The triangle inequality says that the sum of any two sides of a triangle is greater than the third side but we do not know by how much is it actually greater. To improve the triangle inequality, we can ask for explicit lower bounds on $a+b-c$ where, $a,b$ and $c$ are the sides of the triangle. For a fixed semi-perimeter $s$ and a fixed area $A$, we can find infinitely many triangles of sides $a,b$ and $c$ and since the bounds has to be independent of specific values $a,b$ and $c$ we can ask for explicit lower bounds in terms of $s$ and $A$.
Conjecture: Let $a \le b \le c$ be the sides of a triangle whose semi-perimeter is $s$ and area is $A$; then, $$ a+b-c \ge \frac{8A^2}{s^3} \tag 1$$ $$ c+a-b \ge \frac{16A^2}{s^3} \tag 2$$ $$ b+c-a \ge \frac{18A^2}{s^3} \tag 3$$
Further, if $a \le b \le c$ are the sides of a right triangle with $a^2+b^2 = c^2$ then the above lower bounds can be improved to $$ a+b-c \ge \frac{(6+4\sqrt{2})A^2}{s^3} \tag 4$$ $$ c+a-b \ge \frac{(11+5\sqrt{5})A^2}{s^3} \tag 5$$ $$ b+c-a \ge \frac{(14+10\sqrt{2})A^2}{s^3} \tag 6$$
with equality occurring only if the triangle is degenerate.
Notice that the RHS of $(1)$ is the smallest hence it represents the unconditional lower bound for the triangle inequality which occurs when we subtract the largest side from the sum of the two smaller side. However, we if subtract the side which is not the largest then the lower bound can be improved depending upon weather we subtract the smallest side or the next smallest side. $(1)$ was proved in the answer to this related question and is actually the first few terms of an asymptotic series in $A/s^2$. Inequalities $(2)$ and $(3)$ were obtained by experimental methods by randomly inscribing billions of triangles in a unit circle and in all likelihood, they have they own asymptotic series.
Question: Can we prove inequality $(2)$ and $(3)$?