# Mathematica is saying both inequality are false [closed]

I'm trying to ask Mathematica to compare 2 different values in inequality but it keeps saying "False". I put the code I'm running below. Thanks in Advance for any hint on what is problematic in my writting !

Reduce[{(a^2 c1 - a^2 c3 + a^2 s θ1 -
a^2 s θ3)/(4 b^2 + 4 b c1 + 4 b c3 + 4 c1 c3 +
4 b s θ1 + 4 c3 s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) > q, a > 0,
1 > p > 0, s > 0, 1 > b > 0,
0 > θ1 > θ2 > θ3 > 1, c1 > 0, c2 > 0,
c3 > 0}, {c2, c3}] /.
q -> (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(4 b^2 +
4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 + 4 c3 s θ2 +
4 b s θ3 + 4 c2 s θ3 + 4 s^2 θ2 θ3)

(*False*)

• I think you left out some code at the start. Are you trying to Reduce? – flinty Sep 7 at 11:59
• Yes indeed sorry it missed a part of the code that I just replaced. I'm indeed trying to Reduce. – Xavier Koch Sep 7 at 13:57
• That's the issue: 0 > \[Theta]1 > \[Theta]2 > \[Theta]3 > 1  , a typo. Zero is not greater than one. – Akku14 Sep 7 at 15:17
• Thanks a lot, it makes sense now for the False ! – Xavier Koch Sep 7 at 15:24
• The issue can too be 0 > \[Theta]1 > \[Theta]2 > \[Theta]3 >- 1 . That much less effort and a closer match to an error. – Steffen Jaeschke Sep 7 at 16:27

You are setting

0 > θ1 > θ2 > θ3 > 1


Which cannot be, as you are asking the numbers to be smaller than 0 and larger than 1 at the same time.

Correct form should be:

 1 > θ1 > θ2 > θ3 > 0


0 < p < 1 && 0 < θ2 < 1 && θ2 < θ1 < 1 && 0 < θ3 < θ2 && c1 > 0 && s > 0 && 0 < b < 1 && a > 0 &&  0 < c2 < c1 + s θ1 - s θ2 && c3 > 0

   0 > θ1 > θ2 > θ3 >- 1

Reduce[{(a^2 c1 - a^2 c3 + a^2 s θ1 -
a^2 s θ3)/(4 b^2 + 4 b c1 + 4 b c3 + 4 c1 c3 +
4 b s θ1 + 4 c3 s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) > q, a > 0,
1 > p > 0, s > 0, 1 > b > 0,
0 > θ1 > θ2 > θ3 > -1, c1 > 0, c2 > 0,
c3 > 0}, {c2, c3}] /.
q -> (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(4 b^2 +
4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 + 4 c3 s θ2 +
4 b s θ3 + 4 c2 s θ3 + 4 s^2 θ2 θ3)

(0 < p < 1 && -1 < θ1 < 0 && -1 < θ3 < θ1 &&
0 < b < 1 &&
0 < s < -(b/θ3) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) <= -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
c3 > 0) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
0 < s < -(b/θ3) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && -(a^2/(4 b + 4 c1 + 4 s θ1)) < (
a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
0 < c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s == -(b/θ3) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) <= -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
c3 > 0) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s == -(b/θ3) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) > -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
0 < c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 &&
0 < b < 1 && -(b/θ3) <
s <= -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
0 < c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 &&
0 < b < 1 && -(b/θ3) <
s <= -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 &&
0 < b < 1 && -(b/θ3) <
s <= -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) <= (
a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) <= -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 &&
0 < b < 1 && -(b/θ3) <
s <= -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > 0 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) > -(a^2/(
4 b + 4 c1 + 4 s θ1)) &&
c2 > 0 && -b - s θ3 <
c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
0 < c1 < -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
0 < c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
0 < c1 < -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
0 < c1 < -b - s θ1 &&
a > 0 && (a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) <= (
a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) <= -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
0 < c1 < -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) > -(a^2/(
4 b + 4 c1 + 4 s θ1)) &&
c2 > 0 && -b - s θ3 <
c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
0 < c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) + (
4 s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) < (
a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > -b - s θ1 &&
a > 0 && (a^2 c1 + a^2 s θ1 - a^2 s θ3)/(
4 b^2 + 4 b c1 + 4 b s θ1 + 4 b s θ3 +
4 c1 s θ3 + 4 s^2 θ1 θ3) <= (
a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) <= -(a^2/(
4 b + 4 c1 + 4 s θ1)) && c2 > 0 &&
c3 > -b - s θ3) || (0 < p < 1 && -1 < θ1 <
0 && -1 < θ3 < θ1 && 0 < b < 1 &&
s > -(b/θ1) && θ3 < θ2 < θ1 &&
c1 > -b - s θ1 &&
a > 0 && (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3)/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) > -(a^2/(
4 b + 4 c1 + 4 s θ1)) &&
c2 > 0 && -b - s θ3 <
c3 < (a^2 c1 + a^2 s θ1 - a^2 s θ3 - (
4 b^2 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b c1 (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ1 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 b s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 c1 s θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3) - (
4 s^2 θ1 θ3 (a^2 c2 - a^2 c3 + a^2 s θ2 -
a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +
4 s^2 θ2 θ3))/(a^2 + (
4 b (a^2 c2 - a^2 c3 + a^2 s θ2 - a^2 s θ3))/(
4 b^2 + 4 b c2 + 4 b c3 + 4 c2 c3 + 4 b s θ2 +
4 c3 s θ2 + 4 b s θ3 + 4 c2 s θ3 +