MathDB

W. 33

Part of 2019 Jozsef Wildt International Math Competition

Problems(1)

Prove this inequality made by 4 sequences

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

5/19/2020
Let 0<1q≤1p<10 < \frac{1}{q} \leq \frac{1}{p} < 1 and 1p+1q=1\frac{1}{p}+\frac{1}{q}=1. Let uku_k, vkv_k, aka_k and bkb_k be non-negative real sequences such as uk2>akpu^2_k > a^p_k and vk>bkqv_k > b^q_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(lp(uk+vk)2−(ak+bk)p))1p≥(∑k=1n(uk2−akp))1p(∑k=1n(vk2−bkp))1p\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=M1M2+m1m22m1M1m2M2l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}
inequalitiesSummationSequences