How to prove this general inequality $\displaystyle a\left(\frac{\sin{x}}{x}\right)^m+b\left(\frac{\tan{x}}{x}\right)^n>a+b$

Asked Apr 14 '14 at 11:34

Active Apr 20 '14 at 18:34

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If $$m,n<0,\;a,b>0,\;a\left[\left(\dfrac{2}{\pi}\right)^m-1\right]\ge b,\;am\le 2bn$$ show that $$a\left(\dfrac{\sin{x}}{x}\right)^m+b\left(\dfrac{\tan{x}}{x}\right)^n>a+b,\qquad\forall x\in\left(0,\dfrac{\pi}{2}\right)$$

I know this is Wilker inequality $$\left(\dfrac{\sin{x}}{x}\right)^2+\dfrac{\tan{x}}{x}>2$$ and these papers are GENERALIZED WILKER inequality: 1 and 2. But my problem is more general. Thank you for your help.