$ and $δ\in (0,1]$, respectively. We obtain the sharp upper bound for $|γ_n|$ when $n=1,2,3$ and $f$ belongs to the classes $\F(c)$ and $\G(δ)$, respectively. The paper concludes with the following two conjectures: \begin{itemize} \item If $f\in\F (-1/2)$, then $ \displaystyle |γ_n|\le \frac{1}{n}\left(1-\frac{1}{2^{n+1}}\right)$ for $n\ge 1$, and $$ \sum_{n=1}^{\infty}|γ_{n}|^{2} \leq \frac{π^2}{6}+\frac{1}{4} ~{\rm Li\,}_{2}\left(\frac{1}{4}\right) -{\rm Li\,}_{2}\left(\frac{1}{2}\right), $$ where ${\rm Li}_2(x)$ denotes the dilogarithm function. \item If $f\in \G(δ)$, then $ \displaystyle |γ_n|\,\leq \,\fracδ{2n(n+1)}$ for $n\ge 1$. \end{itemize}

math.CV↗
arXiv2018

Logarithmic coefficients of the inverse of univalent functions

Let $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ \sum_{n=2}^{\infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse coefficients $Γ_n$ of $F$ are defined by the formula $\log\left(F(w)/w\right)\,=\,2\sum_{n=1}^{\infty}Γ_n(F)w^n$. % In this paper, we determine the logarithmic inverse coefficients bound of $F$ for the class In this paper, we first determine the sharp bound for the absolute value of $Γ_n(F)$ when $f$ belongs to $\es$ and for all $n \geq 1$. This result motivates us to carry forward similar problems for some of its important geometric subclasses. In some cases, we have managed to solve this question completely but in some other cases it is difficult to handle for $n\geq 4$. For example, in the case of convex functions $f$, we show that the logarithmic inverse coefficients $Γ_n(F)$ of $F$ satisfy the inequality \[ |Γ_n(F)|\,\le \, \frac{1}{2n} \mbox{ for } n\geq 1,2,3 \] and the estimates are sharp for the function $l(z)=z/(1-z)$. Although this cannot be true for $n\ge 10$, it is not clear whether this inequality could still be true for $4\leq n\leq 9$.

math.CV↗
arXiv2017

Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions

Let ${\mathcal U}(λ)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The logarithmic coefficients $γ_n$ of $f$ are defined by the formula $\log(f(z)/z)=2\sum_{n=1}^\infty γ_nz^n$. In a recent paper, the present authors proposed a conjecture that if $f\in {\mathcal U}(λ)$ for some $0<λ\leq 1$, then $|a_n|\leq \sum_{k=0}^{n-1}λ^k$ for $n\geq 2$ and provided a new proof for the case $n=2$. One of the aims of this article is to present a proof of this conjecture for $n=3, 4$ and an elegant proof of the inequality for $n=2$, with equality for $f(z)=z/[(1+z)(1+λz)]$. In addition, the authors prove the following sharp inequality for $f\in{\mathcal U}(λ)$: $$\sum_{n=1}^{\infty}|γ_{n}|^{2} \leq \frac{1}{4}\left(\frac{π^{2}}{6}+2{\rm Li\,}_{2}(λ)+{\rm Li\,}_{2}(λ^{2})\right), $$ where ${\rm Li}_2$ denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of $\mathcal S$.

math.CV↗