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The functions $f_a$ considered are (up to a multiplicative constant) the Fourier transforms of even tent functions. The real linear combinations of even tent functions yield all even continuous and pi …

Partial answer. If $g$ is known to be a non-negative function, Karameta Tauberian theorem relates the behavior of $G = \int_0^\cdot g$ at infinity (resp. at 0) to the behavior of $\mathcal{L}g$ at $0$ …

For $-1<h<1$, $$(1+h)^{3/2}+(1-h)^{3/2} = 2 \sum_{n=0}^{+\infty}{3/2 \choose 2n}h^{2n},$$ where $${3/2 \choose 2n} = \prod_{k=1}^ {2n} \frac{5/2-k}{k}.$$ For $n \ge 1$, since $(-1)^{2n-2}=1$, we get $ …

Consider the norm $\Vert\cdot\Vert$ on $\mathbb{C}^2$ defined by $\Vert z \Vert^2 := |z_1|^2+|z_1+z_2|^2$. Then $$\left\Vert \left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = 1,$$ $$\le …

If my computations are correct, there is a root of the form $x=1+\sin^2(\frac\theta2)$ with: $$\theta \in \left(\frac{k\pi}N,\frac{k\pi}N+\frac\pi{2N}\right) \text{ if }k < \frac{N}3-\frac12$$ $$\thet …

Possible way to find such a function $f$ Since $$\prod_{k=1}^n \frac{ke}{n} = n!\Big(\frac{e}{n}\Big)^n \sim \sqrt{2\pi n} \text{ as } n \to +\infty,$$ it is sufficient to find a continuous function $ …

This question would be possibly at a better place on MathStack Exchange. Yet, once the statement of the question is corrected (the functions $y_n$ need to be defined on $\mathbb{R_+}$ and not only on …

I complete the reformulation given by Andrea Marino and give another counterexample. First, the inequality of the beginning can be written $$\forall x \in (0,1), \quad \frac{f'(x)}{1-f^2(x)} \ge \frac …