Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T04:54:48.516Z Has data issue: false hasContentIssue false

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.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).

References

Avkhadiev, F. G. and Wirths, K.-J., Schwarz–Pick Type Inequalities (Birkhäuser, Basel, 2009).10.1007/978-3-0346-0000-2Google Scholar
de Branges, L., ‘A proof of the Bieberbach conjecture’, Acta Math. 154 (1985), 137152.10.1007/BF02392821Google Scholar
Brannan, D. A. and Kirwan, W. E., ‘On some classes of bounded univalent functions’, J. Lond. Math. Soc. (2) 2(1) (1969), 431443.10.1112/jlms/s2-1.1.431Google Scholar
Duren, P., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259 (Springer, New York, 1983).Google Scholar
Elhosh, M. M., ‘On the logarithmic coefficients of close-to-convex functions’, J. Aust. Math. Soc. (Ser. A) 60 (1996), 16.Google Scholar
FitzGerald, C. H. and Pommerenke, Ch., ‘The de Branges theorem on univalent functions’, Trans. Amer. Math. Soc. 290 (1985), 683690.10.1090/S0002-9947-1985-0792819-9Google Scholar
FitzGerald, C. H. and Pommerenke, Ch., ‘A theorem of de Branges on univalent functions’, Serdica 13(1) (1987), 2125.Google Scholar
Girela, D., ‘Logarithmic coefficients of univalent functions’, Ann. Acad. Sci. Fenn. Math. 25 (2000), 337350.Google Scholar
Goodman, A. W., Univalent Functions, Vol. 1–2 (Mariner, Tampa, FL, 1983).Google Scholar
Janowski, W., ‘Some extremal problem for certain families of analytic functions’, Ann. Polon. Math. 28 (1973), 297326.10.4064/ap-28-3-297-326Google Scholar
Kayumov, I. R., ‘On Brennan’s conjecture for a special class of functions’, Math. Notes 78(3–4) (2005), 498502.10.1007/s11006-005-0149-1Google Scholar
Libera, R. J., ‘Univalent 𝛼-spiral functions’, Canad. J. Math. 19 (1967), 449456.10.4153/CJM-1967-038-0Google Scholar
Miller, S. S. and Mocanu, P. T., Differential Subordinations: Theory and Applications, Vol. 225 (Marcel Dekker, New York, 2000).10.1201/9781482289817Google Scholar
Obradović, M., Ponnusamy, S. and Wirths, K.-J., ‘Coefficient characterizations and sections for some univalent functions’, Sib. Math. J. 54(4) (2013), 679696.10.1134/S0037446613040095Google Scholar
Obradović, M., Ponnusamy, S. and Wirths, K.-J., ‘Logarithmic coefficients and a coefficient conjecture for univalent functions’, Monatsh. Math. 185(3) (2018), 489501.10.1007/s00605-017-1024-3Google Scholar
Pommerenke, Ch., Univalent Functions (Vandenhoeck and Ruprecht, Göttingen, 1975).Google Scholar
Ponnusamy, S., Sahoo, S. K. and Yanagihara, H., ‘Radius of convexity of partial sums of functions in the close-to-convex family’, Nonlinear Anal. 95 (2014), 219228.Google Scholar
Prokhorov, D. V. and Szynal, J., ‘Inverse coefficients for (𝛼, 𝛽)-convex functions’, Ann. Univ. Mariae Curie-Sklodowska Sect. A 35(15) (1981), 125143.Google Scholar
Rogosinski, W., ‘On the coefficients of subordinate functions’, Proc. Lond. Math. Soc. (3) 48(2) (1943), 4882.Google Scholar
Roth, O., ‘A sharp inequality for the logarithmic coefficients of univalent functions’, Proc. Amer. Math. Soc. 135 (2007), 20512054.10.1090/S0002-9939-07-08660-1Google Scholar
Špaček, L., ‘Contribution à la théorie des fonctions univalentes’, Časop Pěst. Mat.-Fys. 62 (1933), 1219.Google Scholar
Stankiewicz, J., ‘Quelques problèmes extrèmaux dans les classes des fonctions 𝛼-angulairement étoilées’, Ann. Univ. Mariae Curie-Slodowska Sect. A 20 (1966), 5975.Google Scholar
Umezawa, T., ‘Analytic functions convex in one direction’, J. Math. Soc. Japan 4 (1952), 194202.10.2969/jmsj/00420194Google Scholar