PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{4}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
659,385 boxes
·
329,693 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{4}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
655,217 boxes
·
327,609 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{4}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
661,597 boxes
·
330,799 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
785,639 boxes
·
392,820 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{4}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
1,079,207 boxes
·
539,604 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{7}{16}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
37 boxes
·
19 leaves
· report
PROVED
certified enclosure
known theorem
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{1}{16} + 0.01\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
Kumar & Verma 2025 (Filomat)
10,987 boxes
·
5,494 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{7}{16}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{8}\)
over Bean S*_𝔅,
\(\varphi(z)=\sqrt{1+\tanh z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = 1\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
17 boxes
·
9 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{4}\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
59 boxes
·
30 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
46,672,735 boxes
·
23,336,368 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
564,869 boxes
·
282,435 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
546,085 boxes
·
273,043 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = 1\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
33 boxes
·
17 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = 1\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Bell S*_B,
\(\varphi(z)=e^{e^{z}-1}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{11}{9}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
23 boxes
·
12 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{7}{9}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
161 boxes
·
81 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{2}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
559,217 boxes
·
279,609 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{2}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
577,471 boxes
·
288,736 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{2}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
1,142,741 boxes
·
571,371 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{7}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{7}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{3}\)
over Cardioid S*_C,
\(\varphi(z)=1+\tfrac{4z}{3}+\tfrac{2z^{2}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = 1\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
17 boxes
·
9 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{4}\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
59 boxes
·
30 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
46,485,173 boxes
·
23,242,587 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
1,493,955 boxes
·
746,978 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
1,455,661 boxes
·
727,831 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = 1\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
33 boxes
·
17 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = 1\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Cardioid S*_℘,
\(\varphi(z)=1+z\,e^{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{5}{6}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
37 boxes
·
19 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{7}{12}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
153 boxes
·
77 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
1,583,769 boxes
·
791,885 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
1,300,267 boxes
·
650,134 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
1,485,565 boxes
·
742,783 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{7}{6}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
13 boxes
·
7 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{7}{6}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Cissoid S*_{cs},
\(\varphi(z)=1+\frac{z}{(1-z)(1+z/3)}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{4}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
240,455 boxes
·
120,228 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{4}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
212,983 boxes
·
106,492 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{4}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
212,261 boxes
·
106,131 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
212,153 boxes
·
106,077 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{4}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
248,973 boxes
·
124,487 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{17}{48}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
73 boxes
·
37 leaves
· report
PROVED
certified enclosure
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{1}{16} + 0.01\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
9,943 boxes
·
4,972 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{17}{48}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{8}\)
over Cosh-sqrt,
\(\varphi(z)=\cosh\sqrt{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{4}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
45 boxes
·
23 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
505,493 boxes
·
252,747 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
491,725 boxes
·
245,863 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
574,155 boxes
·
287,078 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{5}{4}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
7 boxes
·
4 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{5}{4}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Crescent,
\(\varphi(z)=z+\sqrt{1+z^{2}}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{8}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
928,369 boxes
·
464,185 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{8}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
770,713 boxes
·
385,357 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{3}{8}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
697,357 boxes
·
348,679 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{8}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
825,453 boxes
·
412,727 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{3}{8}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
1,225,845 boxes
·
612,923 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{27}{32}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
13 boxes
·
7 leaves
· report
PROVED
certified enclosure
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{5}{32} + 0.01\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
9,983 boxes
·
4,992 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{27}{32}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{3}{16}\)
over Epicycloid S*_{3ℒ},
\(\varphi(z)=1+\tfrac{3z}{4}+\tfrac{z^{3}}{4}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{7}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
574,341 boxes
·
287,171 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{7}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
873,609 boxes
·
436,805 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{3}{7}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
758,381 boxes
·
379,191 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{7}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
978,541 boxes
·
489,271 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{3}{7}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
1,779,045 boxes
·
889,523 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{54}{49}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{54}{49}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{3}{14}\)
over Epicycloid S*_{6ℒ},
\(\varphi(z)=1+\tfrac{6z}{7}+\tfrac{z^{6}}{7}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{4}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
45 boxes
·
23 leaves
· report
PROVED
sharp ✓
1 in literature
closed-form engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
505,493 boxes
·
252,747 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
491,725 boxes
·
245,863 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
574,155 boxes
·
287,078 leaves
· report
PROVED
sharp ✓
1 in literature
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{5}{4}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
7 boxes
·
4 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{5}{4}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Exponential S*_e,
\(\varphi(z)=e^{z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{5}{12}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
530,163 boxes
·
265,082 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{5}{12}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
855,995 boxes
·
427,998 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{5}{12}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
750,353 boxes
·
375,177 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{5}{12}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
947,769 boxes
·
473,885 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{5}{12}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
1,620,345 boxes
·
810,173 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{25}{24}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
7 boxes
·
4 leaves
· report
PROVED
certified enclosure
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{3}{16} + 0.02\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
9,671 boxes
·
71,753 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{25}{24}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{5}{24}\)
over Four-leaf S*_{4L},
\(\varphi(z)=1+\tfrac{5z}{6}+\tfrac{z^{5}}{6}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over janowski_A0.5_B-0.5
491,725 boxes
·
245,863 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{5}{8}\)
over janowski_A0.75_B-0.25
81 boxes
·
41 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over janowski_A0.75_B-0.25
593,153 boxes
·
296,577 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over janowski_A0.75_B-0.25
424,987 boxes
·
212,494 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over janowski_A0.75_B-0.25
441,597 boxes
·
220,799 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over janowski_A0.75_B-0.25
767,533 boxes
·
383,767 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{11}{8}\)
over janowski_A0.75_B-0.25
7 boxes
·
4 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{4}\)
over janowski_A1_B-0.5
604,613 boxes
·
302,307 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{3}{4}\)
over janowski_A1_B-0.5
90,330,087 boxes
·
45,165,044 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over janowski_A1_B0
429,353 boxes
·
214,677 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over janowski_A1_B0
510,565 boxes
·
255,283 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{4}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
212,427 boxes
·
106,214 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{4}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
211,615 boxes
·
105,808 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{4}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
213,339 boxes
·
106,670 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
252,715 boxes
·
126,358 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{4}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
345,703 boxes
·
172,852 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{7}{16}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
37 boxes
·
19 leaves
· report
PROVED
certified enclosure
known theorem
4 in literature
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{1}{16} + 0.01\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
Lee, Ravichandran & Supramaniam 2013 (#455)
10,563 boxes
·
5,282 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{7}{16}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{8}\)
over Lemniscate S*_L,
\(\varphi(z)=\sqrt{1+z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{10}\)
over limacon_0.3
330,771 boxes
·
165,386 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{10}\)
over limacon_0.3
282,521 boxes
·
141,261 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{10}\)
over limacon_0.3
264,957 boxes
·
132,479 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over limacon_0.5
593,153 boxes
·
296,577 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over limacon_0.5
441,597 boxes
·
220,799 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{3}{4}\)
over limacon_0.707
2,775 boxes
·
1,388 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = sqrt(2)/2\)
over limacon_0.707
616,483 boxes
·
308,242 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
429,353 boxes
·
214,677 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
358,741 boxes
·
179,371 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
510,565 boxes
·
255,283 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{2}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Nephroid S*_{Ne},
\(\varphi(z)=1+z-\tfrac{z^{3}}{3}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{3}{4}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
45 boxes
·
23 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
36,194,331 boxes
·
18,097,166 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
1,336,707 boxes
·
668,354 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
1,286,205 boxes
·
643,103 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
1,485,119 boxes
·
742,560 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{5}{4}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
7 boxes
·
4 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{5}{4}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Non-convex S*_{nc},
\(\varphi(z)=\frac{1+z}{\cos z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{4}\)
over order_0.25
656,729 boxes
·
328,365 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over order_0.5
564,869 boxes
·
282,435 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{4}\)
over order_0.75
13,986,531 boxes
·
6,993,266 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over order_0.75
378,521 boxes
·
189,261 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{1}{4}\)
over order_0.75
330,577 boxes
·
165,289 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = (8/3 + 32/pi**2)/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
45 boxes
·
23 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = (48 + 8*pi**2)/(3*pi**4)\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
873 boxes
·
437 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = 4/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
545,823 boxes
·
272,912 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = 4/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
418,917 boxes
·
209,459 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = 4/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
464,471 boxes
·
232,236 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = (-8/3 + 96/pi**2)/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
45 boxes
·
23 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = (288 - 8*pi**2)/(3*pi**4)\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = 2/pi**2\)
over Parabolic S_p,
\(\varphi(z)=1+\tfrac{2}{\pi^{2}}\log^{2}\tfrac{1+\sqrt z}{1-\sqrt z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
25,673,687 boxes
·
12,836,844 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
1,130,073 boxes
·
565,037 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
946,145 boxes
·
473,073 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
1,330,029 boxes
·
665,015 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
53,035,473 boxes
·
26,517,737 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{2}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Petal S*_ρ,
\(\varphi(z)=1+\sinh^{-1}z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0\, a_2^{2}| = 9/2 - 3*sqrt(2)\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
81 boxes
·
41 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = 15/4 - 5*sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
3,357 boxes
·
1,679 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = -1/2 + sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
57,578,165 boxes
·
28,789,083 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = -1/2 + sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
291,131 boxes
·
145,566 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3-a_2^{2}| = -1/2 + sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
255,267 boxes
·
127,634 leaves
· report
PROVED
sharp ✓
2 in literature
dag engine
\(\max |a_3 - 2\, a_2^{2}| = -1/2 + sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
277,173 boxes
·
138,587 leaves
· report
PROVED
certified enclosure
dag engine
\(|a_2a_4-a_3^{2}| \le 3/4 - sqrt(2)/2 + 0.01\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
11,111 boxes
·
5,556 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = -1/2 + sqrt(2)/2\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = -1/4 + sqrt(2)/4\)
over Rational S*_R,
\(\varphi(z)=1+\frac{z(k+z)}{k(k-z)},\ k=1+\sqrt2\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{4}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
215,915 boxes
·
107,958 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{4}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
212,427 boxes
·
106,214 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{4}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
212,217 boxes
·
106,109 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
212,947 boxes
·
106,474 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{4}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
253,387 boxes
·
126,694 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{8}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
59 boxes
·
30 leaves
· report
PROVED
certified enclosure
known theorem
1 in literature
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{1}{16} + 0.01\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
Riaz, Raza & Thomas 2022 (Forum Math.)
9,399 boxes
·
4,700 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{8}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{8}\)
over Sigmoid S*_{SG},
\(\varphi(z)=\frac{2}{1+e^{-z}}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
429,353 boxes
·
214,677 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
358,741 boxes
·
179,371 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
510,565 boxes
·
255,283 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{2}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Sine S*_{\sin},
\(\varphi(z)=1+\sin z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
18 in literature
dag engine
\(\max |a_3 - 0\, a_2^{2}| = 3\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
18 in literature
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = 2\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
33 boxes
·
17 leaves
· report
PROVED
sharp ✓
18 in literature
closed-form engine
\(\max |a_3-\tfrac12 a_2^{2}| = 1\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
18 in literature
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = 1\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
765,983 boxes
·
382,992 leaves
· report
PROVED
sharp ✓
known theorem
18 in literature
dag engine
\(\max |a_3-a_2^{2}| = 1\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
Fekete–Szegő 1933 (classical)
3,113,875 boxes
·
1,556,938 leaves
· report
PROVED
sharp ✓
18 in literature
dag engine
\(\max |a_3 - 2\, a_2^{2}| = 5\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
known theorem
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = 5\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
Reduces to Fekete–Szegő (μ=2); |A₃| ≤ 5 sharp for S*
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
known theorem
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{2}\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
|γₙ| ≤ 1/n for S* (classical)
0 boxes
·
0 leaves
· report
PROVED
certified enclosure
known theorem
dag engine
\(|a_2a_3-a_4| \le 2 + 1e-06\)
over Starlike S*,
\(\varphi(z)=\frac{1+z}{1-z}\)
Generalized Zalcman, Ravichandran–Verma 2016 (|aₙaₘ−a_{n+m−1}| ≤ (n−1)(m−1) = 2)
6,619 boxes
·
3,310 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
25,673,687 boxes
·
12,836,844 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
1,130,073 boxes
·
565,037 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
946,145 boxes
·
473,073 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
1,330,029 boxes
·
665,015 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
53,035,473 boxes
·
26,517,737 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{2}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Strip S*_τ,
\(\varphi(z)=1+\arctan z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{4}\)
over strongly_0.25
281,885 boxes
·
140,943 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{4}\)
over strongly_0.25
243,289 boxes
·
121,645 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{4}\)
over strongly_0.25
211,915 boxes
·
105,958 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over strongly_0.5
491,725 boxes
·
245,863 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{3}{4}\)
over strongly_0.75
631,565 boxes
·
315,783 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{3}{4}\)
over strongly_0.75
1,080,961 boxes
·
540,481 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{1}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{1}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
429,353 boxes
·
214,677 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{1}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
358,741 boxes
·
179,371 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{1}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
510,565 boxes
·
255,283 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |a_3-a_2^{2}| = \tfrac{1}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{3}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
5 boxes
·
3 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{3}{2}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{4}\)
over Tanh S*_{\tanh},
\(\varphi(z)=1+\tanh z\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0\, a_2^{2}| = \tfrac{2}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
493,041 boxes
·
246,521 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.25\, a_2^{2}| = \tfrac{2}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
384,493 boxes
·
192,247 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-\tfrac12 a_2^{2}| = \tfrac{2}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
338,481 boxes
·
169,241 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 0.75\, a_2^{2}| = \tfrac{2}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
412,621 boxes
·
206,311 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3-a_2^{2}| = \tfrac{2}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
699,993 boxes
·
349,997 leaves
· report
PROVED
sharp ✓
dag engine
\(\max |a_3 - 2\, a_2^{2}| = \tfrac{24}{25}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
9 boxes
·
5 leaves
· report
PROVED
certified enclosure
dag engine
\(|a_2a_4-a_3^{2}| \le \tfrac{1}{6} + 0.02\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
10,935 boxes
·
68,943 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |A_3|=|2a_2^{2}-a_3| = \tfrac{24}{25}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
0 boxes
·
0 leaves
· report
PROVED
sharp ✓
closed-form engine
\(\max |\gamma_2| = \tfrac{1}{5}\)
over Three-leaf,
\(\varphi(z)=1+\tfrac{4z}{5}+\tfrac{z^{4}}{5}\)
0 boxes
·
0 leaves
· report