Prove that $16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0$

Solution 1:

$\color{brown}{\textbf{Preliminary notes.}}$

Dividing of the given inequality $(1)$ to $\;4a^4\;$ allows to present it in the equivalent form of $$\left(2\text{ sinc } a - \text{ sinc}^2\,\dfrac a2\right)^2 \le \dfrac{1+\text{sinc }2a}2,\tag1$$ or $\;f_1(a)\le f_2(a),\;$ where $$f_1(x) = \left(2\text{ sinc } x - \text{ sinc}^2\,\dfrac x2\right)^2,\quad f_2(x)\le \dfrac{1+\text{sinc }2x}2.\tag2$$

It suffices to prove the inequalities $$f_1(x) \le (1-\,^1\!/_6\,x^2)^2\le f_2(x),\quad \text{if}\quad x\in\left[0,\dfrac\pi2\right],\tag3$$ and $$f_1(x)\le \dfrac{4\pi-3}{8\pi}\le f_2(x),\quad \text{if}\quad x\in\left(\dfrac\pi2,\infty\right).\tag4$$

We can obtain the derivative $$f_1'(x)= -\dfrac{8}{x^5}\,\sin x\left(\tan \dfrac x2-x\right)\bigg((x^2-2) \cos x - 2 x \sin x+2\bigg)\tag5$$

The root of the derivative ar $\;x=r\approx\dfrac73\;$ corresponds to the root of $\;f_1.$

The root of the derivative near $\;x=m\approx 4\;$ corresponds to the secondary maximum of $\;f_1.$

$\color{brown}{\mathbf{\text{Interval }\left[0,\dfrac\pi2\right].}}$

Taking in account inequality $\;x^2<3,\;$ one can get $$2\text{ sinc } x-\text{ sinc }^2\,\dfrac x2 = 2\left(1-\dfrac16x^2+\dfrac1{5!}x^4-\dfrac1{7!}x^6+\dots\right) - \left(1-\dfrac16\left(\dfrac x2\right)^2+\dfrac1{5!}\left(\dfrac x2\right)^4+\dots\right)^2 $$$$ \le 2-\dfrac13x^2+\dfrac{x^4}{60} - \left(1-\dfrac{x^2}{24}\right)^2 \le 1-\dfrac14x^2+\dfrac1{36}x^4 \le 1-\dfrac14x^2+\dfrac1{12}x^2= 1-\dfrac{x^2}6, $$

$$f_1(x)\le(1-\,^1\!/_6 x^2)^2.$$

Om the other hand, $$\dfrac12\big(1+\text{ sinc } 2x\big)- (1-\,^1\!/_6 x^2)^2 = 1-\dfrac1{12}\,(2x)^2+\dfrac1{2\cdot5!}(2x)^4-\dfrac1{2\cdot7!}(2x)^6+\dots -1 + \dfrac13x^2 - \dfrac1{36}x^4 $$$$ \ge\left(\dfrac1{15}-\dfrac1{36}\right)x^4\ge 0, $$$$ f_2(x)\ge (1+\,^1\!/_6x^2)^2. $$ Therefore, inequality $(3)$ is proved.

$\color{brown}{\mathbf{\text{Interval }\left(\dfrac\pi2,\infty\right).}}$

Since $$f_1\left(\dfrac\pi2\right)=\left(\dfrac4\pi-\dfrac8{\pi^2}\right)^2 <\dfrac14<\dfrac{4\pi-3}{8\pi},$$ then $$f_1(x)\le \dfrac{4\pi-3}{8\pi},\quad \text{if}\quad x\in\left(\dfrac\pi2,r\right].$$

At the same time, $$f'_1(4.0)>0,\;f'_1(4.1)<0\quad\Rightarrow\quad m\in(4.0,4.1).$$ Then $$f_1(m) - \dfrac{4\pi-3}{8\pi} < (\text{ sinc}^2 2 - 2\text{ sinc }4.1)^2 - \dfrac{4\pi-3}{8\pi} < 0,$$ and $$f_1(x)\le \dfrac{4\pi-3}{8\pi},\quad \text{if}\quad x\in\left(\dfrac\pi2,\infty\right].\tag6$$

The least value of $\;\text{ sinc } x\;$ is situated near $\;x=\dfrac{3\pi}2,\;$ wherein $$\text{ sinc }\dfrac{3\pi}2 = -\dfrac2{3\pi},\quad \text{ sinc }\dfrac{4\pi}3 = -\dfrac{3\sqrt3}{8\pi}>\text{ sinc }\dfrac{3\pi}2,\quad \text{ sinc }\dfrac{5\pi}3 = -\dfrac{3\sqrt3}{10\pi}>\text{ sinc }\dfrac{3\pi}2.$$ Taking in account unimodality of $\;\text{ sinc }x\;$ in$\;\left[\dfrac{4\pi}3,\dfrac{5\pi}3\right],\;$ the least value of $\;\text{ sinc }x\;$ belongs to this interval. Therefore, $$\text{ sinc }x>-\dfrac{\sin\dfrac{3\pi}2}{\dfrac{4\pi}3}=-\dfrac3{4\pi},$$ $$f_2(x)>\dfrac{4\pi-3}{8\pi }.\tag7$$ Therefore, inequality $(4)$ is proved.

$\color{brown}{\textbf{Done!}}$