I'm sorry to repost this thread, but I put it in the pre-univ algebra section and I had no suggestions.

There's a proof I need to complete which requires that the following inequality holds:

where:

I have made several plots and in all of them the inequality holds. Nonetheless, I need to finish this problem theoretically. I have already made the proof for the case of , and also for the case where , but these are straightforward, and the real deal comes when I want to solve for the general case.

I know . I also know that . Finally, I also know that the case where both sides of the inequality are maximums occurs when and .

I hope somebody here can help me with this problem. Best regards.