974 CHAPTER 27. ORLITZ SPACES

and this reduces toφ (t)−φ (a)

t−a≤ φ (x)−φ (a)

x−a.

Also these difference quotients are bounded below by

φ (a)−φ (a−1)1

= φ (a)−φ (a−1) .

Let

A≡ inf{

φ (t)−φ (a)t−a

: t ∈ (a,b)}.

Then A is some finite real number. Similarly there exists a real number B such that for allt ∈ (a,b) ,

B≥ φ (b)−φ (t)b− t

.

Now let a≤ s < t ≤ b. Then

φ (t)−φ (s)t− s

≥ φ (t)−θφ (a)− (1−θ)φ (t)t− s

where θ is such that θa+(1−θ) t = s. Thus

θ =t− st− t1

and so the above implies

φ (t)−φ (s)t− s

≥ t− st− t1

φ (t)−φ (a)t− s

=φ (t)−φ (a)

t− t1≥ A.

Similarly,

φ (t)−φ (s)t− s

≤ θφ (b)+(1−θ)φ (s)−φ (s)t− s

=t− sb− s

φ (b)−φ (s)t− s

≤ B.

It follows|φ (t)−φ (s)| ≤ (|A|+ |B|) |t− s|

and this proves the lemma.The following is like the inequality, st ≤ t p/p+ sq/q, important in the study of Lp

spaces.

Proposition 27.1.3 If A is an N function, then so is à and

A(t) = max{

ts− Ã(s) : s≥ 0}, (27.1.4)