Bivariate homogeneous functions of two parameters: Monotonicity, convexity, comparisons, and functional inequalities

Abstract

The bivariate homogeneous functions of two parameters are also called the bivariate means of two parameters. In the paper, the authors survey some results published since 2005 about monotonicity, logarithmic convexity, and Schur convexity of the bivariate homogeneous functions of two parameters, review the Minkowski, Hölder, Chebyshev, and Hermite–Hadamard type inequalities for the bivariate homogeneous functions of two parameters, and exhibit comparisons of the bivariate homogeneous functions of two parameters. Applying these results, the authors derive and remark some nice inequalities for the bivariate homogeneous functions of two parameters.