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Prove this inequality made by 4 sequences

Source: 2019 Jozsef Wildt International Math Competition-W. 33

May 19, 2020

inequalitiesSummationSequences

Problem Statement

Let

0 < \frac{1}{q} \leq \frac{1}{p} < 1

and

\frac{1}{p}+\frac{1}{q}=1

. Let

u_k

,

v_k

,

a_k

and

b_k

be non-negative real sequences such as

u^2_k > a^p_k

and

v_k > b^q_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

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

where

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

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