Endpoint validity for the Wiman–Valiron inequalities

Determine whether either endpoint holds in general: namely, whether the estimate with only the leading factor A(t)√log(A(t)) (corresponding to L≡1) is valid, or whether the iterated-logarithm estimate remains valid when δ=0.

Background

The paper proves Wiman–Valiron estimates outside sets of finite logarithmic measure with a leading factor A(t)√log(A(t)) and additional iterated-logarithm factors, where the final exponent is 1+δ with δ>0. The sharpness example shows that the leading factor itself cannot generally be reduced, but it does not resolve whether the additional factors are necessary.

The remark identifies two possible endpoint improvements: removing all additional iterated-logarithm factors by taking L≡1, or setting δ=0 in the stated iterated-logarithm estimate. Neither endpoint satisfies the integrability condition required by the paper’s method, so the validity of either endpoint for general unbounded analytic functions remains unresolved.

References

In fact, eq:sharp_count shows that this example satisfies $$M(f,t)\leq C A(t)\sqrt{\log\, (A(t))}\,,$$ for every $t$ sufficiently close to $1$, without additional logarithmic factors. This stronger endpoint corresponds to $L\equiv1$, whereas setting $\delta=0$ in eq:disk_iterated retains the iterated-logarithm factors. Neither choice satisfies eq:L_integral. This limits the method without deciding whether either endpoint holds in general.

eq:sharp_count:

M(f,t)∼2π A(t)log⁡ (A(t)), as t↑1.M(f,t)\sim2\sqrt\pi\,A(t)\sqrt{\log\, (A(t))},\qquad \text{ as } t\uparrow1.

eq:disk_iterated:

M(f,t)≤C A(t)(log⁡ (A(t)))1/2(∏j=2n−1log⁡j (A(t)))(log⁡n (A(t)))1+δ, for any t∈(0,1)∖E.M(f,t)\leq C \, A(t)\bigl(\log\, (A(t))\bigr)^{1/2} \left(\prod_{j=2}^{n-1}\log_j\, (A(t))\right) \bigl(\log_n\, (A(t))\bigr)^{1+\delta}, \qquad \text{ for any } t\in(0,1)\setminus E.

eq:L_integral:

∫y0∞yyL(y)<∞,\int_{y_0}^{\infty}\frac{ y}{yL(y)}<\infty,

— Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure  (2609.28814 - Maciá, 23 Sep 2026) in Remark following Proposition 5.1, Section 5.1 (The leading logarithmic exponent)