inequality problem

[singular]

I'm trying to show for (y+x) > 0 and y>0 that


[0.5x²a(a+1)(y+x)^a-1 - x(a+1)(y+x)^a + (y+x)^a+1 - y^a+1 ] / x > 0

where a is any real number larger than or equal to 1.

Simulations in matlab indicates that it holds. I tried binomial expansion but did not get far with it. Any suggestions much appreciated.

 

[singular]

Observation: if a=x=y=1 (meets all conditions) then :
[0.5x²a(a+1)(y+x)^a-1 - x(a+1)(y+x)^a + (y+x)^a+1 - y^a+1 ] / x = 0

Any reason why you're not giving x<>0 as condition?

The inequality holds for x > -y. I need to proof it that way. The problem is relatively simple for individual cases of a=1,2,3.. but not for the general case where a can be non-integer.