MathDB

Sequence inequality

Source: Baltic Way 2001

November 17, 2010

inequalitiesalgebra proposedalgebra

Problem Statement

Let

a_0,a_1,a_2,\ldots

be a sequence of positive real numbers satisfying

i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}

for

i=1, 2, \ldots

Furthermore, let

x

and

y

be positive reals, and let

b_i=xa_i+ya_{i-1}

for

i=1, 2, \ldots

Prove that the inequality

i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}

holds for all integers

i\ge 2

.

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