How to prove that a 2 + b 2 − 2 a + b − 2 &lt; 1 + b if a &lt; b and a , b &gt; 1?

Question

How to prove that                     a        2            +              b        2            −      2              a      +      b      −      2        &amp;lt;  1  +  b if   a  &amp;lt;  b and   a  ,  b  &amp;gt;  1?

Giancarlo Phelps · Accepted Answer

Let   a  =  1  +  x and   b  =  1  +  yThus,  1  +  b  −                    a        2            +              b        2            −      2              a      +      b      −      2        =  2  +  y  −                    x        2            +              y        2            +      2      x      +      2      y              x      +      y        =            x      (      y      −      x      )              x      +      y        &amp;gt;  0.

smetuwh · Accepted Answer

your inequality is equivalent to      a    2    +      b    2    −  2  &amp;lt;  (  1  +  b  )  (  a  +  b  −  2  )and this is equivalent to  (  1  −  a  )  (  a  −  b  )  &amp;gt;  0this is true!

How to prove that (a^2 + b^2 -2)/(a + b - 2) < 1 + b if a<b and a,b>1?

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