By M. A. Krasnosel'skii, Yz. B. Rtuickii
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Extra info for Convex Functions and Orlicz Spaces
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Then, for arbitrary e > 0, the function uBR(u) increases PROOF . It is sufficient to show that the function h(u) = uB/2R(u) has a positive derivative for large values of u. But this follows from (7 . 6) since monotonically to infinity. h ' (u) = ; U(B/2 ) - lR(u) + uB/2r(u) = u(e/2 ) - lR(u ) [ ; + KR(U) ] . LEMMA 7 . 2. Suppose R(u) is a function from the class that the functions u� oc R (u) , vtJ PRtJ-l(v) ( oc , P > 1, im * such � + pI = 1 ) � are the principal parts of the N-functions M(u) and Nl (V) , respectively .
2. P(u) . PROOF . P(u) . P ( � ) . 8. A necessary and sUfficient condition that the N function M (u) satisfy the LJ 2-condition is that the complementary N-function N (v) to it satisfy the inequality N (v) v - -- < N ( v';j k ------= -'-' v' v (6. 1 3) for large values of the argument, where k is some constant. Proof of necessity. 11 2 (u) < M(klU) for large values of u. From this it follows that , for large values of the argument , we have (6. 1 4) where M-l (V) is the function inverse to M (u) .
A necessary and sufficient condition �hat the com plementary function M(u) to the N-function N(v) satisfy the A 2condition is that there exist constants 1 > 1 and V o � 0 such that 1 N (v) '::;;- N (l v) 21 (v � vo). 9) PROOF. 9) is satisfied. We set Nl (V ) { l j(2l)}N(lv) . 5) , the complementary N-function M l (U) to N l (V) is defined by the equality M l (U) { l j(2l ) }M (2u ) . 9) can be written in the form N(v) '::;; N1 (v) . It follows from this, in virtue of Theorem 2. 1 , that = = = 26 CHAPTER I , § 4 M 1 (u) � M (u) or, equivalently, that M(2u) � 2lM(u) for large values of the argument .
Convex Functions and Orlicz Spaces by M. A. Krasnosel'skii, Yz. B. Rtuickii
by Mark
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