Research Problems in Function Theory — Problem 5.44

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.For $\alpha,\beta>0$ and $|x|=1$, define coefficients by the analytic branches normalized to equal $1$ at $z=0$: \[(1+xz)^\alpha(1-z)^{-\beta}=\sum_{m\ge0}A_m^{\alpha,\beta}(x)z^m. \] Equivalently, $A_m^{\alpha,\beta}(x)=\sum_{j=0}^m\binom\alpha j x^j(\beta)_{m-j}/(m-j)!$, where $\binom\alpha j=\alpha(\alpha-1)\cdots(\alpha-j+1)/j!$, $(\beta)_k=\beta(\beta+1)\cdots(\beta+k-1)$, and both empty products are $1$. For $\alpha=3/2$ and $\beta=1/100$ one has $A_3^{\alpha,\beta}(1)=-95583/2000000<0$, so a bound by this signed number already fails at $x=1$.

Statement recorded in the supplied source.

For $0<\alpha<1$ and $\beta=1$, does \[|A_{2n+1}^{\alpha,1}(x)|\le |A_{2n+1}^{\alpha,1}(1)|\qquad(n\ge2,\ |x|=1)\] always hold? More generally, investigate the same odd-coefficient domination for arbitrary $\alpha,\beta>0$: \[|A_{2n+1}^{\alpha,\beta}(x)|\le |A_{2n+1}^{\alpha,\beta}(1)|\qquad(n\ge2,\ |x|=1).\] The source separately poses the literal first general odd-coefficient inequality \[|A_3^{\alpha,\beta}(x)|\le A_3^{\alpha,\beta}(1)\qquad(\alpha,\beta>0,\ |x|=1),\] with a signed right-hand side; this historical subsidiary inequality is distinct from the modulus comparison above. [S2]

Short English statement

Does the coefficient at $x=1$ maximize the modulus of the relevant odd coefficients, first for the original $\alpha$ range and then for arbitrary positive $\alpha$ and $\beta$? The separately posed cubic-coefficient inequality uses a signed right-hand side and fails for some positive parameters.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305044 (AMR-022-5044). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.44, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.