Coefficient Inequalities of q-Bi-Univalent Mappings Associated with q-Hyperbolic Tangent Function

Abstract

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings.

Certain Quantum Operator Related to Generalized Mittag–Leffler Function

“…There are many authors who have studied the operators of quantum calculus through many diverse applications in geometric function theory; e.g., Attiya et al [8] studied differential operators related to the q-Raina function, Ibrahim [9], Al-shbeil et al [10] and Karthikeyan et al [11] studied the q-convolution of a certain class of analytic functions related to the quantum differential operator in GFT, Ismail et al [12] and Riaz et al [13] studied starlike functions defined by q-fractional derivatives, Shaba et al [14] studied coefficient inequalities of q-bi-univalent associated with q-hyperbolic tangent functions, Al-Shaikh et al [15] studied a class of close-to-convex functions defined by a quantum difference operator, Sitthiwirattham et al [16] studied Maclaurin's coefficients inequalities for convex functions in q-calculus, Al-Shaikhm [17] studied some classes of analytic functions associated with a Salagean quantum differential operator, and Tang et al [18] studied the Hankel and Toeplitz determinant for certain subclasses of multivalent q-starlike functions.…”