Need a proof (real analysis)

[Categories+Sheaves]

Conjecture 2 can be used to prove conjecture 1 through induction.

Aww. Well, I just threw together an analytic proof using a,b>=1
WOLOG spz. b>=a>=1
Noting that Int(gx1-g on [0,1]) = 1 <--everyday integration
(a+b-1)g = (b)g + Int(gx1-g on [b,b+a-1])
(a)g = Int(gx1-g on [0,a]) = 1 + Int(gx1-g on [1,a])
And then Int(gx1-g on [1,a]) >= Int(gx1-g on [b,b+a-1]) for the usual reason, and the result follows.

It's to prove a lower bound for a function on a network that gives the 'energy' of the network.

Does the function have a name?
-----edited because CliveStaples has a sharper eye than I do-----

Also: shorter proof.
(a+b-1)g = (b)g + Int(gx1-g on [b,b+a-1]) =< (b)g + Int(gx1-g on [1,a]) = (b)g + (a)g -1
With the middle inequality coming form concavity.