I need to prove that the following inequality is true only if t=0, t=1 or a=b>0:
I have no idea how to proceed.
The inequality is true for each 1<t<1 also! it is just the main ingredient for proving Holder inequality. Since log is incresing, the inequality is equivalent to
Now apply basic properties of the logarithm to the left hand side and use that log is concave to conclude.