Quantum calculus or q-calculus plays an important role in hypergeometric series, quantum physics, operator theory, approximation theory, sobolev spaces, geometric functions theory and others. But role of q-calculus in the theory of harmonic univalent functions is quite new. In this paper, we make an attempt to connect quantum calculus and harmonic univalent starlike functions. In particular, we introduce and investigate the properties of q-harmonic functions and q-harmonic starlike functions of order ?.

In this paper, we consider a subclass of normalized analytic functions associated with the hyperbolic secant function. We compute the sharp bounds on third- and fourth-order Hermitian–Toeplitz determinants for functions in this class. Moreover, we determine the bounds on second- and third-order Hankel determinants, as well as on the generalized Zalcman conjecture. We examine a Briot–Bouquet-type differential subordination involving the Bernardi integral operator. Finally, we obtain a univalent solution to the Briot–Bouquet differential equation, and discuss the majorization property for such function classes.

We create two Sakaguchi-type function classes that are starlike and convex with respect to their symmetric points, including a q-difference operator, which may have symmetric or assymetric properties, in the open unit disc. We first obtain sufficient coefficient bounds for these functions. In view of these bounds, we obtain quasi-Hadamard products and several partial sums for these function classes. Moreover, the special values of the parameters provided the corresponding consequences of the partial sums.

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot–Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot–Bouquet differential equations and observe that these solutions are the best dominant of the Briot–Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral operator for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions.

We determine the necessary and sufficient convolution conditions for the starlike functions on the open unit disk and related to some geometric aspects of the function $ \tanh z $. We also determine sharp bounds on second and third order Hermitian-Toeplitz determinants for such functions. Further, we compute estimates on some initial coefficients and the Hankel determinants of third and fourth order. In addition, using the concept of Briot-Bouquet type differential subordination, we establish a subordination inclusion involving Bernardi integral operator.