This article investigated the inclusion properties of Touchard polynomials $Y_m(p,z)$ and generalized Bessel functions of the first kind ${\mathcal{G}}_a(z)$, along with their associated convolution and integral operators, within the analytic function classes ${\mathcal{R}}_{s,d}^w(\delta )$ and ${\mathcal{M}}_{b,s}^w(\delta )$. Using coefficient bounds of certain function classes, we derived sufficient conditions for the convolution operators ${\mathcal{Y}}_m^p(\psi ,z)$ and ${\mathcal{E}}_{l_a,c}(\psi ,z)$, and the integral operators ${\mathfrak{Y}}_m(p,z)$ and ${\mathfrak{G}}_a(z)$ to belong to various subclasses of starlike and convex functions. Furthermore, inclusion results among these subclasses were obtained, thereby extending and unifying several existing results in geometric function theory.

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $C^{\ast }$-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.