The aim in the present work is to introduce and study new subclasses of analytic functions that are defined by using the generalized classes of Janowski functions combined with the $(n,m)$-symmetrical functions, that generalize many others defined by different authors. We gave a representation theorem for these classes, certain inherently properties, while covering and distortion properties are also pointed out.

The purpose of this paper is to define new classes of analytic functions by amalgamating the concepts of q-calculus, Janowski type functions and $(x,y)$-symmetrical functions. We use the technique of convolution and quantum calculus to investigate the convolution conditions which will be used as a supporting result for further investigation in our work, we deduce the sufficient conditions, P$\acute{o}$lya-Schoenberg theorem and the application. Finally motivated by definition of the neighborhood, we give analogous definition of neighborhood for the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, and then investigate the related neighborhood results, which are also pointed out.

In our present work, the concepts of symmetrical functions and the concept of spirallike Janowski functions are combined to define a new class of analytic functions. We give a structural formula for functions in ${\mathcal{S}}^{\eta ,\mu }(F,H,\lambda )$, a representation theorem, the radius of starlikeness estimates, covering and distortion theorems and integral mean inequalities are obtained.

In this article, the present study employs the utilization of the concepts pertaining to $(x,y)$-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, thereby further scrutinizing the convolution properties of these newly defined classes.

In this paper, we introduce new classes of functions defined within the open unit disk by integrating the concepts of $(\lambda ,\gamma )$-symmetrical functions, generalized Janowski functions, and quantum calculus. We derive a structural formula and a representation theorem for the class ${\mathcal{S}}_q^{\lambda ,\gamma }(x,y,z)$. Utilizing convolution techniques and quantum calculus, we investigate convolution conditions supported by examples and corollary, establishing sufficient conditions. Additionally, we derive properties related to coefficient estimates, which further elucidate the characteristics of the defined function classes.

In this paper, we investigate the properties of functions belonging to the classes $ \psi(\eta)\in\mathcal{\overline{T}}^{j, k}_q(A, B) $ and $ \psi(\eta)\in\mathcal{\overline{K}}^{j, k}_q(A, B) $, these functions are defined within the context of $ q $-calculus and $ (j, k) $-symmetrical functions. We employ convolution techniques and quantum calculus to explore the convolution conditions, which will serve as foundational results for further studies in our work Furthermore, we establish conditions for membership in $ \psi(\eta)\in\mathcal{\overline{T}}^{j, k}_q(A, B) $ and present an example demonstrating the application of these results to rational functions.