Better bound for Hardy-Littlewood maximal function

I have one question about Theorem 1 on page 3 in this notes http://www.cims.nyu.edu/~chou/notes/harmonic.pdf Is there any better bound $\frac{A}{\alpha^{t}}\|f\|_{1}$ for some $t>1$ in b)? What is the greatest such $t$?

• Testing your estimate on the characteristic of a ball, you see that when $\alpha\rightarrow+\infty$ you get a contradiction. – Gian Maria Dall'Ara Jan 8 '14 at 20:13