Exploring a distinct group of analytical functions linked with Bernoulli's Lemniscate using the q-derivative

Isra Al-Shbeil; Timilehin Gideon Shaba; Alina Alb Lupas; Reem K Alhefthi

doi:10.1016/j.heliyon.2024.e34095

Exploring a distinct group of analytical functions linked with Bernoulli's Lemniscate using the q-derivative

Heliyon. 2024 Jul 9;10(14):e34095. doi: 10.1016/j.heliyon.2024.e34095. eCollection 2024 Jul 30.

Authors

Isra Al-Shbeil¹, Timilehin Gideon Shaba², Alina Alb Lupas³, Reem K Alhefthi⁴

Affiliations

¹ Department of Mathematics, University of Jordan, Amman 11942, Jordan.
² Department of Physical Sciences, Mathematics Programme, Landmark University, Omu-Aran 251103, Nigeria.
³ Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania.
⁴ Department of Mathematics, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia.

Abstract

This research presents a new group of mathematical functions connected to Bernoulli's Lemniscate, using the q-derivative. Expanding on previous studies, the research concentrates on determining coefficient approximations, the Fekete-Szego functional, Zalcman inequality, Krushkal inequality, along with the second and third Hankel determinants for this recently established collection of functions. Additionally, the study derives the Fekete-Szego inequality for the function $\frac{ξ}{f (ξ)}$ and obtains the inverse function $f^{- 1} (ξ)$ for this specific class. This research advances our understanding in this area and suggests for further exploration.

Keywords: Bounded turning function; Fekete-Szego estimates; Hankel determinant; Krushkal inequality; Zalcman inequality; primary, 30C45, 30C50, 30C80; q-derivative operator; secondary, 11B65, 47B38.