## Hadamard three lines lemma

Let stripe , is analytic in and continuous to the boundary of . Suppose for some and . Define . Then for , which means is a convex function.

First let us prove if and , then .

Fix , define on , then as . By the maximum principle, must occur on the edge of .

for

Let , we get for .

For the general case, define , then on . And has growth rate no more than some , so apply the above result, we know that , which means .

Let , we get the conclusion.

can not grow too fast, otherwise this theorem fails.

Consider on . Easy to verify that on . But on , grows extremely fast.

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