On a converse inequality for maximal functions in Orlicz spaces

Let ${\displaystyle \Phi \left(t\right)={ʃ}_0^{t}a\left(s\right)ds}$ and ${\displaystyle \Psi \left(t\right)={ʃ}_0^{t}b\left(s\right)ds}$, where a(s) is a positive continuous function such that ${\displaystyle {ʃ}_1^{\infty }a\frac{s}{s}ds=\infty }$ and b(s) is quasi-increasing and ${\displaystyle \underset{s\to \infty }{lim}b\left(s\right)=\infty }$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants ${\displaystyle c_1}$ and ${\displaystyle s_0}$ such that ${\displaystyle {ʃ}_1^{s}a\frac{t}{t}dt\ge c_{1}b\left(c_{1}s\right)}$ for all ${\displaystyle s\ge s_0}$; (jj) there exist positive constants ${\displaystyle c_2}$ and ${\displaystyle c_3}$ such that ${\displaystyle {ʃ}_0^{2\pi }\Psi \left(\frac{c_2}{{|⨍|}_{}}|⨍\left(x\right)|\right)dx\le c_3+c_3{ʃ}_0^{2\pi }\Phi \left(\frac{1}{{|⨍|}_{}}\right)Mf\left(x\right)dx}$ for all ${\displaystyle ⨍\in L^1\left(\right)}$.