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As you suggest, let me consider the case $f \equiv 1$. Without loss of generality, assume also that $a = 0$ and $b = 1$. Let $\sigma := o(1)/o(0) \in (0,1)$. The problem becomes $$ \sup_{v \in C^1([0, …

Problem statement Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$. Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\omeg …

Since $f$ is monotically decreasing to $f(+\infty) = 0$, $f$ is nonnegative. Hence, to prove that $\int_a^{+\infty} f$ converges, it is sufficient to prove that, $\int_a^b f$ is uniformly bounded abov …