Mathematical Inequalities & Applications Volume 9, Number 2 (2006), 219–224.

AN ALTERNATIVE NOTE ON THE SCHUR–CONVEXITY OF THE EXTENDED MEAN VALUES

HUAN-NAN SHI, SHAN-HE WU AND FENG QI

Abstract. he monotonicity, the Schur-convexity and the Schur-geometrically convexity with variables $(x,y)$  in $\mathbb{R}_{++}^{2}$  for fixed $a$ of the generalized exponent mean $I_a (x,y)$ is proved. Besides, the monotonicity with parameters $a$ in $\mathbb{R}$ for fixed $(x,y)$ of $I_a (x,y)$ is discussed by using the hyperbolic composite function. Furthermore, some new inequalities are obtained.

Mathematics subject classification (2000): 26B25, 26E60, 26D07, 26D20.

Key words and phrases: Schur-convexity, extended mean values.

AN ALTERNATIVE NOTE ON THE SCHUR–CONVEXITY.pdf