# Which of the following conditions should be weaker?

Let $$g:[0, \infty) \rightarrow \mathbb{R}$$ and $$h:[0, \infty) \rightarrow \mathbb{R}$$ be nonnegative functions let us put $$\phi(y)=g(y) h(y)$$. Assume that the following condition is satisfied:

$$h$$ is nondecreasing and $$\phi$$ is strictly increasing with $$\phi([0, \infty))=[0,+\infty)$$.

Now on some work relating to the convergence of random variables I've been using the above condition along with the following condition which I'll refer to as A1:

There exists a constant $$b>0$$ such that $$\sum_{i=1}^{n} \frac{1}{h^{2}(i)} \leq b \frac{n}{h^{2}(n)}.$$

And I've been wondering if any of the two following conditions were weaker than A1:

A2: There exists $$p \ge 2$$, a constant $$b>0$$ such that $$\sum_{i=1}^{n} \frac{1}{h^{p}(i)} \leq b \frac{n}{h^{p}(n)}.$$

A3: There exists $$p \le 2$$, a constant $$b>0$$ such that $$\sum_{i=1}^{n} \frac{1}{h^{p}(i)} \leq b \frac{n}{h^{p}(n)}.$$

At the moment I'm reading on "slowly varying functions". I believe some of the tools there might help me reach a conclusion. If anyone knows any references or have any sort of comments that would be helpful then I would gladly appreciate.

$$\def\N{\mathbb{N}}\def\paren#1{\left(#1\right)}$$Here it is assumed that A1 to A3 are all to hold for any $$n \in \N_+$$. Since these conditions on $$h$$ are easily seen to be irrelevant to that on $$φ$$, and $$h(t) > 0$$ for $$t > 0$$, then define $$a_n = \dfrac{1}{(h(n))^2}$$ for $$n \in \N_+$$ and A1 to A3 are equivalent to:\begin{align*} &A_1\colon && \exists c > 0,\ \forall n \in \N_+,\ \frac{1}{n} \sum_{k = 1}^n a_k \leqslant ca_n,\\ &A_2\colon && \exists c > 0,\ q \geqslant 1,\ \forall n \in \N_+,\ \paren{ \frac{1}{n} \sum_{k = 1}^n a_k^q }^{\frac{1}{q}} \leqslant ca_n,\\ &A_3\colon && \exists c > 0,\ q \leqslant 1,\ \forall n \in \N_+,\ \paren{ \frac{1}{n} \sum_{k = 1}^n a_k^q }^{\frac{1}{q}} \leqslant ca_n, \end{align*} respectively.
Now, the inequality of power means shows that for any $$x_1, \cdots, x_n > 0$$,$$f(t) = \begin{cases} \min(x_1, \cdots, x_n); & t = -∞\\ \paren{ \dfrac{1}{n} \sum\limits_{k = 1}^n x_k^t }^{\frac{1}{t}}; & t ≠ 0\\ \paren{ \prod\limits_{k = 1}^n x_k }^{\frac{1}{n}}; & t = 0\\ \max(x_1, \cdots, x_n); & t = +∞ \end{cases}$$ is strictly increasing with respect to $$t$$ on $$[-∞, +∞]$$ unless all $$x_k$$'s are equal, thus in general$$A_2 \implies A_1 \implies A_3,$$ i.e. A3 is the weakest among the three.