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.
 A: $\def\N{\mathbb{N}}\def\paren#1{\left(#1\right)}$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.
