This paper extends the results of theorem 4 in Ky Fan\+\{[1]\}. For proper contraction operator A defined in the complex Hibert space H and analytic function f defined in the unit disk △ the majorization principle is established. Compared with [5]

in this paper the condition which A is normal operator is canceled and

the corresponding results are improved. Further

for functions f which belong to some subclass of H(△)

the value of ‖f′(A)‖ is estimated. Moreover

the majorization principle of subordinate relation also is discussed and

the superior radius criteria\+\{[7]\} of univalent functions given by Shah Goluzin are extended to operator functions.