Subclasses of λ -bi-pseudo-starlike functions with respect to symmetric points based on shell-like curves

Abstract

In this paper we define the subclass $\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z))$ of the class $\Sigma$ of bi-univalent functions defined in the unit disk, called $\lambda$-bi-pseudo-starlike, with respect to symmetric points, related to shell-like curves connected with Fibonacci numbers. We determine the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions $f\in\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z)).$ Further we determine the Fekete-Szeg\"o result for the function class $\mathcal{PSL}^\lambda_{s,\Sigma}(\alpha,\tilde{p}(z))$ and for the special cases $\alpha=0$, $\alpha=1$ and $\tau =-0.618$ we state corollaries improving the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$.

Keywords

• analytic functions
• bi-univalent
• shell-like curve
• fibonacci numbers
• starlike functions