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Existence of constant for concave functions

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October 30, 2010

functionalgebra unsolvedalgebra

Problem Statement

A sequence

(a_n)_0^N

of real numbers is called concave if

2a_n\ge a_{n-1} + a_{n+1}

for all integers

n, 1 \le n \le N - 1

.

(a)

Prove that there exists a constant

C >0

such that

\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)

for all concave positive sequences

(a_n)^N_0

(b)

Prove that

(1)

holds with

C = \frac{3}{4}

and that this constant is best possible.

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