Given a function f with f + f ″ ≥ 0 , show that f ( x ) + f ( x + π ) ≥ 0 for all x.Note that for sine and cosine both inequalities become equations. It seems reasonable to look at f + f ″ = g , but the resulting expressions seem inconclusive.

Question

Given a function f with   f  +      f    ″    ≥  0 , show that   f  (  x  )  +  f  (  x  +  π  )  ≥  0 for all x.Note that for sine and cosine both inequalities become equations. It seems reasonable to look at   f  +      f    ″    =  g , but the resulting expressions seem inconclusive.

Kumamotors · Accepted Answer

Step 1                    f        (        x        )        +        f        (        x        +        π        )                            =                  ∫          0                      π                                    (          sin          ⁡          (          t          )          f          (          x          +          t          )          −          cos          ⁡          (          t          )                      f            ′                    (          x          +          t          )          )                d        t                                          =                  ∫          0                      π                          sin        ⁡        (        t        )        f        (        x        +        t        )        d        t        −                  ∫          0                      π                          cos        ⁡        (        t        )                  f          ′                (        x        +        t        )        d        t                                          =                  ∫          0                      π                          sin        ⁡        (        t        )        f        (        x        +        t        )        d        t        −                  (          −                      ∫            0                          π                                sin          ⁡          (          t          )                      f            ″                    (          x          +          t          )          d          t          )                                                  =                  ∫          0                      π                                    (          sin          ⁡          (          t          )          f          (          x          +          t          )          +          sin          ⁡          (          t          )                      f            ″                    (          x          +          t          )          )                d        t        ≥        0

Given a function f with f + f ″ </msup> ≥ 0

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