A Novel Family of Starlike Functions Involving Quantum Calculus and a Special Function

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is q→1−. Also, the study of integral as well as differential operators has remained a significant field of inquiry from the early developments of function theory. In the present article, a subclass Sscμ,q of functions being analytic in D=z∈C:z<1 is introduced. The definition of Sscμ,q involves the concepts of subordination, that of q-derivative and q-Ruscheweyh operators. Since coefficient estimates and coefficient functionals provide insights into different geometric properties of analytic functions, for this newly defined subclass, we investigate coefficient estimates up to a4, in which both bounds for |a2| and |a3| are sharp, while that of |a4| is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to T32 (sharp in some cases) and sufficient condition are obtained. Several consequences derived from our above-mentioned findings are also part of the discussion.