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Tag Archives: Inequality
Let be a continuous function such that
We note that
On the other hand by Cauchy – Schwartz we have that
The result follows.
Let be a concave function. Prove that
Since is concave , it holds that
By setting and we get that . Thus,
and the exercise is complete.
Let be positive real numbers satisfying the following equality
We begin by stating a lemma:
Lemma: Let be positive real numbers, then:
Now, making use of the lemma we have that:
Making use of the Cauchy – Schwartz inequality we have that
The inequality now follows.
Let be positive numbers other than such that . Prove that
The Engels form of the Cauchy – Schwartz inequality gives us:
and the inequality is proven.
Let be three positive real numbers such that . Prove that
By AM – GM we have,
Hence and the exercise is complete.