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How can I prove the following inequality:

$$\forall p\in[0,1], \theta\in [0,π/2];\cos^p(θ)≤\cos(pθ)$$

Please help me to complete this proof . I suppose $g(θ)=\cos^p(θ)-\cos(pθ) $ and I want show that $g(θ)≤0$. And I use derivative and integral theorems like mean value theorem ,…. but i couldn’t solve it.

  • 0
    Your notation confused me. Are you talking about $\cos(\theta^p)-\cos(p\theta)$ or $(\cos\theta)^p-\cos(p\theta)$ or something else?2012-10-30
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    This expression is meaningless in every mathematics I've ever studied: $$cos〖(θ)^p≤cos(pθ) 〗$$ What do you intend that to mean?2012-10-30
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    I define this function and will show that this function is smaller than zero ( (cos(θ))^p≤cos(pθ) ))by use of integral or derivation or other ways2012-10-30
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    Please use LaTeX.2012-10-30
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    @ThomasAndrews: I think the OP entered the function here exactly as it is written in a worksheet for example in Maple or Matlab. Any way he 'd better fix it.2012-10-30
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    @maisamhedyelloo: Did you mean $\cos^p(\theta)\leq\cos(p\theta)$?2012-10-30
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    I believe this inequality can be proved starting with De Moivre's Identity.2012-10-30

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