Using Maple we show that all four zeros of the 4^th degree polynomial

  lie in the disk 

> g:='g': z:='z':

g := z -> z^4 - 7*z - 1:

`g(z) ` = g(z);

> Zn := sort([fsolve(g(z), z, complex)]):

`The complex roots of  g(z) ` = g(z);

z1 := subs(z=Zn[1],z): z[1] = z1;

z2 := subs(z=Zn[2],z): z[2] = z2;

z3 := subs(z=Zn[3],z): z[3] = z3;

z4 := subs(z=Zn[4],z): z[4] = z4;

> print(abs(z[1]),`< 2  `, abs(z1)<2, evalb(evalf(abs(z1))<2));

print(abs(z[2]),`< 2  `, abs(z2)<2, evalb(evalf(abs(z2))<2));

print(abs(z[3]),`< 2  `, abs(z3)<2, evalb(evalf(abs(z3))<2));

print(abs(z[4]),`< 2  `, abs(z4)<2, evalb(evalf(abs(z4))<2));

> f:='f': z:='z':

f := z ->z^4:

`f(z) ` = f(z);

> A:='A': B:='B': s:='s': t:='t': y1:='y1': y1:='y1':

A := z -> abs(f(z)-g(z)):  `|f(z)-g(z)| ` = A(z);

B := z -> abs(f(z)):       `|f(z)| ` = B(z);

y1 := t -> A(2*exp(I*t)): `|f(z(t))-g(z(t))| ` = y1(t);

y2 := t -> B(2*exp(I*t)): `|f(z(t))| ` = y2(t);