In the current article, we consider certain subfamilies ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$ of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family ${\mathcal{S}}_e^{\ast }$ in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions.

Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete–Szegö type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its q-extension for multivalent functions. Furthermore, we will show that the class defined in this paper, along with the obtained results, generalizes many known works available in the literature.

The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class ${\mathcal{S}}_{sin}^{*}$, with the quantity $\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ subordinated to $1+sinz$, we obtain an estimate on the initial coefficient $a_4$ and an upper bound of the third Hankel determinant. For functions in the class ${\mathcal{BT}}_{sin}$, with $f^{\prime }\left(z\right)$ lie in an eight-shaped domain in the right-half plane, we prove that its upper bound of third Hankel determinant is $\frac{1}{16}$. All the results are proven to be sharp.

In the present paper, we aimed to discuss certain coefficient-related problems for the inverse functions associated with a bounded turning functions class subordinated with the exponential function. We calculated the bounds of some initial coefficients, the Fekete–Szegö-type inequality, and the estimation of Hankel determinants of second and third order. All of these bounds were proven to be sharp.

By making use of the concept of basic (or q-) calculus, many subclasses of analytic and symmetric q-starlike functions have been defined and studied from different viewpoints and perspectives. In this article, we introduce a new class of meromorphic multivalent close-to-convex functions with the help of a q-differential operator. Furthermore, we investigate some useful properties such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity for this new subclass.