Table 4 Checking conditions (18)\(_{r,s}\) for Example 6

\(\frac{r+3}{2}\le s\le 2r-3\)

r + s

\((p+3,\frac{q(r+3)+4r-6}{2(q+1)} )\)

α,β,γ ≥ 0

r ≥ 4, s ≥ 2r − 2

r + s

(p + 4,2r − 2 + q)

Q,α ≥ 0 ≥ γ,R

1 ≤ r ≤ 3, s ≥ 5

r + s

\((\frac{p+3}{p+1},q+5 )\)

Q,α ≥ 0 ≥ γ,R

s ≥ 4, r ≥ 2s − 2

r + s

(2s − 2 + p,q + 4)

γ ≥ 0 ≥ α,Q,R

1 ≤ s ≤ 3, r ≥ 5

r + s

\((p+5,\frac{q+3}{q+1} )\)

γ ≥ 0 ≥ α,Q,R

s = 1, \(r\le \frac{2n-3}{3}\)

\(\frac{3r+3+q}{2}\)

(p + 2,1)

Δ<0

s = 1, \(\frac{2n-2}{3}\le r\le n-3\)

\(\frac{2q(r+3)+3r+2}{2(q+1)}\)

(p + 4,1)

P,R ≥ 0

s = 1, r = n − 2

r + 2

(p + 3,1)

α,β,Q,R ≤ 0

s = 1, r = n − 1

r + 1

(r,1)

α = 0

r = 1, \(s\le \frac{2n-3}{3}\)

\(\frac{3s+3+p}{2}\)

(1,q + 2)

Δ<0

r = 1, \(\frac{2n-2}{3}\le s\le n-3\)

\(\frac{2p(s+3)+3s+2}{2(p+1)}\)

(1,q + 4)

P,R ≥ 0

r = 1, s = n − 2

s + 2

(1,q + 3)

P,Q ≥ 0 ≥ β,γ

r = 1, s = n − 1

s + 1

(1,s)

P,R ≥ 0