A Mixed Hölder and Minkowski Inequality
نویسندگان
چکیده
Hölder’s inequality states that ‖x‖p ‖y‖q − 〈x, y〉 ≥ 0 for any (x, y) ∈ Lp(Ω) × Lq(Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q−2: ‖x‖p ‖y‖q − 〈x, y〉 ≥ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] ≥ 0 if p ∈ (1, 2], 0 ≤ ‖x‖p ‖y‖q − 〈x, y〉 ≤ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] if p ≥ 2. A similar result holds for complex valued functions with Re(〈x, y〉) substituting for 〈x, y〉. We obtain these inequalities from some stronger (though slightly more involved) ones.
منابع مشابه
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