# Check an inequality if one condition holds

A = Abs[-1 + 1/(1 + β)] + Abs[-((1 + 2 α + Sqrt[1 + 4 α + 4 α β])/(2 α + 2 α β))]

Is A <1 if Abs[β]<Abs[1+4 α] ?

How can I check the above statement in Mathematica? Does the TrueQ and If functions work?

• What is assumed about $\alpha$ and $\beta$? Nov 14 '15 at 10:20
• Clearly, it is not true for {α -> 10, β -> 1} or many other instances. Nov 14 '15 at 12:55
• $\alpha$ and $\beta$ are complex numbers. Nov 14 '15 at 12:57
• The Reduce command says what for real parameters the statement is false: Reduce[{A < 1, Abs[[Beta]] < Abs[1 + 4 [Alpha]]}, {[Alpha], [Beta]}, Reals] Nov 14 '15 at 14:34

maybe this give you some idea:

A[α_, β_] :=
Abs[-1 + 1/(1 + β)] +
Abs[-((1 + 2 α +
Sqrt[1 + 4 α + 4 α β])/(2 α +
2 α β))];
f[a_, b_] := If[(Abs[b] < Abs[1 + 4 a]) && (A[a, b] < 1), {a, b}];
Tally@Flatten@
Table[f[RandomInteger[{1, 10000}],
RandomInteger[{1, 10000}]], {1000}]