Research Problems in Function Theory — Problem 2.25

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $f$, write \[ M(r,f)=\max_{|z|=r}|f(z)|,\qquad m_0(r,f)=\min_{|z|=r}|f(z)|. \] For nonconstant $f$, its order is $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$; the lower order is defined using $\liminf$ instead. Every constant entire function, including the zero function, has order and lower order zero. In logarithmic modulus ratios, use the extended-real convention $\log0=-\infty$. Let $\rho\ge0$ be finite, with $\rho\notin\{n/2:n=1,2,\ldots\}$. Entire functions $f,g$ are linearly independent if no nonzero pair $u,v\in\mathbb C$ satisfies $uf+vg\equiv0$. Their Wronskian is $W(f,g)=fg'-gf'$.

Statement recorded in the supplied source.

Can linearly independent entire functions with $\rho(f)=\rho(g)=\rho$ have \[\rho\left(W(f,g)\right)<\rho?\] [S2] Update 2.25 corrects an older "distinct" formulation to "linearly independent" and reports Eremenko's negative answer. The positive half-integer orders are explicitly excluded in the recorded question. [S2]

Short English statement

Can the Wronskian grow with strictly smaller order than two independent entire functions of the same finite order, when that order is not a positive half-integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302025 (AMR-022-2025). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.25 and Update 2.25. https://arxiv.org/abs/1809.07200.