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On the homogeneous functions with two parameters and its monotonicity. (English) Zbl 1079.26006

Summary: Suppose \(f(x, y)\) is a positive homogeneous function defined on \(\mathbb U(\subseteq\mathbb R_+\times \mathbb R_+)\), call \[ \left[H_f(a,b;p,q)= \frac{f(a^p,b^p)}{f(a^q,b^q)}\right]^{\frac{1}{p-q}} \] a homogeneous function with two parameters. If \(f(x,y)\) is second order differentiable, then the monotonicity in parameters \(p\) and \(q\) of \(H_f(a,b;p,q)\) depends on the signs of \(I_1 = (\ln f)_{xy}\), for variables \(a\) and \(b\) depends on the sign of \(I_{2a}=[(\ln f)_x \ln(y/x)]_y\) and \(I_{2b} = [(\ln f)_y\ln(x/y)]_x\), respectively. As applications of these results, a series of inequalities for the arithmetic mean, geometric mean, exponential mean, logarithmic mean, power-exponential mean, and exponential-geometric mean are deduced.

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
26D15 Inequalities for sums, series and integrals
26E60 Means
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