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LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS

Published online by Cambridge University Press:  11 March 2019

SAMINATHAN PONNUSAMY*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India email [email protected]
NAVNEET LAL SHARMA
Affiliation:
Department of Mathematics, Amity School of Applied Sciences, Amity University Gurgaon, Manesar, Haryana 122413, India email [email protected]
KARL-JOACHIM WIRTHS
Affiliation:
Institut für Analysis und Algebra, TU Braunschweig, 38106 Braunschweig, Germany email [email protected]

Abstract

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 $n\geq 1$.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The work of the first author is supported by Mathematical Research Impact Centric Support of Department of Science and Technology (DST), India (MTR/2017/000367). The second author thanks the Science and Engineering Research Board, DST, India, for its support by SERB National Post-Doctoral Fellowship (grant no. PDF/2016/001274).

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