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Yang's periodicity conjecture for products of entire functions and derivatives

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Let ff be a transcendental entire function and let kk be a positive integer. For k≥2k\geq 2, suppose that either ff(k)ff^{(k)} is periodic or

ff(k)=h2ff^{(k)}=h^2

for some entire function hh. Yang's periodicity conjecture. In either case, ff must be periodic. The conjecture is motivated by differential equations such as f(z)f′′(z)=−sin⁡2zf(z)f^{\prime\prime}(z)=-\sin^2 z, whose entire solutions are f(z)=±sin⁡zf(z)=\pm\sin z, and by the analogous cosine equation. The source also notes that the corresponding assertion involving ff′ff' need not hold, so the restriction k≥2k\geq 2 in the second alternative is essential.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Yang's periodicity conjecture for products of entire functions and derivatives

    Let ff be a transcendental entire function and let kk be a positive integer. Yang's periodicity conjecture. If k≥1k\geq 1 and ff(k)ff^{(k)} is periodic, then ff must be periodic. The conjecture concerns when periodicity of a differential product forces periodicity of the underlying transcendental entire function. The source separately gives examples showing that the analogous assertion for ff′=h2ff'=h^2 does not imply periodicity, while this periodic-product formulation is stated for all k≥1k\geq 1.

    source: Molla Basir Ahamed, “On the Conjecture of C. C. Yang and periodicity of meromorphic functions”, arXiv:2606.24918 (2026).

Sources & referencesView supporting material

Primary source

Molla Basir Ahamed, “On the Conjecture of C. C. Yang and periodicity of meromorphic functions”, arXiv:2606.24918 (2026).

complex-analysismeromorphic-functionsperiodicity

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