A \(p\)-adic analog of Wirtinger's inequality (Q1923902)

In classical Fourier analysis, for a nonvanishing complex-valued \(C^1\)-function \(f\) on the unit circle one proves that \(f\) has no zeros on some arc whose length can be expressed in terms of Fourier coefficients of \(f\) and \(f'\). In the present paper, the following \(p\)-adic version is proved. (Let \(\mathbb{Q}_p\) be the \(p\)-adic number field, let \(\mathbb{Z}_n\) be its valuation ring, and let \(\mathbb{C}_n\) be the completion of the algebraic closure of \(\mathbb{Q}_p\).) Let \(f:\mathbb{Z}_p\to K\) be nonzero, uniformly differentiable, where \(K\supset \mathbb{Q}_p\) is a tame extension of \(\mathbb{Q}_p\). Then \(f\) has no zeros on some coset of \(p^{M(f)}\mathbb{Z}_p\). Here \(M(f)\) is some well-defined constant expressed in terms of the Fourier transform \(\widehat f\) (which is defined on the `circle' \(\{\omega\in\mathbb{C}_p: \omega^{p^n}=1\) for some \(n\in\mathbb{N}\}\). A connection between \(M(f)\) and the behaviour of \(f'\) is established. Numerous examples are given.