Inequality Proof?

Question

[There was a typo in my question, and it's now revised. Sorry.]

A proof of an inequality?
Given A>0, B>0, x>0, y>0, x+y=1, prove
A^x B^y <= Ax+By+1 using ln(x) <= x-1, and find when the equality holds.

(I could prove a tighter inequality: A^x B^y <= Ax+By using Jensen's inequality and it's no wonder the inequality in the problem is true if this inequality is true. But I'd like to see how the given inequality ln(x)<=x-1 can be used for proving the less tighter ineqaulity and when the equality of it holds.)

Thanks.

Answer 1

As you pointed out, this is a typical application of Jensen's inequality. Actually, this is Jensen's inequality, right?

So far I couldn't find how to prove the given inequality by means of ln(x) <= x-1, but since, as you proved, the tighter inequality A^x B^y <= A x+B y holds, we ALWAYS have A^x + B^y < A x + By + 1. Equality NEVER holds for the given conditions..

Answer 2

ln(A^xB^y)=xln(A)+yln(B) which is <= to x(A-1)+y(B-1) (using ln(...)<= ... - 1) which equals xA+yB-(x+y) which equals xA+yB-1 since x+y=1.