(α,β,γ)-Order and Type in Analytic Functions

• (α,β,γ)-order and type are quantitative measures that refine classical growth invariants by employing control functions and iterated logarithmic/exponential scaling.
• The framework recovers classical growth orders when elementary control functions are chosen, linking multi-level scaling to standard order and type definitions.
• Key properties include invariance under differentiation and sharp asymptotic estimates for solutions of analytic ODEs with analytic coefficients.

The concepts of $(\alpha,\beta,\gamma)$-order and $(\alpha,\beta,\gamma)$-type are quantitative refinements for describing the growth of solutions to complex linear differential equations in the unit disc, particularly those with analytic coefficients of nonclassical, iterated or generalized orders. These notions extend classical maximum modulus order and type, as well as their higher iterates, by employing three control functions $\alpha,\beta,\gamma$ and multi-level logarithmic and exponential scaling. The framework recovers classical growth invariants for standard function choices and yields sharp results on the asymptotics of solutions to analytic coefficient ODEs, including explicit connection to the growth rates of coefficients.

1. Formal Definitions

Let $$\alpha,\beta,\gamma\colon[0,+\infty)\rightarrow[0,+\infty)$$ be three unbounded, nondecreasing functions satisfying normalization and compatibility conditions needed for the theory. For an analytic function $f$ defined on $\Delta = \{z:\, |z|<1\}$, denote by $M(r,f)=\max_{|z|=r}|f(z)|$ the maximum modulus on the circle of radius $r$. Iterated logarithms and exponentials are given by

$$\log^{[k]}x = \log\bigl(\log^{[k-1]}x\bigr),\quad \exp^{[k]}x = \exp\bigl(\exp^{[k-1]}x\bigr).$$

The $(\alpha,\beta,\gamma)$-order is defined as

$$\varrho_{(\alpha,\beta,\gamma),M}[f] = \limsup_{r\to 1^-} \frac{\alpha\!\bigl(\log^{[2]}M(r,f)\bigr)} {\beta\!\bigl(\log\gamma\bigl(\tfrac{1}{1-r}\bigr)\bigr)}.$$

Provided $0<\varrho_{(\alpha,\beta,\gamma),M}[f]<+\infty$, the corresponding $(\alpha,\beta,\gamma)$-type is

$$\tau_{(\alpha,\beta,\gamma),M}[f] = \limsup_{r\to 1^-} \frac{ \exp\left\{\alpha\left(\log^{[2]}M(r,f)\right)\right\} }{ \left[ \exp\left\{\beta\left(\log\gamma\left(\tfrac{1}{1-r}\right)\right)\right\} \right]^{\varrho_{(\alpha,\beta,\gamma),M}[f]} }.$$

Alternative formulations involving additional iterates, such as $\alpha(\log)$, define $(\alpha(\log),\beta,\gamma)$-order and type by systematically substituting $\alpha$ with $\alpha(\log)$ and, if needed, switching to definitions based on Nevanlinna characteristics for meromorphic $f$.

2. Equivalent Formulations and Classical Recovery

Proposition 1.1 establishes the equivalence between $\alpha\big(\log^{[2]}M(r,f)\big)$ and $\alpha(\log)\big(\log^{[3]}M(r,f)\big)$ in the definition, yielding

$$\varrho_{(\alpha(\log),\beta,\gamma),M}[f] = \varrho_{(\alpha(\log),\beta,\gamma)}[f],$$

for analytic $f$. By selecting elementary control functions, the framework reproduces established growth invariants:

$\alpha(x)$	$\beta(x)$	$\gamma(x)$	Classical object
$\log_{p+1}x$	$x$	$x$	$p$-th iterated order $\rho_{M,p}$
$x$	$\log_{q+1}x$	$x$	$[p,q]$-order
$x$	$x$	$x$	Classical order $\rho_M$

For example, setting $\alpha(x)=x$, $\beta(x)=x$, $\gamma(x)=x$ recovers $$\rho_M(f)=\limsup \frac{\log\log M(r,f)}{-\log(1-r)}$$.

3. Principal Properties and Inequalities

A sequence of technical properties characterizes the robustness and applicability of these growth notions:

• Invariance under Differentiation: For meromorphic or analytic $f$,

$$\varrho_{(\alpha,\beta,\gamma)}[f'] = \varrho_{(\alpha,\beta,\gamma)}[f], \qquad \varrho_{(\alpha(\log),\beta,\gamma)}[f'] = \varrho_{(\alpha(\log),\beta,\gamma)}[f].$$

• Logarithmic Derivative Estimates: If $$\varrho_{(\alpha(\log),\beta,\gamma)}[f]=\varrho<\infty$$, for any $k\in\mathbb{N}$ and $\varepsilon>0$, outside a set of $r$ of finite logarithmic measure,

$$m\bigl(r, f^{(k)}/f\bigr) = O\Bigl( \exp\bigl\{\alpha^{-1}\bigl((\varrho+\varepsilon)\,\beta(\log\gamma(\tfrac{1}{1-r}))\bigr)\bigr\} \Bigr).$$

• Upper Bound for Equation Solutions: For nontrivial solutions $f$ to

$$f^{(k)} + A_{k-1}(z) f^{(k-1)} + \dots + A_0(z) f = 0,$$

where $A_j$ are analytic in $\Delta$,

$$\varrho_{(\alpha(\log),\beta,\gamma),M}[f] \le \max_{0\le j\le k-1} \varrho_{(\alpha,\beta,\gamma),M}[A_j].$$

These properties provide invariance and control under linear ODE manipulation and underpin further comparison theorems.

4. Representative Examples

To demonstrate the sharpness and universality of the framework, explicit constructions for arbitrary order and type are exhibited: $$f(z) = \exp\Bigl\{\alpha^{-1}(\tau^{1/\rho}\,\beta(\log\gamma(\frac{1}{1-z})))\Bigr\}$$ satisfies

$$\varrho_{(\alpha,\beta,\gamma),M}[f] = \rho, \qquad \tau_{(\alpha,\beta,\gamma),M}[f] = \tau.$$

Particular cases include:

• For $\alpha(x)=x$, $\beta(x)=x$, $\gamma(x)=x$, $f(z)=\exp\{c/(1-z)^\rho\}$ realizes classical order $\rho$ and type $c$.
• For $\alpha(x)=\log_{p+1}x$, $\beta(x)=x$, $\gamma(x)=x$, functions attain prescribed iterated $p$-order and type.

This shows that the $(\alpha,\beta,\gamma)$-structure encompasses all standard and iterated growth classifications of analytic functions in the unit disc.

5. Main Theorems Governing Solution Growth

Two central theorems relate the $(\alpha,\beta,\gamma)$-order and type of linear differential equation solutions to those of their analytic coefficients. For coefficients $A_j$, denote $\varrho_j = \varrho_{(\alpha,\beta,\gamma),M}[A_j]$, $\varrho_0 = \varrho_{(\alpha,\beta,\gamma),M}[A_0]$.

Theorem 2.1

Suppose

$$\max_{1 \leq j \leq k-1} \varrho_j < \varrho_0 < +\infty.$$

Then every nontrivial solution $f$ of $f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_0f=0$ satisfies

$$\varrho_{(\alpha(\log),\beta,\gamma),M}[f] = \varrho_{(\alpha,\beta,\gamma),M}[A_0].$$

Theorem 2.2

Suppose

$$\max_{1 \leq j \leq k-1} \varrho_j \leq \varrho_0 < +\infty,$$

$$0<\max\Bigl\{\tau_{(\alpha,\beta,\gamma),M}[A_j] : \varrho_j = \varrho_0\Bigr\} < \tau_{(\alpha,\beta,\gamma),M}[A_0] < +\infty.$$

Then every nontrivial solution $f$ of the same equation satisfies

$$\varrho_{(\alpha(\log),\beta,\gamma),M}[f] = \varrho_0.$$

A direct corollary states: when the coefficient with maximal $(\alpha,\beta,\gamma)$-order either strictly dominates or strictly dominates also in type, then every nontrivial solution has $(\alpha(\log),\beta,\gamma)$-order exactly this maximal value.

6. Context and Generalizations

The $(\alpha,\beta,\gamma)$-order and type generalize work by Biswas and the second author, subsuming all classical, iterated, and multidimensional growth order theories for analytic and meromorphic functions. Notions based on Nevanlinna’s characteristic apply when the function is merely meromorphic, and parallel “$M$”-definitions preserve equivalence in analytic settings. The results extend precise asymptotic control to higher-order, nonclassical entire and analytic function spaces, forming the basis for new studies on spectral theory and analytic operator theory in the unit disc (Arrouche et al., 28 Dec 2025).

7. References and Section Mapping

Definitions 1.1–1.3 present the core function-theoretic invariants. Proposition 1.1 gives analytic equivalence statements. Lemmas 3.1–3.9 and Remark 3.1 supply technical estimates, with Theorems 2.1, 2.2, and Corollary 2.1 collecting global growth results. For further details, see "Growth of $(\alpha,\beta,\gamma)$-order solutions of linear differential equations with analytic coefficients in the unit disc" (Arrouche et al., 28 Dec 2025).