I thought this problem was just an homework-level but it turned out neither two of my professors in the area of analysis nor users active in mathstackexchange could give me a complete solution. The user A.S. gave his idea of proof in the comment but I don't know if his idea and conclusion is correct or not. I don't know how to convert his idea into a formal proof and he hasn't reply my further question.

I wonder if my question is a research level one and whether I can migrate it to mathoverflow.