Inclusion properties of hypergeometric type functions and related integral transforms

Let A be the set of all analytic functions f defined on the open unit disk D satisfying f(0) = f'(0) − 1 = 0. Let φ_Ne (z) := 1 + z − z ³ =3 be the recently introduced Carathéodory function which maps the unit circle ∂ D $ \partial \mathbb{D} $ D onto a 2-cusped kidney-shaped curve called nephroid given by ( ( u − 1 ) 2 + v 2 − 4 a ) 3 − 4 v 2 2 = 0. ${{\left( {{(u-1)}^{2}}+{{v}^{2}}-\frac{4}{a} \right)}^{3}}-\frac{4{{v}^{2}}}{2}=0.$ In this paper, we determine the best possible estimate on the real β so that for some analytic p satisfying p(0) = 1 the following subordination-implication holds: 1 + β z p ′ ( z ) p j ( z ) ≺ F ( z ) ⇒ p ( z ) ≺ φ N e ( z ) , j = 0 , 1 , 2 , $$1+\beta \frac{z{{p}^{\prime }}(z)}{{{p}^{j}}(z)}\prec \mathcal{F}(z)\Rightarrow p(z)\prec {{\varphi }_{Ne}}(z),\quad j=0,1,2,$$ where F(z) is some Carathéodory function with special geometries like right/left-half of Bernoulliφs lemniscate, cardioid, lune, eight-shaped, etc. As applications, we establish sufficient conditions for the Ma-Minda family of nephroid starlike functions given by S N e * : = { f ∈ A : z f ′ ( z ) f ( z ) ≺ φ N e ( z ) } . $$\mathcal{S}_{Ne}^{*}:=\left\{ f\in \mathcal{A}:\frac{z{{f}^{\prime }}(z)}{f(z)}\prec {{\varphi }_{Ne}}(z) \right\}.$$

In this article, we construct a framework for analyzing the equilibrium and stability of networked multi-sector economic systems via fixed-point analysis. We represent directional intersectoral dependencies, nonlinear feedback effects, and heterogeneous adjustment dynamics in the model by the coupled and tripled fixed-point theory in the graphically extended suprametric spaces. The graphical structure encodes supply-chain and influence networks, whereas asymmetric and nonuniform interaction strengths are encoded in the suprametric setting. Furthermore, we prove the existence, uniqueness, and convergence of equilibrium solutions under new generalized contraction conditions. We apply the theoretical findings in nonlinear state systems in which prices in interdependent markets are adjusted using integral equations. The results of numerical simulations show consistent convergence, and the sensitivity parameter of the network structure significantly influences the determination of economic stability and speed of adjustment.