MathDB

Let

(a_0,a_1,a_2,...)

and

(b_0,b_1,b_2,...)

be such sequences of non-negative real numbers, that for every integer

i\geqslant 1

holds

a_i^2\leqslant a_{i-1}a_{i+1}

and

b_i^2\leqslant b_{i-1}b_{i+1}

. Define sequence

c_0,c_1,c_2,...

c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.

Prove that for every integer

k\geqslant 1

holds

c_{k}^2\leqslant c_{k-1}c_{k+1}

6.