One-parameter monotone functionals connected with Stieltjes integrals

Using Stieltjes integrals we define one-parameter functionals that are monotone as a function on the parameter. We prove generalizations of some results from the papers:
1) Heinig H. and Maligranda L. Weighted inequalities for monotone and concave functions, Studia Mathematica 116 (2), 133–165 (1995);
2) Avkhadiev F.G. and Kayumov I.R. Comparison theorems of isoperimetric type for moments of compact sets, Collectanea Math. 55 (1), 1–9 (2004).
In contrast to these papers we prove several theorems on monotonicity of integral functionals in the case when integrating functions are not absolutely continuous. In addition, we obtain applications to isoperimetric inequalities.