The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the ${\Delta }_h$ hybrid special polynomials associated with Hermite polynomials denoted by ${}_{{\Delta }_{h}H}{Q}_m(u,v,w;h)$. Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials.

This article aims to introduce degenerate hybrid type Appell polynomials ${}_{\mathcal{H}}{Q}_m(u,v,w;\eta )$ and establishes their quasi-monomial characteristics. Additionally, a number of features of these polynomials are established, including symmetric identities, implicit summation formulae, differential equations, series definition and operational formalism.

The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of ${\Delta }_h$ tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of ${\Delta }_h$ Appell polynomials such as ${\Delta }_h$ Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained.

This investigation introduces a novel family of univalent analytic functions subordinate to lung-shaped domains within the open unit disk. Through rigorous application of subordination theory and systematic analysis, we establish coefficient bounds for the initial five coefficients, derive estimates for Hankel determinants of orders two and three, determine bounds for the first four logarithmic coefficients, and derive the bounds of some Zalcman functionals. The lung-shaped domain is characterized by the subordination condition involving a secant-based function, which maps the unit disk onto a geometrically distinctive region exhibiting bilateral symmetry. All obtained bounds are demonstrated to be sharp through the construction of specific extreme functions.

This paper studies a generalized class of linear operators acting on spaces of analytic functions, defined by $P_n^{\psi ,\phi }\left(f\right)\left(z\right)={\displaystyle \sum _{j=0}^n}{\psi }_j\left(z\right)f^{\left(j\right)}\left(\phi \left(z\right)\right)$, where $\psi =\{{\psi }_0,{\psi }_1,\dots ,{\psi }_n\}\subset H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. This formulation encompasses several classical operators, including composition, weighted composition, differentiation–composition, and the Stević–Sharma operator. We focus on the action of $P_n^{\psi ,\phi }$ from $BMOA$ and analytic Besov spaces ${\mathcal{B}}_p$ into the Bloch space $\mathcal{B}$, and provide necessary and sufficient conditions for boundedness and compactness. These results unify and extend many previously known characterizations and demonstrate the flexibility of the $P_n^{\psi ,\phi }$ framework in the context of analytic operator theory.