Details

Autor(en) / Beteiligte

• PONNUSAMY, SAMINATHAN
• SHARMA, NAVNEET LAL
• WIRTHS, KARL-JOACHIM

Titel

LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS

Ist Teil von

• Journal of the Australian Mathematical Society (2001), 2020-10, Vol.109 (2), p.230-249

Ort / Verlag

Cambridge, UK: Cambridge University Press

Erscheinungsjahr

2020

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let ${\mathcal{S}}$ be the family of analytic and univalent functions $f$ in the unit disk $\mathbb{D}$ with the normalization $f(0)=f^{\prime }(0)-1=0$, and let $\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$ denote the logarithmic coefficients of $f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families ${\mathcal{F}}(c)$ and ${\mathcal{G}}(c)$ of functions $f\in {\mathcal{S}}$ defined by $$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$ for some $c\in (0,3]$ and $c\in (0,1]$, respectively. We obtain the sharp upper bound for $|\unicode[STIX]{x1D6FE}_{n}|$ when $n=1,2,3$ and $f$ belongs to the classes ${\mathcal{F}}(c)$ and ${\mathcal{G}}(c)$, respectively. The paper concludes with the following two conjectures: ∙If $f\in {\mathcal{F}}(-1/2)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$ for $n\geq 1$, and $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$ where $\text{Li}_{2}(x)$ denotes the dilogarithm function.∙If $f\in {\mathcal{G}}(c)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$ for $n\geq 1$.