Quantum-deformed growth geometry in the open unit disk, with applications in subordination, extremal functions, and phase transitions

Quantum–deformation computing refers to a class of computational and analytical frameworks in which classical operators, dynamical rules, or geometric structures are modified through deformation parameters that interpolate between discrete, nonlocal, or quantum–inspired regimes and their classical continuous counterparts. Such deformations are commonly encoded through parameters such as q, which controls noncommutativity or discretization, and auxiliary parameters that regulate scale, memory, or interaction depth. From a mathematical perspective, quantum deformation arises naturally in q–calculus, quantum groups, and noncommutative geometry, where difference operators replace classical derivatives and finite–scale comparisons replace infinitesimal limits [1–3]. In these settings, computation is no longer purely local; instead, system behavior depends on structured interactions across multiple scales. This viewpoint has proven effective in quantum algorithms, signal processing, and deformation–based numerical schemes, where stability and robustness are enhanced by nonlocal coupling [4–8].

Within complex analysis and geometric function theory, quantum deformation enables the systematic replacement of the classical logarithmic derivative zf′(z)/f(z) by scale–dependent growth factors derived from q–difference or nabla operators. Such replacements preserve analytic structure while encoding additional geometric information related to memory, scale sensitivity, and discrete evolution. As the deformation parameter q approaches unity, the classical theory is recovered, ensuring consistency with standard analytic frameworks. The introduction of a secondary deformation parameter τ further enriches this computational paradigm by controlling the depth at which deformed comparisons are made [9–14]. In computational terms, τ regulates how far information propagates across scales, providing a tunable mechanism for balancing locality against global influence. This two–parameter deformation (q, τ) thus defines a flexible computational geometry that is well suited for modeling systems with hierarchical or multi–resolution structure.

Motivated by these considerations, the present work integrates quantum–deformation computing concepts into geometric function theory through the formulation of the (q, τ)–balanced growth class. This approach allows classical notions such as starlikeness, subordination, and extremal behavior to be reinterpreted within a deformed computational geometry, leading to new sharp results, phase transitions, and dominant mappings that have no classical analogue.

and

where the (q, τ)–nabla operator is defined by

Throughout, let 0 < q < 1, τ > 0, and set a≔q^τ and b ≔ q^2τ. Let $D≔{z∈C:|z|<1}$⁠.

represents a scale–dependent radial growth factor, measuring the expansion of the image domain across the finite radial contraction z↦q^τz. Unlike the classical logarithmic derivative, this expression captures geometric behavior over a non-infinitesimal scale. The defining inequality of $BG(q,τ)(α,β)$ constrains this quantity to lie in a parabolic admissible region of the complex plane, rather than a half-plane. The quadratic term penalizes deviations from uniform radial expansion and suppresses excessive angular oscillation. As a result, functions in $BG(q,τ)(α,β)$ generate image domains that remain starlike with respect to the origin while exhibiting enhanced stability against boundary folding and angular distortion. The parameters q and τ encode scale sensitivity and geometric memory, whereas α controls the baseline expansion and β regulates angular damping. This produces a geometric class that is distinct from classical starlike, spiral-like, and leaf-type families.

We denote by $KEXP(q,τ)(α,β)$ the singleton class consisting of $Fα,β(q,τ)$⁠.

Equivalently, r_⋆ is the smallest positive root (if it exists) of

Then $Fα,β(q,τ)$ is starlike in the disk |z| < r_⋆; that is,

Proof. Write $F(z)=Fα,β(q,τ)(z)$⁠. From (1),

Differentiate, we get

A direct computation gives

Hence, we obtain

Fix r ∈ (0, 1) and let |z| = r. We estimate the real part from below using the elementary inequality $R(w)≥−|w|$⁠. From (4),

Now use the standard bounds for |z| = r:

and similarly

Therefore, we have

If 0 < r < r_⋆, then the right-hand side of (5) is positive by (2), hence

and thus for all |z| < r_⋆. By the standard characterization of starlike functions, $RzF′(z)F(z)>0$ implies that F is starlike in |z| < r_⋆. This completes the proof. □

1. If β = 0, then r_⋆ is the smallest positive root of $1−αr(1−ar)2=0,$ which reduces to a quadratic equation in r.

2. If α = 0, then r_⋆ is the smallest positive root of $1−2βr2(1−br2)2=0,$ which reduces to a quadratic equation in r².

Let 0 < q < 1, τ > 0, and set a≔q^τ, b ≔ q^2τ. Define, for α, β ≥ 0,

Moreover, the minimum is attained at angles θ = θ_ρ satisfying

Proof. Write u = ρe^iθ and note

Taking real parts gives the explicit one-variable function (in c = cos θ)

Differentiate:

A direct substitution of c = c_ρ into Φ_ρ(c) yields

Since Φ_ρ is continuous on [−1, 1] and has a critical point at c_ρ (which lies in [−1, 1] for 0 < ρ < 1), and since the boundary values are larger, this value is the global minimum. Hence the stated minimum and extremal angles follow. □

If α ≥ 8a, then $Fα,0(q,τ)$ is not starlike in any disk |z| < r. If 0 ≤ α < 8a, then $Fα,0(q,τ)$ is starlike in the sharp disk |z| < r_⋆, where

and for α = 0 we have r_⋆ = 1. Moreover, sharpness is realized at boundary points

where $θρ⋆$ is as in Lemma 2.8, and

Proof. From $log⁡Fα,0(q,τ)(z)=log⁡z+αz1−az$ we compute

Fix r ∈ (0, 1) and set z = re^iθ. Let u ≔ az, so |u| = ρ≔ar and

By Lemma 2.8,

In particular, letting r ↓ 0 (so ρ ↓ 0) gives the necessary condition 1 − α/(8a) > 0, i.e. α < 8a, for any positive starlikeness radius. Thus if α ≥ 8a, the minimum real part is $≤0$ already for arbitrarily small r, so no disk of starlikeness exists. Assume now 0 < α < 8a. The sharp radius is determined by the boundary equation

Let t = ρ² ∈ (0, 1). Then

hence, this yields

For every 0 < r < r_⋆ we have ρ < ar_⋆ and the above minimum is strictly positive, so $R(zF′/F)>0$ on |z| ≤ r and hence in |z| < r_⋆, proving starlikeness there. Finally, Lemma 2.8 provides an angle $θρ⋆$ at which $R(u/(1−u)2)$ attains its minimum; taking $u⋆=ρ⋆eiθρ⋆$ and z_⋆ = u_⋆/a yields

showing the radius is sharp and the extremal boundary points are explicit. □

If β ≥ 4b, then $F0,β(q,τ)$ is not starlike in any disk |z| < r. If 0 ≤ β < 4b, then $F0,β(q,τ)$ is starlike in the sharp disk |z| < r_⋆, where

and sharpness is attained at points z_⋆ with $bz⋆2=ρ⋆eiθρ⋆$ where $θρ⋆$ is as in Lemma 2.8.

Proof. The identity for zF′/F follows by differentiating $log⁡F0,β(q,τ)(z)=log⁡z+βz21−bz2$⁠. Set w ≔ bz². Then |w| = ρ≔br² and

Apply Lemma 2.8 to conclude

Letting r ↓ 0 yields the necessary condition β < 4b. For 0 < β < 4b, solve the boundary equation

exactly as in Proposition 2.9, obtaining $ρ2=4b/β−14b/β+1$ and $r⋆=ρ/b$⁠. Extremal points arise from the minimizing angles in Lemma 2.8, which make the real part vanish on |z| = r_⋆. □

1. Case β = 0. If 0 ≤ α < 8a, then $Fα,0(q,τ)$ is starlike in the sharp disk |z| < r_⋆, where

Moreover, there exist boundary points z_⋆ with |z_⋆| = r_⋆ such that

so the radius r_⋆ is sharp. If α ≥ 8a, then $Fα,0(q,τ)$ is not starlike in any disk |z| < r.

1. Case α = 0. If 0 ≤ β < 4b, then $F0,β(q,τ)$ is starlike in the sharp disk |z| < r_⋆, where

Again, there exist boundary points z_⋆ with |z_⋆| = r_⋆ such that

and the radius is sharp. If β ≥ 4b, then no disk of starlikeness exists.

Proof. We treat the two cases separately.

1. The case β = 0. From

we compute

Fix r ∈ (0, 1) and write z = re^iθ. Let u ≔ az, so |u| = ρ≔ar. Then

A direct harmonic minimization on |u| = ρ yields the exact minimum

and this minimum is attained at angles θ = θ_ρ satisfying

Hence, we have

Starlikeness in |z| < r is equivalent to the positivity of this minimum. Letting r ↓ 0 gives the necessary condition α < 8a. For 0 < α < 8a, the sharp radius r_⋆ is determined by the boundary equation

Solving explicitly yields

At $z⋆=(ρ/a)eiθρ$ the real part vanishes, so the radius is sharp. If α ≥ 8a, the minimum is nonpositive for arbitrarily small r, and no starlike disk exists.

1. The case α = 0. From

we obtain

Let w ≔ bz², so |w| = ρ≔br². Then

Applying the same sharp minimization gives

Positivity near r = 0 requires β < 4b. For 0 < β < 4b, solving the boundary equation

yields

At boundary points z_⋆ with $bz⋆2=ρeiθρ$ the real part vanishes, so the radius is sharp. If β ≥ 4b, no disk of starlikeness exists. □

remains strictly inside the parabolic admissible region Ω_α,β for all |z| < r_⋆ and becomes tangent to its boundary at explicit points z_⋆ with |z_⋆| = r_⋆. These extremal points correspond to critical directions along which radial expansion is weakest and angular curvature is maximal. Consequently, the sharp radius r_⋆ marks the exact geometric threshold at which radial coherence may fail. The extremal functions thus describe the transition from uniformly controlled growth to boundary tangency and demonstrate that the (q, τ)–balanced growth condition is geometrically optimal.

Assume that

If there exists $z0∈D$ such that

then there exists a real number λ ≥ 1 such that

Moreover,

Proof. Define the auxiliary function

which is analytic in $D$ with ϕ(0) = 0. By assumption, |ϕ(z)| attains its maximum on the closed disk $D¯|z0|$ at z = z₀ ≠ 0. By the classical Jack lemma applied to ϕ, there exists λ ≥ 1 such that

Equivalently,

Next, we use the admissibility condition

At z = z₀, maximality of |ϕ| implies that any outward perturbation of z₀ would violate the inequality unless equality holds. Hence, we have

which means w(z₀) ∈ ∂Ω_α,β. Finally, since |ϕ(z₀)| is maximal and ϕ(0) = 0, the argument of ϕ(z₀) must oppose the outward radial direction, yielding

This completes the proof. □

If $f∈BG(q,τ)(α,β)$⁠, then

where

Equivalently,

Proof. Set

By definition of $BG(q,τ)(α,β)$⁠, we have

The function Ψ_α,β is analytic and univalent in $D$ and maps $D$ conformally onto Ω_α,β, with

Assume, for contradiction, that w is not subordinate to Ψ_α,β. Then there exists a point $z0∈D$ such that

Define

which is analytic in |z| < |z₀| and satisfies ϕ(0) = 0. Moreover,

By the classical Jack lemma applied to ϕ, there exists λ ≥ 1 such that

Differentiating w(z) = Ψ_α,β(ϕ(z)) and evaluating at z₀, we obtain

Since $ϕ(z0)∈∂D$ and Ψ_α,β maps $∂D$ onto ∂Ω_α,β, the above identity implies

This contradicts the admissibility inequality

unless equality holds at z₀. Hence, $w(D)⊂Ωα,β$ and

□

Then the following properties hold.

1. Analyticity and normalization. Ψ_α,β is analytic and univalent in $D$, with

1. Mapping property. Ψ_α,β maps $D$ conformally onto the parabolic region

1. Boundary parametrization. For z = e^it, t ∈ (0, 2π),

and the curve Ψ_α,β(e^it) parametrizes ∂Ω_α,β exactly once, with no self-intersections.

1. Inverse function. The inverse map $Ψα,β−1:Ωα,β→D$ is given by

where the principal branch of the square root is taken.

1. Extremal contact. For w ∈ ∂Ω_α,β,

and this equality is attained if and only if w = Ψ_α,β(e^it) for some t ∈ [0, 2π).

Proof.

1. The function z/(1 − z)² is analytic and univalent in $D$⁠, hence so is Ψ_α,β. Normalization follows by direct computation.

2. Let

It is classical that $Φ(D)={u∈C:R(u)>−14}$⁠. The affine transformation

maps this half-plane conformally onto Ω_α,β, yielding the stated mapping property.

1. Since Ψ_α,β is conformal and $∂D$ is mapped to the boundary of the image domain, the parametrization follows directly by substitution z = e^it.

2. Solving (7) for z yields a quadratic equation in (1 − z)⁻¹, whose solution gives the stated inverse formula.

3. Equality in $R(w)≥α−β|w−1|2$ characterizes the boundary of Ω_α,β. Since Ψ_α,β is conformal, the boundary is attained exactly when |z| = 1, completing the proof. □