Following are some interesting problems from The American Mathematical Monthly.

Problem 1

See https://www.tandfonline.com/doi/pdf/10.1080/00029890.2018.1470415.

Let $f$ be a $C^1$ real-valued function on $[0, 1]$, infinitely differentiable at $x=0$, and such that $f^{(n)}(0) = 0$ for every $n \in \mathbb{N}$. If

$$ |xf'(x)| \le C \cdot |f(x)| \text{ for some } C > 0 \text{ and every } x \in [0, 1], $$

then $f(x) = 0$ for every $x \in [0, 1]$.

Hint: Use

$$ (x^{-a-1}f^2(x))' = -(a+1)x^{-a-2}f^2(x) + 2x^{-a-1}f(x)f'(x) $$


Problem 2

See https://www.tandfonline.com/doi/pdf/10.1080/00029890.2018.1507205.

Let $D$ be a convex subset of a real vector space, and $f: D\rightarrow \mathbb{R} \cup \{+\infty\}$ be a radially lower semicontinuous function, that is, $$f(x) \le \liminf\limits_{t \rightarrow 0} f(x + t (y - x)).$$ Then $f$ is convex iff for all $x, y \in D$, there exsits $\lambda = \lambda(x, y) \in (0, 1)$ that satisfies inequality $$f(\lambda x + (1-\lambda) y) \le \lambda f(x) + (1-\lambda) f(y).$$

Hint: Fix $x, y \in D$ and define the set $$ S \triangleq \{\lambda \in [0,1]: f(\lambda x + (1-\lambda) y) \le \lambda f(x) + (1-\lambda) f(y)\}. $$ Prove that
(a) $S$ is a nonempty compact subset of $[0, 1]$.
(b) $(\lambda_1, \lambda_2) \cap S \neq \emptyset$ for all $\lambda_1, \lambda_2 \in S$ with $\lambda_1 < \lambda_2$.


Problem 3

See https://www.tandfonline.com/doi/pdf/10.1080/00029890.2019.1537761.

Let $\{ x_n \}$ be a positive nonincreasing real sequence such that

$$\sum_n x_n = +\infty, ~~~~ \lim_{n \rightarrow \infty} x_n = 0.$$

Denote

$$L \triangleq \liminf_{n \rightarrow \infty} \frac{x_{n+1}}{x_n}.$$

Prove that
(a) if $L > ½$, then there exists a constant $\theta \in (0, 1)$ such that, for each $l > 0$, there is a subsequence $\{ x_{n_k} \}$ for which

$$\sum_k x_{n_k} = l \text{ and } x_{n_k} = \mathcal{O}(\theta^k).$$

(b) if $L > \frac{\sqrt{5} - 1}{2}$, then for each $l > 0$ and for each $\epsilon > 0$, there exists a subsequence $\{x_{n_k}\}$ for which

$$\sum_k x_{n_k} = l \text{ and } x_{n_k} = \mathcal{O}(\sqrt{1 + \epsilon - L}^k).$$

Reference

[1] George Stoica (2018) When Must a Flat Function be Identically 0?, The American Mathematical Monthly, 125:7, 648-649, DOI: 10.1080/00029890.2018.1470415
[2] Paolo Leonetti (2018) A Characterization of Convex Functions, The American Mathematical Monthly, 125:9, 842-844, DOI: 10.1080/00029890.2018.1507205
[3] Paolo Leonetti (2019) Convergence Rates of Subseries, The American Mathematical Monthly, 126:2, 163-167, DOI: 10.1080/00029890.2019.1537761