Article SERIES_5, MML version 4.99.1005

:: SERIES_5:th 1
theorem
for b1, b2 being real positive set holds
4 <= (b1 + b2) * ((1 / b1) + (1 / b2));

:: SERIES_5:th 2
theorem
for b1, b2 being real positive set holds
((b1 |^ 3) * b2) + (b1 * (b2 |^ 3)) <= (b1 |^ 4) + (b2 |^ 4);

:: SERIES_5:th 3
theorem
for b1, b2, b3 being real positive set
      st b1 < b2
   holds 1 < (b2 + b3) / (b1 + b3);

:: SERIES_5:th 4
theorem
for b1, b2 being real positive set
      st b1 < b2
   holds b1 / b2 < sqrt (b1 / b2);

:: SERIES_5:th 5
theorem
for b1, b2 being real positive set
      st b1 < b2
   holds sqrt (b1 / b2) < (b2 + sqrt ((b1 ^2 + (b2 ^2)) / 2)) / (b1 + sqrt ((b1 ^2 + (b2 ^2)) / 2));

:: SERIES_5:th 6
theorem
for b1, b2 being real positive set
      st b1 < b2
   holds b1 / b2 < (b2 + sqrt ((b1 ^2 + (b2 ^2)) / 2)) / (b1 + sqrt ((b1 ^2 + (b2 ^2)) / 2));

:: SERIES_5:th 7
theorem
for b1, b2 being real positive set holds
2 / ((1 / b1) + (1 / b2)) <= sqrt (b1 * b2);

:: SERIES_5:th 8
theorem
for b1, b2 being real positive set holds
(b1 + b2) / 2 <= sqrt ((b1 ^2 + (b2 ^2)) / 2);

:: SERIES_5:th 9
theorem
for b1, b2 being real set holds
b1 + b2 <= sqrt (2 * (b1 ^2 + (b2 ^2)));

:: SERIES_5:th 10
theorem
for b1, b2 being real positive set holds
2 / ((1 / b1) + (1 / b2)) <= (b1 + b2) / 2;

:: SERIES_5:th 11
theorem
for b1, b2 being real positive set holds
sqrt (b1 * b2) <= sqrt ((b1 ^2 + (b2 ^2)) / 2);

:: SERIES_5:th 12
theorem
for b1, b2 being real positive set holds
2 / ((1 / b1) + (1 / b2)) <= sqrt ((b1 ^2 + (b2 ^2)) / 2);

:: SERIES_5:th 13
theorem
for b1, b2 being real set
      st abs b1 < 1 & abs b2 < 1
   holds abs ((b1 + b2) / (1 + (b1 * b2))) <= 1;

:: SERIES_5:th 14
theorem
for b1, b2 being real set holds
(abs (b1 + b2)) / (1 + abs (b1 + b2)) <= ((abs b1) / (1 + abs b1)) + ((abs b2) / (1 + abs b2));

:: SERIES_5:th 15
theorem
for b1, b2, b3, b4 being real positive set holds
1 < (((b1 / ((b1 + b2) + b3)) + (b2 / ((b1 + b2) + b4))) + (b4 / ((b2 + b4) + b3))) + (b3 / ((b1 + b4) + b3));

:: SERIES_5:th 16
theorem
for b1, b2, b3, b4 being real positive set holds
(((b1 / ((b1 + b2) + b3)) + (b2 / ((b1 + b2) + b4))) + (b4 / ((b2 + b4) + b3))) + (b3 / ((b1 + b4) + b3)) < 2;

:: SERIES_5:th 17
theorem
for b1, b2, b3 being real positive set
      st b3 < b1 + b2 & b1 < b2 + b3 & b2 < b1 + b3
   holds 9 / ((b1 + b2) + b3) <= ((1 / ((b1 + b2) - b3)) + (1 / ((b2 + b3) - b1))) + (1 / ((b3 + b1) - b2));

:: SERIES_5:th 18
theorem
for b1, b2, b3, b4 being real positive set holds
(sqrt (b1 * b3)) + sqrt (b2 * b4) <= sqrt ((b1 + b2) * (b3 + b4));

:: SERIES_5:th 19
theorem
for b1, b2, b3, b4 being real positive set holds
(((4 * b1) * b2) * b3) * b4 <= ((b1 * b2) + (b3 * b4)) * ((b1 * b3) + (b2 * b4));

:: SERIES_5:th 20
theorem
for b1, b2, b3 being real positive set holds
3 <= ((b1 / b2) + (b2 / b3)) + (b3 / b1);

:: SERIES_5:th 21
theorem
for b1, b2, b3 being real positive set
      st ((b1 * b2) + (b2 * b3)) + (b3 * b1) = 1
   holds sqrt 3 <= (b1 + b2) + b3;

:: SERIES_5:th 22
theorem
for b1, b2, b3 being real positive set holds
3 <= ((((b1 + b2) - b3) / b3) + (((b2 + b3) - b1) / b1)) + (((b3 + b1) - b2) / b2);

:: SERIES_5:th 23
theorem
for b1, b2 being real positive set holds
((sqrt (b1 * b2)) + (1 / sqrt (b1 * b2))) ^2 <= (b1 + (1 / b1)) * (b2 + (1 / b2));

:: SERIES_5:th 24
theorem
for b1, b2, b3 being real positive set holds
(b3 + b1) + b2 <= (((b1 * b2) / b3) + ((b3 * b2) / b1)) + ((b3 * b1) / b2);

:: SERIES_5:th 25
theorem
for b1, b2, b3 being real set
      st b2 < b1 & b3 < b2
   holds ((b1 * (b2 ^2)) + (b2 * (b3 ^2))) + (b3 * (b1 ^2)) < ((b1 ^2 * b2) + (b2 ^2 * b3)) + (b3 ^2 * b1);

:: SERIES_5:th 26
theorem
for b1, b2, b3 being real positive set
      st b2 < b1 & b3 < b2
   holds b3 / (b1 - b3) < b2 / (b1 - b2);

:: SERIES_5:th 27
theorem
for b1, b2, b3, b4 being real positive set
      st b2 < b1 & b4 < b3
   holds b4 / (b4 + b1) < b3 / (b3 + b2);

:: SERIES_5:th 28
theorem
for b1, b2, b3, b4 being real set holds
(b1 * b2) + (b3 * b4) <= (sqrt (b1 ^2 + (b3 ^2))) * sqrt (b2 ^2 + (b4 ^2));

:: SERIES_5:th 29
theorem
for b1, b2, b3, b4, b5, b6 being real set holds
(((b1 * b2) + (b3 * b4)) + (b5 * b6)) ^2 <= ((b1 ^2 + (b3 ^2)) + (b5 ^2)) * ((b2 ^2 + (b4 ^2)) + (b6 ^2));

:: SERIES_5:th 30
theorem
for b1, b2, b3 being real positive set holds
(((9 * b1) * b2) * b3) / ((b1 ^2 + (b2 ^2)) + (b3 ^2)) <= (b1 + b2) + b3;

:: SERIES_5:th 31
theorem
for b1, b2, b3 being real positive set holds
(b1 + b2) + b3 <= ((sqrt (((b1 ^2 + (b1 * b2)) + (b2 ^2)) / 3)) + sqrt (((b2 ^2 + (b2 * b3)) + (b3 ^2)) / 3)) + sqrt (((b3 ^2 + (b3 * b1)) + (b1 ^2)) / 3);

:: SERIES_5:th 32
theorem
for b1, b2, b3 being real positive set holds
((sqrt (((b1 ^2 + (b1 * b2)) + (b2 ^2)) / 3)) + sqrt (((b2 ^2 + (b2 * b3)) + (b3 ^2)) / 3)) + sqrt (((b3 ^2 + (b3 * b1)) + (b1 ^2)) / 3) <= ((sqrt ((b1 ^2 + (b2 ^2)) / 2)) + sqrt ((b2 ^2 + (b3 ^2)) / 2)) + sqrt ((b3 ^2 + (b1 ^2)) / 2);

:: SERIES_5:th 33
theorem
for b1, b2, b3 being real positive set holds
((sqrt ((b1 ^2 + (b2 ^2)) / 2)) + sqrt ((b2 ^2 + (b3 ^2)) / 2)) + sqrt ((b3 ^2 + (b1 ^2)) / 2) <= sqrt (3 * ((b1 ^2 + (b2 ^2)) + (b3 ^2)));

:: SERIES_5:th 34
theorem
for b1, b2, b3 being real positive set holds
sqrt (3 * ((b1 ^2 + (b2 ^2)) + (b3 ^2))) <= (((b2 * b3) / b1) + ((b3 * b1) / b2)) + ((b1 * b2) / b3);

:: SERIES_5:th 35
theorem
for b1, b2 being real positive set
      st b1 + b2 = 1
   holds 9 <= ((1 / (b1 ^2)) - 1) * ((1 / (b2 ^2)) - 1);

:: SERIES_5:th 36
theorem
for b1, b2 being real positive set
      st b1 + b2 = 1
   holds 17 / 4 <= (b1 * b2) + (1 / (b1 * b2));

:: SERIES_5:th 37
theorem
for b1, b2, b3 being real positive set
      st (b1 + b2) + b3 = 1
   holds 9 <= ((1 / b1) + (1 / b2)) + (1 / b3);

:: SERIES_5:th 38
theorem
for b1, b2, b3 being real positive set
      st (b1 + b2) + b3 = 1
   holds 8 <= (((1 / b1) - 1) * ((1 / b2) - 1)) * ((1 / b3) - 1);

:: SERIES_5:th 39
theorem
for b1, b2, b3 being real positive set
      st (b1 + b2) + b3 = 1
   holds 64 <= ((1 + (1 / b1)) * (1 + (1 / b2))) * (1 + (1 / b3));

:: SERIES_5:th 40
theorem
for b1, b2, b3 being real set
      st (b1 + b2) + b3 = 1
   holds 1 / 3 <= (b1 ^2 + (b2 ^2)) + (b3 ^2);

:: SERIES_5:th 41
theorem
for b1, b2, b3 being real set
      st (b1 + b2) + b3 = 1
   holds ((b1 * b2) + (b2 * b3)) + (b3 * b1) <= 1 / 3;

:: SERIES_5:th 42
theorem
for b1, b2, b3 being real positive set
      st b2 < b1 & b3 < b2
   holds ((b1 to_power (b2 + b3)) * (b2 to_power (b1 + b3))) * (b3 to_power (b1 + b2)) < ((b1 to_power (2 * b1)) * (b2 to_power (2 * b2))) * (b3 to_power (2 * b3));

:: SERIES_5:th 43
theorem
for b1, b2 being real positive set
for b3 being Element of NAT
      st 1 <= b3
   holds ((b1 |^ b3) * b2) + (b1 * (b2 |^ b3)) <= (b1 |^ (b3 + 1)) + (b2 |^ (b3 + 1));

:: SERIES_5:th 44
theorem
for b1, b2, b3 being real positive set
for b4 being Element of NAT
      st b1 ^2 + (b2 ^2) = b3 ^2 & 3 <= b4
   holds (b1 |^ (b4 + 2)) + (b2 |^ (b4 + 2)) < b3 |^ (b4 + 2);

:: SERIES_5:th 45
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds (1 + (1 / (b1 + 1))) |^ b1 < (1 + (1 / b1)) |^ (b1 + 1);

:: SERIES_5:th 46
theorem
for b1, b2 being real positive set
for b3, b4 being Element of NAT
      st 1 <= b3 & 1 <= b4
   holds ((b1 |^ b4) + (b2 |^ b4)) * ((b1 |^ b3) + (b2 |^ b3)) <= 2 * ((b1 |^ (b4 + b3)) + (b2 |^ (b4 + b3)));

:: SERIES_5:th 47
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 1 / sqrt (b2 + 1)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 < 2 * sqrt (b2 + 1);

:: SERIES_5:th 48
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 1 / ((b2 + 1) ^2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 <= 2 - (1 / (b2 + 1));

:: SERIES_5:th 49
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 = 1 / ((b3 + 1) ^2)
   holds (Partial_Sums b2) . b1 < 2;

:: SERIES_5:th 50
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 < 1
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 < b2 + 1;

:: SERIES_5:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        0 < b1 . b2 & b1 . b2 < 1
for b2 being Element of NAT holds
   ((Partial_Sums b1) . b2) - b2 <= (Partial_Product b1) . b2;

:: SERIES_5:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st for b3 being Element of NAT holds
        0 < b1 . b3 & b2 . b3 = 1 / (b1 . b3)
for b3 being Element of NAT holds
   0 < (Partial_Sums b2) . b3;

:: SERIES_5:th 53
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
   st for b3 being Element of NAT holds
        0 < b1 . b3 & b2 . b3 = 1 / (b1 . b3)
for b3 being Element of NAT holds
   (b3 + 1) ^2 <= ((Partial_Sums b1) . b3) * ((Partial_Sums b2) . b3);

:: SERIES_5:th 54
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = sqrt b2 & b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 < ((1 / 6) * ((4 * b2) + 3)) * sqrt b2;

:: SERIES_5:th 55
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = sqrt b2 & b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds ((2 / 3) * b2) * sqrt b2 < (Partial_Sums b1) . b2;

:: SERIES_5:th 56
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 + (1 / ((2 * b2) + 1)) &
         b1 . 0 = 1
for b2 being Element of NAT
      st 1 <= b2
   holds (1 / 2) * sqrt ((2 * b2) + 3) < (Partial_Product b1) . b2;

:: SERIES_5:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = sqrt (b2 * (b2 + 1)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (b2 * (b2 + 1)) / 2 < (Partial_Sums b1) . b2;