Prove the inequality a 1 + a 2 + b 1 + b 2 + c 1 + c 2 ≤ 3 3 4 a , b , c are nonnegative real numbers such that a 2 + b 2 + c 2 = 1Prove the inequality a 1 + a 2 + b 1 + b 2 + c 1 + c 2 ≤ 3 3 4 I tried the method of Lagrange multipliers and Jensen's inequality but I have not been proved this inequality

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Prove the inequality       a          1      +              a        2              +      b          1      +              b        2              +      c          1      +              c        2              ≤            3              3              4    a  ,  b  ,  c are nonnegative real numbers such that       a    2    +      b    2    +      c    2    =  1Prove the inequality      a          1      +              a        2              +      b          1      +              b        2              +      c          1      +              c        2              ≤            3              3              4  I tried the method of Lagrange multipliers and Jensen&#039;s inequality but I have not been proved this inequality

gulali4eG · Accepted Answer

Consider the function   f  (  x  )  =            x              1      +      x      with the substitution that   x  =      a    2    ,  y  =      b    2    ,  z  =      c    2     x  =      a    2    ,  y  =      b    2    ,  z  =      c    2  ,     ⟹    x  +  y  +  z  =  1we have a new expression       a          1      +              a        2              +      b          1      +              b        2              +      c          1      +              c        2              ⟶            x              1      +      x        +            y              1      +      y        +            z              1      +      z        ⟶  f  (  x  )  +  f  (  y  )  +  f  (  z  )Differentiating   f  (  z  ) twice, we get negative second derivatives for   0  ≤  x  ≤  1, so it is concaveby Jensen&#039;s inequality, we get   f  (            x      +      y      +      z        3    )  ≥            f      (      x      )        3    +            f      (      y      )        3    +            f      (      z      )        3      ⟹                      1        3                    1      +              1        3              ≥            f      (      x      )        3    +            f      (      y      )        3    +            f      (      z      )        3      ⟹              3              3              4    ≥  f  (  x  )  +  f  (  y  )  +  f  (  z  )  =      a          1      +              a        2              +      b          1      +              b        2              +      c          1      +              c        2

Prove the inequality a/(1+a^2)+ b/(1+b^2)+c/(1+c^2)<=(3 sqrt3)/(4)

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