Logarithmic coefficients of starlike functions connected with k-Fibonacci numbers

Abstract

Let $\mathcal{A}$ denote the class of analytic functions in the open unit disc $\mathbb{U}$ normalized by $f(0)=f^{\prime }(0)-1=0,$ and let $\mathcal{S}$ be the class of all functions $f\in\mathcal{A}$ which are univalent in $\mathbb{U}$. For a function $f\in \mathcal{S}$, the logarithmic coefficients $\delta _{n}\,\left( n=1,2,3,\ldots \right) $ are defined by
$\log \frac{f(z)}{z}=2\sum_{n=1}^{\infty }\delta _{n}z^{n}\qquad \left( z\in\mathbb{U}\right).$
and it is known that $\left\vert \delta _{1}\right\vert \leq 1$ and $\left\vert \delta _{2}\right\vert \leq \frac{1}{2}\left( 1+2e^{-2}\right)=0,635\cdots .$ The problem of the best upper bounds for $\left\vert \delta_{n}\right\vert $ of univalent functions for $n\geq 3$ is still open. Let $\mathcal{SL}^{k}$ denote the class of functions $f\in \mathcal{A}$ such that
$\frac{zf^{\prime }\left( z\right) }{f(z)}\prec \frac{1+\tau _{k}^{2}z^{2}}{1-k\tau _{k}z-\tau _{k}^{2}z^{2}},\quad \tau _{k}=\frac{k-\sqrt{k^{2}+4}}{2}\qquad \left( z\in \mathbb{U}\right).$
In the present paper, we determine the sharp upper bound for $\left\vert\delta _{1}\right\vert ,\left\vert \delta _{2}\right\vert $ and $\left\vert\delta _{3}\right\vert $ for functions $f$ belong to the class $\mathcal{SL}^{k}$ which is a subclass of $\mathcal{S}$. Furthermore, a general formula is given for $\left\vert \delta _{n}\right\vert \,\left( n\in \mathbb{N}\right) $ as a conjecture.

Keywords

• Analytic function
• univalent function
• shell-like function
• logarithmic coefficients
• k-Fibonacci number
• subordination

References

1. Bulut, S., Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers, Math. Slovaca, 71 (4) (2021), 823-830. doi: 10.1515/ms-2021-0023
2. Duren, P. L., Univalent Functions, Grundlehren der Mathematics. Wissenschaften, Bd, Springer-Verlag, NewYork, 1983.
3. Falcon, S., Plaza, A., The k-Fibonacci sequence and the Pascal 2-triangle, Chaos Solitons Fractals, 33 (1) (2007), 38-49. doi: 10.1016/j.chaos.2006.10.022
4. Grenander, U., Szegö, G., Toeplitz forms and their applications, Univ. of California Press, Berkeley, Los Angeles, 1958.
5. Güney, H. Ö., Sokol, J., İlhan, S., Second Hankel determinant problem for some analytic function classes with connected k-Fibonacci numbers, Acta Univ. Apulensis Math. Inform., 54 (2018), 161-174. doi: 10.17114/j.aua.2018.54.11
6. Güney, H. Ö., İlhan, S., Sokol, J., An upper bound for third Hankel determinant of starlike functions connected with k-Fibonacci numbers, Bol. Soc. Mat. Mex. (3), 25 (1) (2019), 117-129. doi: 10.1007/s40590-017-0190-6
7. Kalman, D., Mena, R., The Fibonacci numbers-exposed, Math. Mag., 76 (3) (2003), 167-181. doi: 10.2307/3219318.
8. Kayumov, I. R., On Brennan's conjecture for a special class of functions, Math. Notes, 78 (2005), 498-502. doi: 10.1007/s11006-005-0149-1

9. Raina, R. K., Sokol, J., Fekete-Szegö problem for some starlike functions related to shell-like curves, Math. Slovaca, 66 (1) (2016), 135-140. doi: 10.1515/ms-2015-0123
10. Sokol, J., On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, 175 (23) (1999), 111-116.
11. Sokol, J., Raina, R. K., Yılmaz Özgür, N., Applications of k-Fibonacci numbers for the starlike analytic functions, Hacettepe J. Math. Stat., 44 (1) (2015), 121-127. doi:10.15672/HJMS.2015449091
12. Şiar, Z., Keskin, R., Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat., 42 (3) (2013), 211-222.
13. Yılmaz Özgür, N., Sokol, J., On starlike functions connected with k-Fibonacci numbers, Bull. Malays. Math. Sci. Soc., 38 (1) (2015), 249-258. doi: 10.1007/s40840-014-0016-x.
14. Yılmaz Özgür, N., Uçar, S., Öztunç, Ö., Complex factorizations of the k-Fibonacci and k-Lucas numbers, An. Ştiint. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 62 (1) (2016), 13-20.