Schur convexity and Schur-geometrically concavity of generalized exponent mean. (English) Zbl 1177.26040
Summary: The 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.
MSC:
26D15 | Inequalities for sums, series and integrals |
26A51 | Convexity of real functions in one variable, generalizations |