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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?

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

I know this is not an answer but I am under 50 reputation and con not to post a comment

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