Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities

In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality 2βα+2β(sinxx)α+αα+2β(tanxx)β−1>(<)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{2\beta}{\alpha+2\beta} \biggl(\frac{\sin x}{x} \biggr)^{\alpha}+ \frac{\alpha}{\alpha+2\beta} \biggl(\frac{\tan x}{x} \biggr)^{\beta}-1>(< )0 $$\end{document} holds for all x∈(0,π/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in(0, \pi/2)$\end{document}.