PROPERTIES OF CLASSES OF FUNCTIONS 21 



implies 



(3) ^*'i=™.^. <p*'''^'*'=^ 



A class g = [(p] of functions is said to have the property KM 

 (i = 1, 2, 12) in case each function ^ of g has the property KM- In 

 symbols : 



(4,) S^'"':-:5 9^.3.<p^'^'' (1 = 1,2,12). 



In case of a function <p' and a class W of functions on ^' to 31 

 we use the notations: 



,Ki"SX' ji-.'l'!' /Si^'m' 



(p ; (f ; (p 



We give also the definitions of the properties K'M'iW) ; K:M'm") ; 

 K[M'{'i^l") of a function <p on ^^ = f^'f^" to 31, where m', Wi" are 

 classes of functions on ^', '^V', respectively, to 31. The definitions are 

 relative to a development A' of ^V. 



(5) ^*v-^>'cJ!"). ^ . ^ 5 ( 3 (^', ^") 9 e : 3 : 3 /H, 9 A';„, . 3 . .4 ^; ^ e^iu^M") > 



viz., the function (p has the property KM'CiSl") in case (p is such that 

 there exists a function m' of 50^' and a function n" of 9)c" such that 

 for every e there exists an nie such that the relation 



■^^p'nig 



implies 



A<pI, ^ eAy.y 



i. e., 



^*'='™'<^'"' . = .^9(3 (m', ju") 3 e : =1 : 3 me 9 i^;,w™, 



/yx jri2'2)!'(™") = A-i'l'i'(ili")Aym'(Il(")_ 



A class g = [^ff] of functions on ^'%V' is said to have the property 

 K'M'idJl"), (i = 1, 2, 12), in case each function <p of the class g 

 has the property K'M'CSR"), in symbols: 



(8) g^/.r.(.=") ^ 5 , ^ . :3 . ^*v-»'(.>") i = (1^ 2, 12). 



A class W of functions on "ip to 31 is said to have the property 



Ki'j Ki] K12, 



