Logarithmic coefficients of starlike functions connected with k-Fibonacci numbers

Year 2021, , 910 - 923, 31.12.2021

Serap Bulut

https://doi.org/10.31801/cfsuasmas.808319

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

• 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
• Duren, P. L., Univalent Functions, Grundlehren der Mathematics. Wissenschaften, Bd, Springer-Verlag, NewYork, 1983.
• 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
• Grenander, U., Szegö, G., Toeplitz forms and their applications, Univ. of California Press, Berkeley, Los Angeles, 1958.
• 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
• 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
• Kalman, D., Mena, R., The Fibonacci numbers-exposed, Math. Mag., 76 (3) (2003), 167-181. doi: 10.2307/3219318.
• 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
• 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
• Sokol, J., On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, 175 (23) (1999), 111-116.
• 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
• Şiar, Z., Keskin, R., Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat., 42 (3) (2013), 211-222.
• 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.
• 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.