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On the Homogeneous Functions with Two Parameters and Its Monotonicity  
 
  Authors: Zhen-Hang Yang,  
  Keywords: Homogeneous function with two parameters, $f$-mean with two-parameter, Monotonicity, Estimate for lower and upper bounds.  
  Date Received: 18/05/05  
  Date Accepted: 04/08/05  
  Subject Codes:

Primary 26B35, 26E60; Secondary 26A48, 2

  
  Editors: Peter S. Bullen,  
 
  Abstract:

Suppose $ f(x,y) $is a positive homogeneous function defined on $ mathbb{ U(subseteqq R}_{+}times mathbb{R}_{+})$, call $ H_{f}(a,b;p,q)=left[ frac{{f(a^{p},b^{p})}}{{f(a^{q},b^{q})}}right] ^{frac{1}{p-q}}$ homogeneous function with two parameters. If $ f(x,y) $ is 2nd differentiable, then the monotonicity in parameters $ p$ and $ q$ of $ H_{f}(a,b;p,q)$ depend on the signs of $ I_{1}=(ln f)_{xy}$, for variable $ a$ and $ b$ depend on the sign of $ I_{2a}=left[ {(ln f)_{x}ln (y/x)}right] _{y}$ and $ I_{2b}=left[ {(ln f)_{y}ln (x/y)}right] _{x}$ 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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