(a*b/c)*Sqrt[m1^2 - (p*c/b)^2] -
  2 ArcTan[Sqrt[((p*c/b)^2 - m2^2)/(m1^2 - (p*c/b)^2)]] == 0

m2*b < p < m1*b

a = 2; c = 3; m1 = 1.5; m2 = 1.49;

With[{a = 2, c = 3, m1 = 1.5, m2 = 1.49},
  ContourPlot[
    (a*b/c)*Sqrt[m1^2 - (p*c/b)^2] -
      2 ArcTan[Sqrt[((p*c/b)^2 - m2^2)/(m1^2 - (p*c/b)^2)]] == 0,
    {p, -6, -4}, {b, 9.2, 10.5}]]