MathDB

Let

u_k

v_k

a_k

and

b_k

be non-negative real sequences such as

u_k > a_k

and

v_k > b_k

, where

k = 1, 2,\cdots , n

. If

0 < m_1 \leq u_k \leq M_1

and

0 < m_2 \leq v_k \leq M_2

, then

\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}

where

l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}

W. 32