Subclasses of bi-univalent functions defined by the normalized Le Roy-type Mittag-Leffler function

Shrouq Khaldoon Al-Titi, Basem Aref Frasin

Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan

Abstract: Let $\displaystyle \mathbb{F}_{\varkappa ,\mu }^{(\gamma )}(\xi )=\xi +\sum \limits_{n=1}^{\infty }\left[ \frac{\Gamma (\mu )}{\Gamma (\varkappa n+\mu )} \right] ^{\gamma }\xi ^{n+1}$ be the normalized Le Roy-type Mittag-Leffler function. The purpose of the present paper is to introduce two new subclasses $\mathbb{H}_{\Sigma }^{\gamma }(\varkappa ,\mu ,\lambda ,\tau )$ and $\mathbb{H}_{\Sigma }^{\gamma }(\varkappa ,\mu ,\lambda ,\delta )$ of the function class $\Sigma $ of bi-univalent functions defined by the function $\mathbb{F}_{\varkappa ,\mu }^{(\gamma )}(\xi )$. Furthermore, we find estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for functions in these new subclasses. Also, we solve the Fekete–Szegö functional problem for functions in the classes $\mathbb{H}_{\Sigma }^{\gamma }(\varkappa ,\mu ,\lambda ,\tau )$ and $\mathbb{H}_{\Sigma }^{\gamma }(\varkappa ,\mu ,\lambda ,\delta ).$ Several examples of the main results are also considered.

Keywords: analytic and univalent functions, bi-univalent functions, Le Roy-type Mittag-Leffler function, coefficients bounds.

UDC: 517.5

Received: 10.08.2025
Received in revised form: 27.09.2025
Accepted: 15.11.2025

Language: English