On the equal variables method applied to real variables

As it is known, the equal variables method can be used to create and solve difficult symmetric inequalities in nonnegative variables involving the expressions x1+x2+⋯+xnx1+x2+⋯+xn, xk1+xk2+⋯+xknx1k+x2k+⋯+xnk and f(x1)+f(x2)+⋯+f(xn)f(x1)+f(x2)+⋯+f(xn), where kk is a real constant, and ff is a differentiable function on (0,∞)(0,∞) such that g(x)=f′(x1k−1)g(x)=f′(x1k−1) is strictly convex. In this paper, we extend the equal variables method to real variables.