Applications of the Horadam Polynomials Involving $\lambda$-Pseudo-Starlike Bi-Univalent Functions Associated with a Certain Convolution Operator

Abstract

In this article, we introduce and study a new family
$\mathcal{P}_{\Sigma}(\delta,\lambda,k,\gamma,\alpha,\beta,r)$ of normalized analytic
and $\lambda$-pseudo-starlike bi-univalent functions by using the Horadam polynomials,
which is associated with a certain convolution operator
defined in the open unit disk $\mathbb{U}$.
We establish the bounds for $|a_2|$ and $|a_3|$, where $a_2$, $a_3$ are the initial
Taylor-Maclaurin coefficients. Furthermore, we obtain the Fekete-Szeg\"{o} inequality for
functions in the class $\mathcal{P}_{\Sigma}(\delta,\lambda,k,\gamma,\alpha,\beta,r)$,
which we have introduced here. We also indicate several special cases and consequences
for our results.