Monotonous two-parameter functions for asymptoticly approximating the inequalities involving the inverse tangent functions (Q2240628)

Consider the functions \[F_{1}(t,\alpha)=\frac{15(95-64\cos(\alpha t)-31\cos(\alpha t)^{2})}{\alpha^{2}\left( 578+344\cos(\alpha t)+23\cos (\alpha t)^{2}\right) }\] and \[F_{2}(t,\alpha)=\frac{\sin(\alpha t)\left[ \ln\left( \sin(\alpha t)+1\right) -\ln\left( \cos(\alpha t)\right) \right] }{\alpha^{2}}.\] Theorem 1 of this paper asserts that for all \(0<\alpha_{1}<\alpha_{2}<1\) and \(t\in(0,\pi/2)\) the following inequalities hold: \[F_{1}(t,\alpha_{2})<F_{1}(t,\alpha_{1})<\lim_{a\rightarrow0} F_{1}(t,\alpha)=t^{2}=\lim_{\alpha\rightarrow0}F_{2}(t,0)<F_{2}(t,\alpha _{1})<F_{2}(t,\alpha_{2}).\] Based on it, several inequalities are derived. However, a simple computation shows that \(\lim_{a\rightarrow0}F_{1}(t,\alpha)\) is actually \((32/63)t^{2}.\) Most likely, in the expression of \(F_{1}\) the term \(\cos(\alpha t)^{2}\) should be replaced by \((\cos\alpha t)^{2}.\) Anyway, the paper would have needed a careful revision.