By B. M. Levitan, V. V. Zhikov, L. W. Longdon
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Additional info for Almost Periodic Functions and Differential Equations
First we consider the circle |z| = r. 1) and let PQS denote respectively the points reiθ where θ = − cos−1 ( 2x r ), 2x −1 cos ( r ) and π. 2) is analytic in the region enclosed by the straight line PQ and the circular arc QS P. 3) on the boundary of PQS P and also |φ(0)| ≥ 1. 4) X = Exp(u1 + u2 + . . 5) Let 29 where u1 , u2 , . . , un vary over the box B defined by 0 ≤ u j ≤ B( j = 1, 2, . . , n), and B > 0. We begin with Lemma 1. 8) and 1 2πi I2 = F(z)X z QS P where the lines of integration are the straight lien PQ and the circular arc QS P.
Follows since on QS P we have |φ(z)| = 1 (and so |F(z)| ≤ M) and also |B−n | Xz B QS P 2 dz du1 . . dun | ≤ 2πiz Br n . Lemma 4. We have, | f (0)|k ≤ e2Bnx 30 2 Br n M+ e2Bnx 2π Proof. Follows by Lemmas 1, 2 and 3. PQ |( f (z))k dz |. 11) Some Preliminaries 32 Step 2. 11), we replace | f (z)|k by an integral over a chord P1 Q1 (parallel to PQ) of |w| = 2r, of slightly bigger length with a similar error. Let x1 be any real number with |x1 | ≤ x. 12) Let P1 Q1 R1 be the points 2reiθ x1 , 0 and cos−1 where θ = − cos−1 2r x1 2r .
R. Also the constant 3π/2 is not important in many applications. It is the object of this section to supply a very simple proof in this special case with a larger constant in place of 3π/2. 2. Suppose R ≥ 2, λn = log(n + α) where 0 ≤ α ≤ 1 is fixed and n = 1, 2, . . , R. Let a1 , . . aR be complex numbers. 2) λ − λ m m n m A Theorem of Montgomery and Vaughan 23 where C is an absolute numerical constant which is effective. Remark 1. Instead of the condition λn = log(n + α) we can also work with the weaker condition n(λn+1 − λn ) is both ≫ 1 and ≪ 1.
Almost Periodic Functions and Differential Equations by B. M. Levitan, V. V. Zhikov, L. W. Longdon