Fraction inequalitySuppose we know that a + b + d a ′ + b ′ + d ′ ≤ M and a + c + d a ′ + c ′ + d ′ ≤ M and a a ′ ≤ b b ′ ≤ d d ′ and a a ′ ≤ c c ′ ≤ d d ′ and also that a , b , c , d , a ′ , b ′ , c ′ , d ′ ∈ ( 0 , 1 ]Is the following true? a + b + c + d a ′ + b ′ + c ′ + d ′ ≤ M

Question

Fraction inequalitySuppose we know that                                          a            +            b            +            d                                              a              ′                        +                          b              ′                        +                          d              ′                                      ≤        M                 and                                       a            +            c            +            d                                              a              ′                        +                          c              ′                        +                          d              ′                                      ≤        M            and                              a                      a            ′                          ≤                  b                      b            ′                          ≤                  d                      d            ′                                   and                           a                      a            ′                          ≤                  c                      c            ′                          ≤                  d                      d            ′                              and also that   a  ,  b  ,  c  ,  d  ,      a    ′    ,      b    ′    ,      c    ′    ,      d    ′    ∈  (  0  ,  1  ]Is the following true?                                          a            +            b            +            c            +            d                                              a              ′                        +                          b              ′                        +                          c              ′                        +                          d              ′                                      ≤        M

frisiao · Accepted Answer

No. Take all numbers equal to       1    2  , except for       a    ′  , which is equal to   1. Besides, take   M  =      3    4  . Then            a      +      b      +      c      +      d                      a        ′            +              b        ′            +              c        ′            +              d        ′              =      4    5    .

Suppose we know that (a+b+d)/(a'+b'+d')<= M and (a+c+d)/(a'+c'+d')<= M and (a)/(a')<=(b)/(b')<=(d)/(d') and (a)/(a')<=(c)/(c')<=(d)/(d') and also that a,b,c,d,a′,b′,c′,d′ in (0,1]. Is the following true? (a+b+c+d)/(a'+b'+c'+d')<= M

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