Are these inequalities for primes equivalent? Let $p_n$ be the $n$th prime, let $L$ consist of the primes satisfying $p_{n+2} - 2p_{n+1} + p_{n} > 0$, and let $Q$ consist of the primes satisfying $p_{n+1}^2 < p_{n}p_{n+2}.$  Is $L=Q$?
Background:  The arithmetic-geometric mean inequality implies $Q \subset L$, but the reverse containment, if true, may be harder to prove.
The inequalities have been verified equivalent by computer for the first $4,000,000$ primes (i.e., up to $67,867,967$).  The number of these primes for which the inequalities hold is $1,941,180$, which is more than $48.5$ percent of the total.  Further experimentation leads me to guess that a limiting ratio exists.
It's been checked that for $1<n<4,000,000$, when the two inequalities hold we also have  $$p_{n+1}^2 - p_{n}p_{n+2} > p_{n+2} - 2p_{n+1} + p_{n}.$$          
 A: I am using Anthony Quas's reformulation to restate the problem.  Letting the $n$th
prime gap be given by $d_n= p_{n+1} - p_n$, and given $n$ we relabel $a=d_n$ and
$b=d_{n+1}$, we look at $L$ as the set of $p_{n+1}$ in which $n$ satisfies $b-a\gt 0$
and we look at $Q$ as the set of $p_{n+1}$ in which $0 \lt (b-a)p_{n+1} -ab$.  As
 noted in the post, $p_{n+1}$ not in $L$ readily implies $p_{n+1}$ not in $Q$; it is
natural to ask if $p_{n+1} \in L$ implies $p_{n+1} \in Q$.   The question further
notes that when $p_{n+1}$ is observed to be in $L$ one also has
$ ab - p_{n+1}(b-a) \gt (b-a)$, which I think should be reversed as $7=p_4 \in L$ but
$49 - 55 \lt 11 - 14 + 5$.
If the last inequality is reversed, it says $ab \lt ( p_{n+1} + 1)(b-a)$.  If $p_{n+1} \in Q$, then
clearly this last inequality holds.  Finally as Anthony Quas observes, if
$p_{n+1} \in L\setminus Q$ then $ab \geq (b-a)p_{n+1}$ and $b \gt a$, so one would have $ab \geq 2p_{n+1}$ if $n \gt 1$.
The formulation shows that the basic question is about consecutive prime gaps, and that $L$
is different from $Q$ only when a large gap $d_n$ is greater than the square root of an
adjacent prime $p_n$.  Such large gaps have not been observed for $n \gt 30$ 
(so $p_n \gt 113$), and the stronger inequality $ab \geq 2p_{n+1}$ is also not
observed for $1 \lt n \leq 30$.  The case $n=1$ is left to the reader, as is the
conclusion that $L$ properly containing $Q$ would violate expectations and
many conjectures in prime number theory.
