On Hayman's Problem on the Series Σ f(c_n/a_n) Formed with Second Partial Sums: A Comparison Theorem and a Characterization

Problem 7.31 of Hayman and Lingham's collection Research Problems in Function Theory, proposed by W. K. Hayman, concerns sequences with \(a_1>0\) and \(0\le a_n\le n\) and their partial sums \(b_n=a_1+\dots+a_n\), \(c_n=b_1+\dots+b_n\). It states that \(\sum(a_n/c_n)^\alpha<\infty\) for \(\alpha>\frac12\), and asks for which functions \(f\) the series \(\sum f(c_n/a_n)\) converges, and whether, under a regularity condition on \(f\), it converges whenever \(\sum f(n^2)\) does. We prove: if \(\sqrt x\,f(x)\) is non-negative and non-increasing and \(\sum f(n^2)<\infty\), then \(\sum f(c_n/a_n)\le 6\sum f(n^2)+(2\log(1/a_1)+2+5\log2)f(1)\) for every such sequence, and if \(\sum f(n^2)=\infty\), the series diverges for \(a_n=n\). For an arbitrary real function \(f\), the series converges for all such sequences if and only if \(\int^\infty\sup_{t\ge u}\sqrt t\,|f(t)|\,du/u<\infty\); the answer is the same under the normalizations \(a_n\le1\) and \(b_n\le n\). The proofs rest on the bound \(\sum_{c_n\le Xa_n}(a_n/c_n)^{1/2}<3\log X+2\log(1/a_1)+2+2\log2\) and on explicit sequences along which \(c_n/a_n\) is piecewise constant. Without a condition the comparison fails: there are a non-increasing \(C^\infty\) function, and a non-negative \(C^\infty\) function vanishing at all squares, with \(\sum f(n^2)<\infty\) and \(\sum f(c_n/a_n)=\infty\); the exponent \(\frac12\) cannot be lowered, and \(a_n\le n\) cannot be relaxed to \(a_n\le n\psi(n)\) with \(\psi\) unbounded. The problem quotes as known the case \(f(x)=x^{-\alpha}\) and an analogue for \(c_n/b_n\) attributed to D. Borwein. Borwein's note of 1965, which we read, concerns positive non-decreasing sequences \(b_n\le nk_n\) with \(k_n\ge1\): if \(xf(x)\) is positive and non-increasing and \(\int^\infty f<\infty\), then \(\sum k_nf(k_nc_n/b_n)<\infty\), hence \(\sum f(c_n/b_n)<\infty\) if \(b_n\le Kn\). We extend the latter statement to \(0\le a_n\le Kn\). We did not read Monthly problem 5167 of Barry and Hayman (1964–65), the problem list of Anderson, Barth and Brannan (1977), in which the problem was proposed according to the tables of the collection, the later lists, or Update 7.31 of the 2019 edition of the collection (those we tried were not accessible to us), and the origin of the statement for \(\alpha>\frac12\) is not identified. Our literature search found no earlier answer; a search that finds nothing is not a proof of novelty, no priority is claimed, and the arguments are elementary and may be known to specialists. The computations reported are tests and are not used in the proofs. This is an unrefereed note.