The purpose of this work is to study the relations between the nontangential maximal functions and some other function operators. In Chapter 1, we show that a certain tangential area integral of harmonic function on the upper half space $R^{n+1}$ of $R^n$ is dominated by the nontangential maximal function in $L^8$-mean. This may supplement the $L^p$-boundedness of the $g^*\lambda$-function and the nontangential area integal function for the limiting case. In Chapter 2, we prove that for nonnegative plurisubharmonic function on the unit ball B of $C_n$ the admissible maximal function is dominnated by the radial maximal function in $L^p$-maen. This gives another characterization of the class $M^p$ of holomorpic functions with certain growth condition and its invariance under the compomposition by automorphisms of B. As a consequence of the invariance, all onto-endomrphisms of $M^1$(n = 1) are characterized. In Charter 3, weprove that for nonnegative M-subharmonic functions on B the admissible maximal function is dominanted by the radial maximal function in $L^p$-mean. This domination on M-subharmonic functions implies that on plurisubhamonic functions but we include both proofs because they are interesting themselves.