×

Extension of homogeneous function. (English) Zbl 1234.26047

Summary: We defined and studied the monotonicity of a homogeneous function of the type \(N[a,b;p,q;r,\alpha,\mu]=\left\{\frac{a^p+\mu M_r (a^p,b^p)+\alpha b^p}{a^q+\mu M_r(a^q,b^q)+\alpha b^q} \right\}^{\frac{1}{p-q}}\). Further, we established some valuable inequalities involving various means and some well known means are deduced as particular cases of this homogeneous function.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite