We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that Minc-Sathre and C. P. Chen-G. Wang’s inequality are extended and refined.

H. Minc and L. Sathre, “Some Inequalities Involving ,” Proceedings of the Edinburgh Mathematical Society, Vol. 14, No. 65, 1964, pp. 41-46.
doi:10.1017/S0013091500011214

B.-N. Guo and F. Qi, “Inequalities and Monotonicity for the Ratio of Gamma Functions,” Taiwanese Journal of Mathematics, Vol. 7, No. 2, 2003, pp. 239-247.

D. Kershaw and A. Laforgia, “Monotonicity Results for the Gamma Function,” Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., Vol. 119, 1985, pp. 127-133.

M. Abramowitz and I. A. Stegun (Eds.), “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards,” Applied Mathematics Series, 4th Printing, Washington, Vol. 55, 1965.