In the article under review, the authors examine the Iyengar-Madhava Rao-Nanjundiah inequality $$ \cos ax>\frac{\sin x}{x}>\cos bx,\quad 0MR0044596]. They obtain that the Iyengar--Madhava~Rao--Nanjundiah type inequality $$ \cos \left(\frac{x}{\sqrt 3}+\alpha_rx^r\right)< \frac{\sin x}{x}<\cos\left(\frac{x}{\sqrt 3}+\beta_rx^r\right),\quad x\in(0,\pi /2), \tag2 $$ holds with \roster \item"$\bullet$" the best possible constants ${\beta_3=(2/\pi)^3 (-\pi/(2\sqrt{3})+\arccos(2/\pi))}$ and ${\alpha_3=-1/(90 \sqrt{3})}$ for $r=3$; \item"$\bullet$" the best possible constants $\alpha_r=0$ and $\beta_r=(2/\pi)^r(-\pi/( 2\sqrt{3})+\arccos(2/\pi))$ for $04$. Finally, they draw the conclusion that there is no constant $\beta_{r}$ ($r>4$) for which the right-hand side of inequality (2) holds. \par This article uses L'Hospital's rule for the monotonicity of functions. The authors expand the scope of inequality research and offer a variety of methods.