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Question Number 156448 by MathSh last updated on 11/Oct/21

$$\mathrm{If}0<\mathrm{a}⩽\mathrm{b}<\pi \mathrm{then}\mathrm{prove}\mathrm{that}:$$ $$\frac{\sin \sqrt{\mathrm{ab}}}{\sin \left(\frac{\mathrm{a}+\mathrm{b}}{2}\right)}⩾\frac{{32\mathrm{a}}^2{\mathrm{b}}^2\sqrt{\mathrm{ab}}}{{(\mathrm{a}+\mathrm{b})}^5}$$

Answered by ghimisi last updated on 11/Oct/21

$$f\left(t\right)=\frac{sint}{t^5}\searrow$$ $$\frac{a+b}{2}⩾\sqrt{ab}\Rightarrow f\left(\frac{a+b}{2}\right)⩽f\left(\sqrt{ab}\right)$$

Commented byMathSh last updated on 11/Oct/21

$$\mathrm{Very}\mathrm{nice}\mathrm{dear}\mathrm{Ser},\mathrm{thankyou}$$

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