Notations. Denote by $S^{-}\left\{a_0,a_1,\ldots,a_m\right\}$ the number of strict sign changes in the indicated sequence $\left\{a_j\right\}_{j=0}^{m}$ of real numbers (i.e., when counting the sign changes, zero terms are discarded). Let $Z\left(f\right)$ be the number of zeros (counting their multiplicities as well) of a $C^{\infty}$ real-valued real function $f$ over its definition domain $\operatorname{Dom}\left(f\right)$.
One of J. J. Sylvesters's results can be formulated in the next
Theorem (Sylvester's classic result). Let $\left\{\varphi_{j}\right\}_{j=0}^{m}$ be a sequence of strictly increasing real numbers and assume that the polynomial \begin{equation} p\left(u\right) = \sum_{j=0}^{m} a_j \left(u - \varphi_{j}\right)^{q},~u \in \mathbb{R} \label{eq:q_polynomial} \end{equation} does not vanish identically, where $q\in\mathbb{N}_{\geq 1}$ and the sequence $\left\{a_j\right\}_{j=0}^{m}$ consists of non-zero real numbers. Then, \begin{equation} Z\left(p\right) \leq S^{-}\left\{a_0, a_1, \ldots, a_{m},\left(-1\right)^{q}a_0\right\}. \label{eq:Sylvester_s_classic_inequality} \end{equation}
The theorem above can be found, e.g., in:
- [J. J. Sylvester, 1908. Collected mathematical papers, Vol. II, Cambridge, p. 408];
- [Gy. Pólya, I. J. Schönberg, 1958. Remarks on de la Vallée Poussin means and convex conformal maps of the circle, Pacific Journal of Mathematics, 8(2):295$-$334, p. 299, https://projecteuclid.org/euclid.pjm/1103040105];
- [S. Karlin, 1968. Total positivity, Vol. I, Stanford University Press, California, Lemma 3.1, p. 466].
Question. I would be interested in the validity of a special continuous extension of Sylvester's classic result. Can anyone either prove or disprove the next statement?
Statement to be proved/disproved. Let $\left\{ \varphi_{j}\right\} _{j=0}^{m}$ be a sequence of strictly increasing real numbers and consider the function \begin{equation} p\left( u\right) =\sum_{j=0}^{m}a_{j}\left( \left( u-\varphi_{j}\right) ^{2}\right) ^{q+\alpha},~u\in\mathbb{R},\label{eq:q_alpha_function}% \end{equation} where $q\in\mathbb{N}_{\geq1}$, $\alpha\in\left( 0,1\right) $, while the sequence $\left\{ a_{j}\right\} _{j=0}^{m}$ consists of non-zero real numbers. Then, \begin{equation} Z\left( p\right) \leq S^{-}\left\{ a_{0},a_{1},\ldots,a_{m},a_{0}\right\}. \end{equation}
Remarks. Note that, for $\alpha \in \left\{0,1\right\}$, the statement in the question is true, since in these cases the function $p$ becomes a polynomial, and one can apply Sylvester's classic result for the even powers $2q$ and $2(q+1)$. The function $p$ is $C^\infty$ almost everywhere, which can make it difficult to study the higher order zeros of $p$. It would suffice to prove/disprove the statement by counting the zeros of $p$ as simple points (in the obvious geometric sense) over $\mathbb{R}$.
Thank you for your time and energy for dealing with my question.
