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Inequality for a, b, c, d being reals and am+b=-cm+d=m

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December 6, 2010
inequalitiesfunctioninequalities unsolved

Problem Statement

(a)(a) Prove that for a,b,c,d∈R,m∈[1,+∞)a, b, c, d \in\mathbb{R}, m \in [1,+\infty) with am+b=−cm+d=mam + b =-cm + d = m, (i)a2+b2+c2+d2+(a−c)2+(b−d)2≥4m21+m2, and(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and} (ii)2≤4m21+m2<4.(ii) 2 \le \frac{4m^2}{1+m^2} < 4. (b)(b) Express a,b,c,da, b, c, d as functions of mm so that there is equality in (i).(i).
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