Suppose is a positive homogeneous function defined on , call homogeneous function with two parameters. If is 2nd differentiable, then the monotonicity in parameters and of depend on the signs of , for variable and depend on the sign of and respectively. As applications of these results, a serial of inequalities for arithmetic mean, geometric mean, exponential mean, logarithmic mean, power-Exponential mean and exponential-geometric mean are deduced.
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is a positive homogeneous function defined on 
, call ![$ H_{f}(a,b;p,q)=left[ frac{{f(a^{p},b^{p})}}{{f(a^{q},b^{q})}}right] ^{frac{1}{p-q}}$](images/155_05_JIPAM/img3.gif)
homogeneous function with two parameters. If 
and 
of 
depend on the signs of 
, for variable 
and 
depend on the sign of ![$ I_{2a}=left[ {(ln f)_{x}ln (y/x)}right] _{y}$](images/155_05_JIPAM/img10.gif){kind=small}
and ![$ I_{2b}=left[ {(ln f)_{y}ln (x/y)}right] _{x}$](images/155_05_JIPAM/img11.gif){kind=small}
respectively. As applications of these results, a serial of inequalities for arithmetic mean, geometric mean, exponential mean, logarithmic mean, power-Exponential mean and exponential-geometric mean are deduced.
