Is there an inequality for sinh ⁡ ( x ) which is similar to this inequality cosh ⁡ ( x ) ≤ e x 2 / 2

Question

Is there an inequality for   sinh  ⁡  (  x  ) which is similar to this inequality   cosh  ⁡  (  x  )  ≤      e                  x        2                    /            2

furniranizq · Accepted Answer

Since:  cos  ⁡  (  x  )  =      ∏          n      ≥      0            (    1    −                  4                  x          2                            (        2        n        +        1                  )          2                          π          2                      )  we have:  cosh  ⁡  (  x  )  =      ∏          n      ≥      0            (    1    +                  4                  x          2                            (        2        n        +        1                  )          2                          π          2                      )    ≤  exp  ⁡      (          x      2              ∑              n        ≥        0                    4              (        2        n        +        1                  )          2                          π          2                      )    =      e                  x        2                    /            2        .In a similar fashion, from:  sinh  ⁡  (  x  )  =  x      ∏          n      ≥      1            (    1    +                  x        2                              n          2                          π          2                      )  we get:            sinh      ⁡      x        x    ≤  exp  ⁡      (          x      2              ∑              n        ≥        1                    1                        n          2                          π          2                      )    =      e                  x        2                    /            6        .

sweetymoeyz · Accepted Answer

sinh            ⁡            (            x            )                    x                                    =                  ∑                      k            =            0                    ∞                                      x                          2              k                                            (            2            k            +            1            )            !                                                            ≤                  e                                    x              2                                      /                        6                                                            =                  ∑                      k            =            0                    ∞                                      x                          2              k                                                          6              k                        ⋅            k            !                              which can be proven by induction and for   k  ≥  0,            (      2      k      +      3      )      !              (      2      k      +      1      )      !        =  (  4  k  +  6  )  (  k  +  1  )  ≥  6  (  k  +  1  )  =                    6                  k          +          1                    (      k      +      1      )      !                      6        k            k      !      Therefore,            sinh      ⁡      (      x      )        x    ≤      e                  x        2                    /            6

Is there an inequality for sinh ⁡ ( x ) which is similar to this

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