Express it as
(* http://community.wolfram.com/groups/-/m/t/508791 *)
Clear[vayF, a, b, c, x]
vayF[a_, b_, c_, x_] :=
b*c^(1 \[Minus]
1/a)*((1/x - 1)^(1/a)*(a - 1) - (1/x - 1)^(1/a - 1)) + a /;
0 < a < 1 && 0 < b < 1 && 0 < c < 1 && 0 < x < 1
do a simplification
In[43]:= FullSimplify[b c^(1 - 1/a) ((1/x - 1)^(1/a) (a - 1) - (1/x - 1)^(1/a - 1)) + a]
Out[43]= a + (b c^((-1 + a)/a) (1 + a (-1 + x)) (-1 + 1/x)^(1/a))/(-1 + x)
check the existence of positive values at all
In[42]:= {#, (vayF @@ #) >= 0} & /@ Tuples[{1/99, 99/100}, 4]
Out[42]= {{{1/99, 1/99, 1/99, 1/99}, False}, {{1/99, 1/99, 1/99, 99/100}, False},
{{1/99, 1/99, 99/100, 1/99}, False}, {{1/99, 1/99, 99/100, 99/100}, True},
{{1/99, 99/100, 1/99, 1/99}, False}, {{1/99, 99/100, 1/99, 99/100}, False},
{{1/99, 99/100, 99/100, 1/99}, False}, {{1/99, 99/100, 99/100, 99/100}, True},
{{99/100, 1/99, 1/99, 1/99}, True}, {{99/100, 1/99, 1/99, 99/100}, True},
{{99/100, 1/99, 99/100, 1/99}, True}, {{99/100, 1/99, 99/100, 99/100}, True},
{{99/100, 99/100, 1/99, 1/99}, False}, {{99/100, 99/100, 1/99, 99/100}, False},
{{99/100, 99/100, 99/100, 1/99}, False}, {{99/100, 99/100, 99/100, 99/100}, True}}}
make a picture of the 4-D cube where the positive (resp. negative) corners are, then go along the edges, where the shift happens, then look into the faces .... to get an idea, step by step.