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Prove this inequality based on 4 squences

Source: 2019 Jozsef Wildt International Math Competition-W. 32

May 19, 2020
SummationSequencesinequalities

Problem Statement

Let uku_k, vkv_k, aka_k and bkb_k be non-negative real sequences such as uk>aku_k > a_k and vk>bkv_k > b_k, where k=1,2,⋯ ,nk = 1, 2,\cdots , n. If 0<m1≤uk≤M10 < m_1 \leq u_k \leq M_1 and 0<m2≤vk≤M20 < m_2 \leq v_k \leq M_2, then ∑k=1n(lukvk−akbk)≥(∑k=1n(uk2−ak2))12(∑k=1n(vk2−bk2))12\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}wherel=M1M2+m1m22m1M1m2M2l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}
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