Table 6 Numerical results of \(\mathcal{O}_{j}^{*}\) and \(\Upsilon _{j}\), \(j=1,2,3,4\), for \(\mathfrak{t}\in [0.2,0.85]\) in Example 6.3 when \(\mathbb{G}_{1}(\mathfrak{t}) = 2^{\mathfrak{t}}\), \(\mathbb{G}_{2}(\mathfrak{t})= \mathfrak{t}\), \(\mathbb{G}_{3}(\mathfrak{t}) = \ln \mathfrak{t}\), \(\mathbb{G}_{4}( \mathfrak{t} )= \sqrt{\mathfrak{t}}\)

0.30

4.4298

0.0000

3.6823

2.7630

0.8289

2.7643

3.6643

2.7630

0.40

2.1565

0.0002

1.8284

2.3073

0.4244

2.3117

1.9820

2.3072

0.50

1.3460

0.0007

1.1746

1.9969

0.2912

2.0055

1.3782

1.9967

0.60

0.9321

0.0016

0.8424

1.7650

0.2254

1.7782

1.0636

1.7643

0.70

0.6836

0.0031

0.6432

1.5843

0.1863

1.6023

0.8694

1.5828

0.80

0.5200

0.0055

0.5116

1.4406

0.1602

1.4632

0.7371

1.4378

0.81

0.5067

0.0058

0.5009

1.4279

0.1581

1.4509

0.7261

1.4249

0.82

0.4939

0.0061

0.4905

1.4154

0.1560

1.4389

0.7155

1.4123

0.83

0.4815

0.0065

0.4805

1.4033

0.1540

1.4272

0.7052

1.4000

0.84

0.4696

0.0068

0.4709

1.3913

0.1521

1.4157

0.6952

1.3879

0.85

0.4580

0.0072

0.4615

1.3797

0.1502

1.4045

0.6855

1.3761