Andrica-Iwata's inequality in hyperbolic triangle (Q2911478)

For any Euclidean triangle \(ABC\) the lengths of its sides \(a=|BC|\), \(b=|CA|\), and \(c=|AB|\) satisfy the inequality \(\frac{a}{b+c}\geq\sin\frac{A}{2}\). The authors name it the Andrica-Iwata inequality. They suggest a generalization of the inequality and prove some relations for hyperbolic triangles. NEWLINENEWLINENEWLINENEWLINE Let \(ABC\) be a hyperbolic triangle. Denote by \(AD\) its median. Let the sides of \(ABC\) and \(AD\) have hyperbolic lengths \(a=|BC|\), \(b=|CA|\), \(c=|AB|\), and \(d=|AD|\). NEWLINENEWLINENEWLINENEWLINE The following inequalities are proved in the paper: NEWLINENEWLINENEWLINENEWLINE (1) for an acute-angled or right-angled in its vertex \(A\) triangle \(ABC\): NEWLINE\[NEWLINE\frac{\sinh a}{\sinh b+ \sinh c}<\frac{1}{\sqrt2\cos[(\varepsilon+A)/2]},\quad\text{where}\quad\varepsilon=\pi-(A+B+C)\qquad (\text{Theorem 4});NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE(2)\quad \cosh b+\cosh c>2\cosh\frac{a}{2};NEWLINE\]NEWLINE NEWLINE\[NEWLINE (3)\quad\sinh d>\frac{\cosh b-\cosh c}{2\sinh\frac{a}{2}}\qquad (\text{Theorem 12});NEWLINE\]NEWLINENEWLINENEWLINE\[NEWLINE (4)\quad \sinh a \geq \sqrt{\cosh a-\cosh(b-c)}\qquad (\text{Theorem 14}). NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINE Some interesting corollaries of the theorems are established as well.