This article discusses an iterative transformation method via fuzziness that mixtures the Laplace transform with the iterative iterative method. Using Caputo derivative operator, the proposed technique demonstrates the inherent reliability of fractional fuzzy biological population equations with initial fuzzy conditions. The obtained results to the fuzzy fractional biological equations are more general and apply to a broad variety of problems. A parametric description of the solutions is obtained by translating the fuzzy fractional differential equation into an equivalent system of corresponding fractional differential equations. The proposed method is numerically tested against crisp solutions and those produced by other methods, demonstrating that it is a convenient and remarkably accurate way to solve a tool for solving a wide variety of physics and engineering problems.

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 is to characterize isometries of multiplication operators and weighted composition operators on the nth weighted-type Banach spaces {V_n : n ∈ N} of analytic functions on the open unit disk of which the Bloch space and the Zygmund space are particular cases at n = 1, 2. We give characterizations of the symbols ψ and φ for which the multiplication operator M_ψ and the weighted composition operator W_ψ,φ are surjective isometries. Moreover, we show that generalized weighted composition operators are not isometric on V_n.

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.

Let $H\left(\mathbb{D}\right)$ be the set of analytic functions on the open unit disk $\mathbb{D}$. For $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, define the iterated weighted-type Banach space ${\mathcal{V}}_n:=\left\{f\in H\left(\mathbb{D}\right):{sup}_{z\in \mathbb{D}}{(1-|z|}^2\left)\right|f^{\left(n\right)}\left(z\right)|<\infty \right\}.$ In this work, we study the boundedness and the essential norm of products of Volterra-type operators and composition operators on iterated weighted-type Banach spaces.